Subgroups — C₄₇×#23882 #26063580

Order 12032.

1866 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3484	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3025	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3021	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#3003	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2010	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2008	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1910	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1904	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1891	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1795	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#585	C₂²	1	1	yes	characteristic, complemented
13	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic, complemented
15	C₄₇×#577	—	2	4	no	—
16	C₄₇×#576	—	2	4	no	—
17	C₄₇×#575	—	2	2	no	—
18	C₄₇×#575	C₂²	1	2	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
20	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	—
21	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	complemented
22	C₄₇×#568	C₂²	1	1	yes	characteristic, complemented
23	C₄₇×#546	—	2	2	no	—
24	C₄₇×#546	—	2	2	no	—
25	D₄²×C₄₇	—	2	2	no	—
26	D₄²×C₄₇	—	2	2	no	—
27	C₄₇×#535	C₂²	1	1	yes	characteristic
28	C₄₇×#535	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#534	—	2	2	no	—
30	C₄₇×#525	C₂²	1	1	yes	characteristic
31	C₄₇×#525	C₂²	1	1	yes	characteristic, complemented
32	C₄₇×#525	C₂²	1	2	yes	complemented
33	C₂×#120×C₄₇	—	2	2	no	—
34	C₄₇×#518	C₂²	1	1	yes	characteristic, complemented
35	C₄₇×#458	—	2	2	no	—
36	C₄₇×B(4,𝔽₂)	—	2	2	no	—
37	C₄₇×#454	—	2	2	no	—
38	C₄₇×#453	—	2	2	no	—
39	C₄₇×#450	—	2	4	no	—
40	C₄₇×#447	—	2	4	no	—
41	C₄₇×#421	C₂²	1	2	yes	complemented
42	C₄₇×#421	C₂²	1	2	yes	complemented
43	C₄₇×#418	C₂²	1	2	yes	complemented
44	C₄₇×#418	C₂²	1	2	yes	complemented
45	C₄₇×#413	C₂²	1	1	yes	characteristic, complemented
46	C₄₇×#410	C₂²	1	2	yes	complemented
47	C₄₇×#410	C₂²	1	1	yes	characteristic, complemented
48	C₄₇×#407	C₂²	1	2	yes	complemented
49	C₄₇×#344	C₂²	1	2	yes	complemented
50	C₄₇×H(3,ℤ/4)	C₂²	1	2	yes	complemented
51	C₂⁵×C₄₇	—	2	1	no	—
52	2₋¹⁺⁴×C₄₇	—	2	2	no	—
53	2₋¹⁺⁴×C₄₇	—	2	1	no	—
54	H(5,𝔽₂)×C₄₇	—	2	4	no	—
55	H(5,𝔽₂)×C₄₇	—	2	2	no	—
56	H(5,𝔽₂)×C₄₇	—	2	2	no	—
57	H(5,𝔽₂)×C₄₇	—	2	1	no	—
58	H(5,𝔽₂)×C₄₇	—	4	2	no	—
59	C₂×#41×C₄₇	—	2	1	no	—
60	C₂×#41×C₄₇	—	2	2	no	—
61	C₂×#41×C₄₇	—	4	2	no	—
62	C₂×#41×C₄₇	D₄	1	2	yes	complemented
63	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×#41×C₄₇	C₂³	1	2	yes	—
65	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₄₇	—	2	4	no	—
67	C₂²×D₄×C₄₇	—	2	2	no	—
68	C₂²×D₄×C₄₇	—	2	1	no	—
69	C₂²×D₄×C₄₇	—	2	2	no	—
70	C₂²×D₄×C₄₇	—	2	2	no	—
71	C₂²×D₄×C₄₇	—	2	4	no	—
72	C₂²×D₄×C₄₇	—	2	1	no	—
73	C₂²×D₄×C₄₇	—	4	2	no	—
74	C₂²×D₄×C₄₇	D₄	1	2	yes	complemented
75	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
78	#137×C₄₇	—	2	4	no	—
79	#137×C₄₇	—	2	2	no	—
80	#137×C₄₇	—	2	2	no	—
81	AGL(1,ℤ/8)×C₄₇	—	2	8	no	—
82	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
83	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
84	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
85	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
86	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
87	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
88	AGL(1,ℤ/8)×C₄₇	—	4	2	no	—
89	#135×C₄₇	—	2	8	no	—
90	#135×C₄₇	—	2	4	no	—
91	C₂×SD₁₆×C₄₇	—	2	4	no	—
92	C₂×SD₁₆×C₄₇	—	2	2	no	—
93	C₂×SD₁₆×C₄₇	D₄	1	4	yes	complemented
94	C₂×D₈×C₄₇	—	2	4	no	—
95	C₂×D₈×C₄₇	—	2	2	no	—
96	C₂×D₈×C₄₇	D₄	1	4	yes	complemented
97	#131×C₄₇	—	2	4	no	—
98	C₂×M₁₆×C₄₇	—	2	2	no	—
99	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
101	#127×C₄₇	—	2	1	no	—
102	#127×C₄₇	D₄	1	2	yes	complemented
103	#124×C₄₇	—	2	1	no	—
104	#124×C₄₇	D₄	1	2	yes	complemented
105	#123×C₄₇	—	2	4	no	—
106	#123×C₄₇	—	2	1	no	—
107	#123×C₄₇	—	2	2	no	—
108	#123×C₄₇	D₄	1	2	yes	—
109	C₂²⋊Q₈×C₄₇	—	2	1	no	—
110	C₂²⋊Q₈×C₄₇	—	2	1	no	—
111	C₂²⋊Q₈×C₄₇	—	2	2	no	—
112	#121×C₄₇	—	2	4	no	—
113	#121×C₄₇	—	2	4	no	—
114	#121×C₄₇	—	2	1	no	—
115	#121×C₄₇	—	2	1	no	—
116	#121×C₄₇	—	2	2	no	—
117	#121×C₄₇	—	2	2	no	—
118	#121×C₄₇	—	2	2	no	—
119	#120×C₄₇	—	2	4	no	—
120	#120×C₄₇	—	2	4	no	—
121	#120×C₄₇	—	2	4	no	—
122	#120×C₄₇	—	2	1	no	—
123	#120×C₄₇	—	2	2	no	—
124	#120×C₄₇	—	2	2	no	—
125	#120×C₄₇	—	2	2	no	—
126	#120×C₄₇	D₄	1	2	yes	—
127	C₄×D₄×C₄₇	—	2	2	no	—
128	C₄×D₄×C₄₇	—	2	2	no	—
129	C₄×D₄×C₄₇	—	2	2	no	—
130	C₄×D₄×C₄₇	—	2	2	no	—
131	C₄×D₄×C₄₇	—	4	2	no	—
132	#117×C₄₇	C₂³	1	2	yes	—
133	#117×C₄₇	C₂³	1	1	yes	characteristic
134	#117×C₄₇	C₂³	1	1	yes	characteristic
135	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
136	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
137	#104×C₄₇	—	2	4	no	—
138	#104×C₄₇	—	4	2	no	—
139	#102×C₄₇	—	2	4	no	—
140	#102×C₄₇	—	2	4	no	—
141	#101×C₄₇	—	2	1	no	—
142	#100×C₄₇	—	2	1	no	—
143	#99×C₄₇	—	2	2	no	—
144	#99×C₄₇	—	2	4	no	—
145	C₂²⋊C₈×C₄₇	—	2	4	no	—
146	C₂⁴×C₄₇	—	2	1	no	—
147	C₂⁴×C₄₇	—	2	4	no	—
148	C₂⁴×C₄₇	—	2	2	no	—
149	C₂⁴×C₄₇	—	2	1	no	—
150	C₂⁴×C₄₇	—	4	4	no	—
151	C₂⁴×C₄₇	—	8	4	no	—
152	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
153	#41×C₄₇	—	2	2	no	—
154	#41×C₄₇	—	2	2	no	—
155	#41×C₄₇	—	2	4	no	—
156	#41×C₄₇	—	2	4	no	—
157	#41×C₄₇	—	2	2	no	—
158	#41×C₄₇	—	2	2	no	—
159	#41×C₄₇	—	2	2	no	—
160	#41×C₄₇	—	2	2	no	—
161	#41×C₄₇	—	2	2	no	—
162	#41×C₄₇	—	2	2	no	—
163	#41×C₄₇	—	4	2	no	—
164	#41×C₄₇	—	4	4	no	—
165	#41×C₄₇	—	4	2	no	—
166	#41×C₄₇	—	4	4	no	—
167	#41×C₄₇	—	4	2	no	—
168	#41×C₄₇	—	4	2	no	—
169	#41×C₄₇	—	4	2	no	—
170	C₂×Q₈×C₄₇	—	2	2	no	—
171	C₂×Q₈×C₄₇	—	4	2	no	—
172	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
173	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
174	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	complemented
175	C₂×D₄×C₄₇	—	2	2	no	—
176	C₂×D₄×C₄₇	—	2	4	no	—
177	C₂×D₄×C₄₇	—	2	2	no	—
178	C₂×D₄×C₄₇	—	2	4	no	—
179	C₂×D₄×C₄₇	—	2	2	no	—
180	C₂×D₄×C₄₇	—	2	2	no	—
181	C₂×D₄×C₄₇	—	2	1	no	—
182	C₂×D₄×C₄₇	—	2	2	no	—
183	C₂×D₄×C₄₇	—	2	4	no	—
184	C₂×D₄×C₄₇	—	2	4	no	—
185	C₂×D₄×C₄₇	—	2	2	no	—
186	C₂×D₄×C₄₇	—	2	1	no	—
187	C₂×D₄×C₄₇	—	2	2	no	—
188	C₂×D₄×C₄₇	—	2	1	no	—
189	C₂×D₄×C₄₇	—	2	2	no	—
190	C₂×D₄×C₄₇	—	2	2	no	—
191	C₂×D₄×C₄₇	—	4	2	no	—
192	C₂×D₄×C₄₇	—	4	4	no	—
193	C₂×D₄×C₄₇	—	4	8	no	—
194	C₂×D₄×C₄₇	—	4	4	no	—
195	C₂×D₄×C₄₇	—	4	2	no	—
196	C₂×D₄×C₄₇	—	4	8	no	—
197	C₂×D₄×C₄₇	—	4	4	no	—
198	C₂×D₄×C₄₇	—	4	4	no	—
199	C₂×D₄×C₄₇	—	4	2	no	—
200	C₂×D₄×C₄₇	—	4	4	no	—
201	C₂×D₄×C₄₇	—	4	2	no	—
202	C₂×D₄×C₄₇	—	8	8	no	—
203	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
204	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
205	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
206	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
208	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
209	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	complemented
210	C₂²×C₄×C₄₇	—	2	4	no	—
211	C₂²×C₄×C₄₇	—	2	2	no	—
212	C₂²×C₄×C₄₇	—	2	1	no	—
213	C₂²×C₄×C₄₇	—	2	1	no	—
214	C₂²×C₄×C₄₇	—	2	1	no	—
215	C₂²×C₄×C₄₇	—	2	2	no	—
216	C₂²×C₄×C₄₇	—	2	2	no	—
217	C₂²×C₄×C₄₇	—	2	2	no	—
218	C₂²×C₄×C₄₇	—	4	2	no	—
219	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
221	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
222	Q₁₆×C₄₇	—	2	4	no	—
223	SD₁₆×C₄₇	—	2	4	no	—
224	SD₁₆×C₄₇	—	2	4	no	—
225	SD₁₆×C₄₇	—	2	8	no	—
226	SD₁₆×C₄₇	—	4	4	no	—
227	D₈×C₄₇	—	2	4	no	—
228	D₈×C₄₇	—	2	8	no	—
229	D₈×C₄₇	—	4	4	no	—
230	M₁₆×C₄₇	—	2	4	no	—
231	M₁₆×C₄₇	—	2	2	no	—
232	M₁₆×C₄₇	—	2	2	no	—
233	M₁₆×C₄₇	—	2	2	no	—
234	M₁₆×C₄₇	—	2	2	no	—
235	C₂×C₈×C₄₇	—	2	4	no	—
236	C₂×C₈×C₄₇	—	2	2	no	—
237	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	complemented
238	C₄⋊C₄×C₄₇	—	2	2	no	—
239	C₄⋊C₄×C₄₇	—	2	2	no	—
240	C₄⋊C₄×C₄₇	—	2	1	no	—
241	C₄⋊C₄×C₄₇	—	2	1	no	—
242	C₄⋊C₄×C₄₇	—	2	2	no	—
243	C₄⋊C₄×C₄₇	—	4	2	no	—
244	C₂²⋊C₄×C₄₇	—	2	2	no	—
245	C₂²⋊C₄×C₄₇	—	2	4	no	—
246	C₂²⋊C₄×C₄₇	—	2	4	no	—
247	C₂²⋊C₄×C₄₇	—	2	2	no	—
248	C₂²⋊C₄×C₄₇	—	2	2	no	—
249	C₂²⋊C₄×C₄₇	—	2	1	no	—
250	C₂²⋊C₄×C₄₇	—	2	2	no	—
251	C₂²⋊C₄×C₄₇	—	2	4	no	—
252	C₂²⋊C₄×C₄₇	—	2	2	no	—
253	C₂²⋊C₄×C₄₇	—	4	2	no	—
254	C₂²⋊C₄×C₄₇	—	4	2	no	—
255	C₂²⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
256	C₄²×C₄₇	—	2	1	no	—
257	C₄²×C₄₇	—	2	2	no	—
258	C₄²×C₄₇	C₂×D₄	1	2	yes	—
259	C₂³×C₄₇	—	2	1	no	—
260	C₂³×C₄₇	—	2	2	no	—
261	C₂³×C₄₇	—	2	2	no	—
262	C₂³×C₄₇	—	2	4	no	—
263	C₂³×C₄₇	—	2	2	no	—
264	C₂³×C₄₇	—	2	1	no	—
265	C₂³×C₄₇	—	2	2	no	—
266	C₂³×C₄₇	—	4	2	no	—
267	C₂³×C₄₇	—	4	4	no	—
268	C₂³×C₄₇	—	4	8	no	—
269	C₂³×C₄₇	—	4	8	no	—
270	C₂³×C₄₇	—	4	2	no	—
271	C₂³×C₄₇	—	4	2	no	—
272	C₂³×C₄₇	—	8	8	no	—
273	C₂³×C₄₇	—	8	2	no	—
274	C₂³×C₄₇	—	8	2	no	—
275	C₂³×C₄₇	—	8	4	no	—
276	C₂³×C₄₇	—	8	4	no	—
277	C₂³×C₄₇	—	8	8	no	—
278	C₂³×C₄₇	#120	1	2	yes	—
279	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
280	Q₈×C₄₇	—	2	2	no	—
281	Q₈×C₄₇	—	2	2	no	—
282	Q₈×C₄₇	—	2	1	no	—
283	Q₈×C₄₇	—	2	1	no	—
284	Q₈×C₄₇	—	2	2	no	—
285	Q₈×C₄₇	—	4	2	no	—
286	D₄×C₄₇	—	2	4	no	—
287	D₄×C₄₇	—	2	2	no	—
288	D₄×C₄₇	—	2	1	no	—
289	D₄×C₄₇	—	2	2	no	—
290	D₄×C₄₇	—	2	1	no	—
291	D₄×C₄₇	—	2	4	no	—
292	D₄×C₄₇	—	2	2	no	—
293	D₄×C₄₇	—	2	2	no	—
294	D₄×C₄₇	—	2	4	no	—
295	D₄×C₄₇	—	4	4	no	—
296	D₄×C₄₇	—	4	2	no	—
297	D₄×C₄₇	—	4	4	no	—
298	D₄×C₄₇	—	4	8	no	—
299	D₄×C₄₇	—	4	4	no	—
300	D₄×C₄₇	—	4	4	no	—
301	D₄×C₄₇	—	4	4	no	—
302	D₄×C₄₇	—	4	2	no	—
303	D₄×C₄₇	—	4	2	no	—
304	D₄×C₄₇	—	4	2	no	—
305	D₄×C₄₇	—	4	1	no	—
306	D₄×C₄₇	—	4	1	no	—
307	D₄×C₄₇	—	8	4	no	—
308	D₄×C₄₇	—	8	4	no	—
309	D₄×C₄₇	—	8	4	no	—
310	D₄×C₄₇	#120	1	4	yes	—
311	C₂×C₄×C₄₇	—	2	2	no	—
312	C₂×C₄×C₄₇	—	2	2	no	—
313	C₂×C₄×C₄₇	—	2	2	no	—
314	C₂×C₄×C₄₇	—	2	2	no	—
315	C₂×C₄×C₄₇	—	2	1	no	—
316	C₂×C₄×C₄₇	—	2	1	no	—
317	C₂×C₄×C₄₇	—	2	2	no	—
318	C₂×C₄×C₄₇	—	2	2	no	—
319	C₂×C₄×C₄₇	—	4	4	no	—
320	C₂×C₄×C₄₇	—	4	4	no	—
321	C₂×C₄×C₄₇	—	4	2	no	—
322	C₂×C₄×C₄₇	—	4	1	no	—
323	C₂×C₄×C₄₇	—	4	1	no	—
324	C₂×C₄×C₄₇	—	4	2	no	—
325	C₂×C₄×C₄₇	—	4	2	no	—
326	C₂×C₄×C₄₇	—	4	2	no	—
327	C₂×C₄×C₄₇	—	4	2	no	—
328	C₂×C₄×C₄₇	—	4	2	no	—
329	C₂×C₄×C₄₇	—	8	4	no	—
330	C₂×C₄×C₄₇	#120	1	2	yes	—
331	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
332	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
333	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
335	C₈×C₄₇	—	2	4	no	—
336	C₈×C₄₇	—	4	2	no	—
337	#23882	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
338	C₂²×C₄₇	—	2	2	no	—
339	C₂²×C₄₇	—	2	4	no	—
340	C₂²×C₄₇	—	2	2	no	—
341	C₂²×C₄₇	—	2	4	no	—
342	C₂²×C₄₇	—	4	1	no	—
343	C₂²×C₄₇	—	4	2	no	—
344	C₂²×C₄₇	—	4	4	no	—
345	C₂²×C₄₇	—	4	1	no	—
346	C₂²×C₄₇	—	4	2	no	—
347	C₂²×C₄₇	—	4	2	no	—
348	C₂²×C₄₇	—	4	4	no	—
349	C₂²×C₄₇	—	4	1	no	—
350	C₂²×C₄₇	—	4	2	no	—
351	C₂²×C₄₇	—	4	1	no	—
352	C₂²×C₄₇	—	4	2	no	—
353	C₂²×C₄₇	—	8	4	no	—
354	C₂²×C₄₇	—	8	2	no	—
355	C₂²×C₄₇	—	8	4	no	—
356	C₂²×C₄₇	—	8	16	no	—
357	C₂²×C₄₇	—	16	4	no	—
358	C₂²×C₄₇	—	16	2	no	—
359	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
360	C₂²×C₄₇	D₄²	1	2	yes	—
361	C₄×C₄₇	—	2	2	no	—
362	C₄×C₄₇	—	4	2	no	—
363	C₄×C₄₇	—	4	2	no	—
364	C₄×C₄₇	—	4	1	no	—
365	C₄×C₄₇	—	4	1	no	—
366	C₄×C₄₇	—	4	2	no	—
367	C₄×C₄₇	—	8	2	no	—
368	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
369	C₄×C₄₇	#534	1	1	yes	characteristic
370	C₄×C₄₇	D₄²	1	2	yes	—
371	#3585	C₂×C₄₇	1	1	yes	characteristic, complemented
372	#3484	C₂×C₄₇	1	1	yes	characteristic, complemented
373	#3025	C₂×C₄₇	1	1	yes	characteristic, complemented
374	#3021	C₂×C₄₇	1	1	yes	characteristic, complemented
375	#3003	C₂×C₄₇	1	2	yes	complemented
376	#2010	C₂×C₄₇	1	2	yes	complemented
377	#2008	C₂×C₄₇	1	2	yes	complemented
378	#1910	C₂×C₄₇	1	1	yes	characteristic, complemented
379	#1904	C₂×C₄₇	1	2	yes	complemented
380	#1891	C₂×C₄₇	1	1	yes	characteristic, complemented
381	#1795	C₂×C₄₇	1	1	yes	characteristic, complemented
382	C₂×C₄₇	—	2	1	no	—
383	C₂×C₄₇	—	2	2	no	—
384	C₂×C₄₇	—	4	2	no	—
385	C₂×C₄₇	—	4	4	no	—
386	C₂×C₄₇	—	4	2	no	—
387	C₂×C₄₇	—	8	2	no	—
388	C₂×C₄₇	—	8	1	no	—
389	C₂×C₄₇	—	8	2	no	—
390	C₂×C₄₇	#2403	1	1	yes	characteristic
391	#585	C₂²×C₄₇	1	1	yes	characteristic, complemented
392	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	1	yes	characteristic
393	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic, complemented
394	#577	—	2	4	no	—
395	#576	—	2	4	no	—
396	#575	—	2	2	no	—
397	#575	C₂²×C₄₇	1	2	yes	complemented
398	C₂×AGL(1,ℤ/8)	—	2	2	no	—
399	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	—
400	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	complemented
401	#568	C₂²×C₄₇	1	1	yes	characteristic, complemented
402	#546	—	2	2	no	—
403	#546	—	2	2	no	—
404	D₄²	—	2	2	no	—
405	D₄²	—	2	2	no	—
406	#535	C₂²×C₄₇	1	1	yes	characteristic
407	#535	C₂²×C₄₇	1	1	yes	characteristic, complemented
408	#534	—	2	2	no	—
409	#525	C₂²×C₄₇	1	1	yes	characteristic
410	#525	C₂²×C₄₇	1	1	yes	characteristic, complemented
411	#525	C₂²×C₄₇	1	2	yes	complemented
412	C₂×#120	—	2	2	no	—
413	#518	C₂²×C₄₇	1	1	yes	characteristic, complemented
414	#458	—	2	2	no	—
415	B(4,𝔽₂)	—	2	2	no	—
416	#454	—	2	2	no	—
417	#453	—	2	2	no	—
418	#450	—	2	4	no	—
419	#447	—	2	4	no	—
420	#421	C₂²×C₄₇	1	2	yes	complemented
421	#421	C₂²×C₄₇	1	2	yes	complemented
422	#418	C₂²×C₄₇	1	2	yes	complemented
423	#418	C₂²×C₄₇	1	2	yes	complemented
424	#413	C₂²×C₄₇	1	1	yes	characteristic, complemented
425	#410	C₂²×C₄₇	1	1	yes	characteristic, complemented
426	#410	C₂²×C₄₇	1	2	yes	complemented
427	#407	C₂²×C₄₇	1	2	yes	complemented
428	#344	C₂²×C₄₇	1	2	yes	complemented
429	H(3,ℤ/4)	C₂²×C₄₇	1	2	yes	complemented
430	C₄₇	#23882	1	1	yes	characteristic, direct product factor, Sylow
431	C₂⁵	—	2	1	no	—
432	2₋¹⁺⁴	—	2	2	no	—
433	2₋¹⁺⁴	—	2	1	no	—
434	H(5,𝔽₂)	—	2	4	no	—
435	H(5,𝔽₂)	—	2	2	no	—
436	H(5,𝔽₂)	—	2	2	no	—
437	H(5,𝔽₂)	—	2	1	no	—
438	H(5,𝔽₂)	—	4	2	no	—
439	C₂×#41	—	2	1	no	—
440	C₂×#41	—	2	2	no	—
441	C₂×#41	—	4	2	no	—
442	C₂×#41	D₄×C₄₇	1	2	yes	complemented
443	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
444	C₂×#41	C₂³×C₄₇	1	2	yes	—
445	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
446	C₂²×D₄	—	2	4	no	—
447	C₂²×D₄	—	2	2	no	—
448	C₂²×D₄	—	2	1	no	—
449	C₂²×D₄	—	2	2	no	—
450	C₂²×D₄	—	2	2	no	—
451	C₂²×D₄	—	2	4	no	—
452	C₂²×D₄	—	2	1	no	—
453	C₂²×D₄	—	4	2	no	—
454	C₂²×D₄	D₄×C₄₇	1	2	yes	complemented
455	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
456	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
457	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
458	#137	—	2	4	no	—
459	#137	—	2	2	no	—
460	#137	—	2	2	no	—
461	AGL(1,ℤ/8)	—	2	8	no	—
462	AGL(1,ℤ/8)	—	2	4	no	—
463	AGL(1,ℤ/8)	—	2	2	no	—
464	AGL(1,ℤ/8)	—	2	4	no	—
465	AGL(1,ℤ/8)	—	2	4	no	—
466	AGL(1,ℤ/8)	—	2	2	no	—
467	AGL(1,ℤ/8)	—	2	4	no	—
468	AGL(1,ℤ/8)	—	4	2	no	—
469	#135	—	2	8	no	—
470	#135	—	2	4	no	—
471	C₂×SD₁₆	—	2	4	no	—
472	C₂×SD₁₆	—	2	2	no	—
473	C₂×SD₁₆	D₄×C₄₇	1	4	yes	complemented
474	C₂×D₈	—	2	4	no	—
475	C₂×D₈	—	2	2	no	—
476	C₂×D₈	D₄×C₄₇	1	4	yes	complemented
477	#131	—	2	4	no	—
478	C₂×M₁₆	—	2	2	no	—
479	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
480	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
481	#127	—	2	1	no	—
482	#127	D₄×C₄₇	1	2	yes	complemented
483	#124	—	2	1	no	—
484	#124	D₄×C₄₇	1	2	yes	complemented
485	#123	—	2	4	no	—
486	#123	—	2	1	no	—
487	#123	—	2	2	no	—
488	#123	D₄×C₄₇	1	2	yes	—
489	C₂²⋊Q₈	—	2	1	no	—
490	C₂²⋊Q₈	—	2	1	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	#121	—	2	4	no	—
493	#121	—	2	4	no	—
494	#121	—	2	1	no	—
495	#121	—	2	1	no	—
496	#121	—	2	2	no	—
497	#121	—	2	2	no	—
498	#121	—	2	2	no	—
499	#120	—	2	4	no	—
500	#120	—	2	4	no	—
501	#120	—	2	4	no	—
502	#120	—	2	1	no	—
503	#120	—	2	2	no	—
504	#120	—	2	2	no	—
505	#120	—	2	2	no	—
506	#120	D₄×C₄₇	1	2	yes	—
507	C₄×D₄	—	2	2	no	—
508	C₄×D₄	—	2	2	no	—
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	—	2	2	no	—
511	C₄×D₄	—	4	2	no	—
512	#117	C₂³×C₄₇	1	2	yes	—
513	#117	C₂³×C₄₇	1	1	yes	characteristic
514	#117	C₂³×C₄₇	1	1	yes	characteristic
515	C₂×C₂²⋊C₄	—	2	4	no	—
516	C₂×C₂²⋊C₄	—	2	2	no	—
517	#104	—	2	4	no	—
518	#104	—	4	2	no	—
519	#102	—	2	4	no	—
520	#102	—	2	4	no	—
521	#101	—	2	1	no	—
522	#100	—	2	1	no	—
523	#99	—	2	2	no	—
524	#99	—	2	4	no	—
525	C₂²⋊C₈	—	2	4	no	—
526	C₂⁴	—	2	1	no	—
527	C₂⁴	—	2	4	no	—
528	C₂⁴	—	2	2	no	—
529	C₂⁴	—	2	1	no	—
530	C₂⁴	—	4	4	no	—
531	C₂⁴	—	8	4	no	—
532	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
533	#41	—	2	2	no	—
534	#41	—	2	2	no	—
535	#41	—	2	4	no	—
536	#41	—	2	4	no	—
537	#41	—	2	2	no	—
538	#41	—	2	2	no	—
539	#41	—	2	2	no	—
540	#41	—	2	2	no	—
541	#41	—	2	2	no	—
542	#41	—	2	2	no	—
543	#41	—	4	2	no	—
544	#41	—	4	4	no	—
545	#41	—	4	2	no	—
546	#41	—	4	4	no	—
547	#41	—	4	2	no	—
548	#41	—	4	2	no	—
549	#41	—	4	2	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	4	2	no	—
552	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
553	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
554	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	complemented
555	C₂×D₄	—	2	2	no	—
556	C₂×D₄	—	2	4	no	—
557	C₂×D₄	—	2	2	no	—
558	C₂×D₄	—	2	4	no	—
559	C₂×D₄	—	2	2	no	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	—	2	4	no	—
565	C₂×D₄	—	2	2	no	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	2	2	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	—	2	2	no	—
571	C₂×D₄	—	4	2	no	—
572	C₂×D₄	—	4	4	no	—
573	C₂×D₄	—	4	8	no	—
574	C₂×D₄	—	4	4	no	—
575	C₂×D₄	—	4	2	no	—
576	C₂×D₄	—	4	8	no	—
577	C₂×D₄	—	4	4	no	—
578	C₂×D₄	—	4	4	no	—
579	C₂×D₄	—	4	2	no	—
580	C₂×D₄	—	4	4	no	—
581	C₂×D₄	—	4	2	no	—
582	C₂×D₄	—	8	8	no	—
583	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
584	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
585	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
586	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
587	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
588	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
589	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	complemented
590	C₂²×C₄	—	2	4	no	—
591	C₂²×C₄	—	2	2	no	—
592	C₂²×C₄	—	2	2	no	—
593	C₂²×C₄	—	2	1	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	1	no	—
596	C₂²×C₄	—	2	2	no	—
597	C₂²×C₄	—	2	2	no	—
598	C₂²×C₄	—	4	2	no	—
599	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
600	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
601	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
602	Q₁₆	—	2	4	no	—
603	SD₁₆	—	2	4	no	—
604	SD₁₆	—	2	4	no	—
605	SD₁₆	—	2	8	no	—
606	SD₁₆	—	4	4	no	—
607	D₈	—	2	4	no	—
608	D₈	—	2	8	no	—
609	D₈	—	4	4	no	—
610	M₁₆	—	2	4	no	—
611	M₁₆	—	2	2	no	—
612	M₁₆	—	2	2	no	—
613	M₁₆	—	2	2	no	—
614	M₁₆	—	2	2	no	—
615	C₂×C₈	—	2	4	no	—
616	C₂×C₈	—	2	2	no	—
617	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	complemented
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	2	no	—
623	C₄⋊C₄	—	4	2	no	—
624	C₂²⋊C₄	—	2	2	no	—
625	C₂²⋊C₄	—	2	4	no	—
626	C₂²⋊C₄	—	2	4	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	2	no	—
629	C₂²⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	4	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	4	2	no	—
634	C₂²⋊C₄	—	4	2	no	—
635	C₂²⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
636	C₄²	—	2	1	no	—
637	C₄²	—	2	2	no	—
638	C₄²	C₂×D₄×C₄₇	1	2	yes	—
639	C₂³	—	2	1	no	—
640	C₂³	—	2	2	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	2	4	no	—
643	C₂³	—	2	2	no	—
644	C₂³	—	2	1	no	—
645	C₂³	—	2	2	no	—
646	C₂³	—	4	2	no	—
647	C₂³	—	4	4	no	—
648	C₂³	—	4	8	no	—
649	C₂³	—	4	8	no	—
650	C₂³	—	4	2	no	—
651	C₂³	—	4	2	no	—
652	C₂³	—	8	8	no	—
653	C₂³	—	8	2	no	—
654	C₂³	—	8	2	no	—
655	C₂³	—	8	4	no	—
656	C₂³	—	8	4	no	—
657	C₂³	—	8	8	no	—
658	C₂³	#120×C₄₇	1	2	yes	—
659	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
660	Q₈	—	2	2	no	—
661	Q₈	—	2	2	no	—
662	Q₈	—	2	1	no	—
663	Q₈	—	2	1	no	—
664	Q₈	—	2	2	no	—
665	Q₈	—	4	2	no	—
666	D₄	—	2	4	no	—
667	D₄	—	2	2	no	—
668	D₄	—	2	1	no	—
669	D₄	—	2	2	no	—
670	D₄	—	2	1	no	—
671	D₄	—	2	4	no	—
672	D₄	—	2	2	no	—
673	D₄	—	2	2	no	—
674	D₄	—	2	4	no	—
675	D₄	—	4	4	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	4	no	—
678	D₄	—	4	8	no	—
679	D₄	—	4	4	no	—
680	D₄	—	4	4	no	—
681	D₄	—	4	4	no	—
682	D₄	—	4	2	no	—
683	D₄	—	4	2	no	—
684	D₄	—	4	2	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	8	4	no	—
688	D₄	—	8	4	no	—
689	D₄	—	8	4	no	—
690	D₄	#120×C₄₇	1	4	yes	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	2	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	4	no	—
701	C₂×C₄	—	4	2	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	4	2	no	—
705	C₂×C₄	—	4	2	no	—
706	C₂×C₄	—	4	2	no	—
707	C₂×C₄	—	4	2	no	—
708	C₂×C₄	—	4	2	no	—
709	C₂×C₄	—	8	4	no	—
710	C₂×C₄	#120×C₄₇	1	2	yes	—
711	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
712	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
713	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
714	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
715	C₈	—	2	4	no	—
716	C₈	—	4	2	no	—
717	C₂²	—	2	2	no	—
718	C₂²	—	2	4	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	2	4	no	—
721	C₂²	—	4	1	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	4	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	4	4	no	—
728	C₂²	—	4	1	no	—
729	C₂²	—	4	2	no	—
730	C₂²	—	4	1	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	8	4	no	—
733	C₂²	—	8	2	no	—
734	C₂²	—	8	4	no	—
735	C₂²	—	8	16	no	—
736	C₂²	—	16	4	no	—
737	C₂²	—	16	2	no	—
738	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
739	C₂²	D₄²×C₄₇	1	2	yes	—
740	C₄	—	2	2	no	—
741	C₄	—	4	2	no	—
742	C₄	—	4	2	no	—
743	C₄	—	4	1	no	—
744	C₄	—	4	1	no	—
745	C₄	—	4	2	no	—
746	C₄	—	8	2	no	—
747	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
748	C₄	C₄₇×#534	1	1	yes	characteristic
749	C₄	D₄²×C₄₇	1	2	yes	—
750	C₂	—	2	1	no	—
751	C₂	—	2	2	no	—
752	C₂	—	4	2	no	—
753	C₂	—	4	4	no	—
754	C₂	—	4	2	no	—
755	C₂	—	8	2	no	—
756	C₂	—	8	1	no	—
757	C₂	—	8	2	no	—
758	C₂	C₄₇×#2403	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄₇×#3585	C₂	semidirect
2. C₄₇×#3484	C₂	semidirect
3. C₄₇×#3025	C₂	semidirect
4. C₄₇×#3021	C₂	semidirect
5. C₄₇×#3003	C₂	semidirect
6. C₄₇×#2010	C₂	semidirect
7. C₄₇×#2008	C₂	semidirect
8. C₄₇×#1910	C₂	semidirect
9. C₄₇×#1904	C₂	semidirect
10. C₄₇×#1891	C₂	semidirect
11. C₄₇×#1795	C₂	semidirect
12. C₄₇×#585	C₂²	semidirect
14. C₂³×D₄×C₄₇	C₂²	semidirect
18. C₄₇×#575	C₂²	semidirect
21. C₂×AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
22. C₄₇×#568	C₂²	semidirect
28. C₄₇×#535	C₂²	semidirect
31. C₄₇×#525	C₂²	semidirect
32. C₄₇×#525	C₂²	semidirect
34. C₄₇×#518	C₂²	semidirect
41. C₄₇×#421	C₂²	semidirect
42. C₄₇×#421	C₂²	semidirect
43. C₄₇×#418	C₂²	semidirect
44. C₄₇×#418	C₂²	semidirect
45. C₄₇×#413	C₂²	semidirect
46. C₄₇×#410	C₂²	semidirect
47. C₄₇×#410	C₂²	semidirect
48. C₄₇×#407	C₂²	semidirect
49. C₄₇×#344	C₂²	semidirect
50. C₄₇×H(3,ℤ/4)	C₂²	semidirect
62. C₂×#41×C₄₇	D₄	semidirect
74. C₂²×D₄×C₄₇	D₄	semidirect
93. C₂×SD₁₆×C₄₇	D₄	semidirect
96. C₂×D₈×C₄₇	D₄	semidirect
102. #127×C₄₇	D₄	semidirect
104. #124×C₄₇	D₄	semidirect
174. C₂×Q₈×C₄₇	C₂×D₄	semidirect
209. C₂×D₄×C₄₇	C₂×D₄	semidirect
237. C₂×C₈×C₄₇	C₂×D₄	semidirect
337. #23882	C₄₇	direct (N × H)
371. #3585	C₂×C₄₇	semidirect
372. #3484	C₂×C₄₇	semidirect
373. #3025	C₂×C₄₇	semidirect
374. #3021	C₂×C₄₇	semidirect
375. #3003	C₂×C₄₇	semidirect
376. #2010	C₂×C₄₇	semidirect
377. #2008	C₂×C₄₇	semidirect
378. #1910	C₂×C₄₇	semidirect
379. #1904	C₂×C₄₇	semidirect
380. #1891	C₂×C₄₇	semidirect
381. #1795	C₂×C₄₇	semidirect
391. #585	C₂²×C₄₇	semidirect
393. C₂³×D₄	C₂²×C₄₇	semidirect
397. #575	C₂²×C₄₇	semidirect
400. C₂×AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
401. #568	C₂²×C₄₇	semidirect
407. #535	C₂²×C₄₇	semidirect
410. #525	C₂²×C₄₇	semidirect
411. #525	C₂²×C₄₇	semidirect
413. #518	C₂²×C₄₇	semidirect
420. #421	C₂²×C₄₇	semidirect
421. #421	C₂²×C₄₇	semidirect
422. #418	C₂²×C₄₇	semidirect
423. #418	C₂²×C₄₇	semidirect
424. #413	C₂²×C₄₇	semidirect
425. #410	C₂²×C₄₇	semidirect
426. #410	C₂²×C₄₇	semidirect
427. #407	C₂²×C₄₇	semidirect
428. #344	C₂²×C₄₇	semidirect
429. H(3,ℤ/4)	C₂²×C₄₇	semidirect
430. C₄₇	#23882	direct (N × H)
442. C₂×#41	D₄×C₄₇	semidirect
454. C₂²×D₄	D₄×C₄₇	semidirect
473. C₂×SD₁₆	D₄×C₄₇	semidirect
476. C₂×D₈	D₄×C₄₇	semidirect
482. #127	D₄×C₄₇	semidirect
484. #124	D₄×C₄₇	semidirect
554. C₂×Q₈	C₂×D₄×C₄₇	semidirect
589. C₂×D₄	C₂×D₄×C₄₇	semidirect
617. C₂×C₈	C₂×D₄×C₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₇×#3585	2
C₄₇×#3484	2
C₄₇×#3025	2
C₄₇×#3021	2
C₄₇×#3003	2
C₄₇×#2010	2
C₄₇×#2008	2
C₄₇×#1910	2
C₄₇×#1904	2
C₄₇×#1891	2
C₄₇×#1795	2
#23882	47

Subgroups — C47×#23882 #26063580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23882 #26063580