Subgroups — C₄₇×#23873 #26063571

Order 12032.

1686 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3586	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3027	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3019	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3018	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2009	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2004	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1914	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1906	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1884	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1796	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#585	C₂²	1	1	yes	characteristic, complemented
13	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	1	yes	characteristic, complemented
15	C₄₇×#578	—	2	4	no	—
16	C₄₇×#577	—	2	4	no	—
17	C₄₇×#575	—	2	2	no	—
18	C₄₇×#575	C₂²	1	2	yes	—
19	C₄₇×#575	C₂²	1	2	yes	—
20	C₂×#137×C₄₇	—	2	2	no	—
21	C₂×#137×C₄₇	C₂²	1	2	yes	complemented
22	C₄₇×#568	C₂²	1	1	yes	characteristic, complemented
23	C₄₇×#548	—	2	2	no	—
24	C₄₇×#546	—	2	2	no	—
25	C₄₇×#546	—	2	4	no	—
26	C₄₇×#536	—	2	1	no	—
27	C₄₇×#536	C₂²	1	1	yes	characteristic, complemented
28	C₄₇×#535	—	2	1	no	—
29	C₄₇×#535	C₂²	1	2	yes	—
30	C₄₇×#535	C₂²	1	1	yes	characteristic
31	C₄₇×#532	—	2	1	no	—
32	C₄₇×#532	C₂²	1	1	yes	characteristic
33	C₂×#124×C₄₇	—	2	1	no	—
34	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic, complemented
35	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic, complemented
36	C₄₇×B(4,𝔽₂)	—	2	4	no	—
37	C₄₇×#456	—	2	1	no	—
38	C₄₇×#455	—	2	4	no	—
39	C₄₇×#455	—	2	1	no	—
40	C₄₇×#454	—	2	1	no	—
41	C₄₇×#453	—	2	4	no	—
42	C₄₇×#453	—	2	1	no	—
43	C₄₇×#421	C₂²	1	2	yes	complemented
44	C₄₇×#421	C₂²	1	2	yes	complemented
45	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
46	C₄₇×#419	C₂²	1	2	yes	complemented
47	C₄₇×#413	C₂²	1	1	yes	characteristic, complemented
48	C₂×#100×C₄₇	C₂²	1	2	yes	complemented
49	C₄₇×#410	C₂²	1	2	yes	complemented
50	C₄₇×#410	C₂²	1	1	yes	characteristic, complemented
51	C₄₇×#344	C₂²	1	2	yes	complemented
52	C₄₇×#339	C₂²	1	2	yes	complemented
53	2₋¹⁺⁴×C₄₇	—	2	4	no	—
54	2₋¹⁺⁴×C₄₇	—	2	2	no	—
55	2₋¹⁺⁴×C₄₇	—	2	1	no	—
56	2₋¹⁺⁴×C₄₇	—	4	2	no	—
57	H(5,𝔽₂)×C₄₇	—	2	2	no	—
58	H(5,𝔽₂)×C₄₇	—	2	2	no	—
59	H(5,𝔽₂)×C₄₇	—	2	4	no	—
60	H(5,𝔽₂)×C₄₇	—	2	1	no	—
61	H(5,𝔽₂)×C₄₇	—	4	2	no	—
62	C₂×#41×C₄₇	—	2	1	no	—
63	C₂×#41×C₄₇	—	2	1	no	—
64	C₂×#41×C₄₇	—	2	1	no	—
65	C₂×#41×C₄₇	—	2	2	no	—
66	C₂×#41×C₄₇	—	2	2	no	—
67	C₂×#41×C₄₇	—	4	2	no	—
68	C₂×#41×C₄₇	D₄	1	2	yes	complemented
69	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×#41×C₄₇	C₂³	1	2	yes	—
71	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₄₇	—	2	1	no	—
74	C₂²×Q₈×C₄₇	D₄	1	2	yes	complemented
75	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₄₇	—	2	2	no	—
77	C₂²×D₄×C₄₇	—	2	1	no	—
78	C₂²×D₄×C₄₇	—	2	1	no	—
79	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
80	#137×C₄₇	—	2	8	no	—
81	#137×C₄₇	—	2	4	no	—
82	#137×C₄₇	—	2	2	no	—
83	#137×C₄₇	—	2	4	no	—
84	#137×C₄₇	—	2	2	no	—
85	#137×C₄₇	—	2	2	no	—
86	#137×C₄₇	—	2	4	no	—
87	#137×C₄₇	D₄	1	4	yes	complemented
88	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
89	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
90	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
91	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
92	AGL(1,ℤ/8)×C₄₇	D₄	1	4	yes	complemented
93	#135×C₄₇	—	2	8	no	—
94	#135×C₄₇	—	2	4	no	—
95	#135×C₄₇	—	4	2	no	—
96	C₂×Q₁₆×C₄₇	—	2	4	no	—
97	C₂×Q₁₆×C₄₇	—	2	2	no	—
98	C₂×SD₁₆×C₄₇	—	2	4	no	—
99	C₂×SD₁₆×C₄₇	—	2	2	no	—
100	#131×C₄₇	—	2	4	no	—
101	C₂×M₁₆×C₄₇	—	2	2	no	—
102	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
103	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
104	C₄⋊Q₈×C₄₇	—	2	1	no	—
105	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic, complemented
106	#127×C₄₇	—	2	1	no	—
107	#127×C₄₇	—	2	2	no	—
108	#127×C₄₇	D₄	1	1	yes	characteristic, complemented
109	#124×C₄₇	—	2	4	no	—
110	#124×C₄₇	—	2	1	no	—
111	#124×C₄₇	—	2	2	no	—
112	#124×C₄₇	—	2	1	no	—
113	#124×C₄₇	—	2	2	no	—
114	#124×C₄₇	—	2	1	no	—
115	#124×C₄₇	—	2	1	no	—
116	#124×C₄₇	D₄	1	1	yes	characteristic, complemented
117	#124×C₄₇	D₄	1	1	yes	characteristic, complemented
118	#123×C₄₇	—	2	4	no	—
119	#123×C₄₇	—	2	2	no	—
120	C₂²⋊Q₈×C₄₇	—	2	4	no	—
121	C₂²⋊Q₈×C₄₇	—	2	2	no	—
122	#121×C₄₇	—	2	4	no	—
123	#121×C₄₇	—	2	4	no	—
124	#121×C₄₇	—	2	2	no	—
125	#121×C₄₇	—	2	2	no	—
126	#121×C₄₇	—	4	2	no	—
127	#120×C₄₇	—	2	4	no	—
128	#120×C₄₇	—	2	4	no	—
129	#120×C₄₇	D₄	1	4	yes	—
130	C₄×Q₈×C₄₇	—	2	1	no	—
131	C₄×D₄×C₄₇	—	2	4	no	—
132	C₄×D₄×C₄₇	—	2	1	no	—
133	C₄×D₄×C₄₇	—	2	2	no	—
134	C₄×D₄×C₄₇	—	2	2	no	—
135	#117×C₄₇	—	2	1	no	—
136	#117×C₄₇	C₂³	1	2	yes	—
137	#117×C₄₇	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
139	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
140	C₂×C₄²×C₄₇	—	2	1	no	—
141	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
142	#104×C₄₇	—	2	4	no	—
143	#104×C₄₇	—	2	4	no	—
144	#104×C₄₇	—	2	2	no	—
145	#104×C₄₇	—	2	2	no	—
146	#103×C₄₇	—	4	2	no	—
147	#101×C₄₇	—	2	1	no	—
148	#100×C₄₇	—	2	1	no	—
149	#100×C₄₇	—	2	4	no	—
150	#99×C₄₇	—	2	2	no	—
151	#99×C₄₇	—	2	4	no	—
152	C₂⁴×C₄₇	—	2	2	no	—
153	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
154	#41×C₄₇	—	2	2	no	—
155	#41×C₄₇	—	2	2	no	—
156	#41×C₄₇	—	2	2	no	—
157	#41×C₄₇	—	2	4	no	—
158	#41×C₄₇	—	2	4	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	—	2	2	no	—
164	#41×C₄₇	—	2	2	no	—
165	#41×C₄₇	—	4	2	no	—
166	#41×C₄₇	—	4	4	no	—
167	#41×C₄₇	—	4	4	no	—
168	#41×C₄₇	—	4	4	no	—
169	#41×C₄₇	—	4	2	no	—
170	#41×C₄₇	—	4	8	no	—
171	#41×C₄₇	—	4	2	no	—
172	#41×C₄₇	—	4	2	no	—
173	#41×C₄₇	—	4	2	no	—
174	#41×C₄₇	—	4	2	no	—
175	#41×C₄₇	—	4	2	no	—
176	#41×C₄₇	C₂×D₄	1	4	yes	—
177	C₂×Q₈×C₄₇	—	2	2	no	—
178	C₂×Q₈×C₄₇	—	2	1	no	—
179	C₂×Q₈×C₄₇	—	2	2	no	—
180	C₂×Q₈×C₄₇	—	2	4	no	—
181	C₂×Q₈×C₄₇	—	2	2	no	—
182	C₂×Q₈×C₄₇	—	2	2	no	—
183	C₂×Q₈×C₄₇	—	4	2	no	—
184	C₂×Q₈×C₄₇	—	4	4	no	—
185	C₂×Q₈×C₄₇	—	4	2	no	—
186	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
187	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
188	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
191	C₂×D₄×C₄₇	—	2	1	no	—
192	C₂×D₄×C₄₇	—	2	2	no	—
193	C₂×D₄×C₄₇	—	2	2	no	—
194	C₂×D₄×C₄₇	—	2	2	no	—
195	C₂×D₄×C₄₇	—	2	2	no	—
196	C₂×D₄×C₄₇	—	2	2	no	—
197	C₂×D₄×C₄₇	—	2	1	no	—
198	C₂×D₄×C₄₇	—	2	2	no	—
199	C₂×D₄×C₄₇	—	2	4	no	—
200	C₂×D₄×C₄₇	—	2	1	no	—
201	C₂×D₄×C₄₇	—	2	1	no	—
202	C₂×D₄×C₄₇	—	2	2	no	—
203	C₂×D₄×C₄₇	—	2	2	no	—
204	C₂×D₄×C₄₇	—	4	4	no	—
205	C₂×D₄×C₄₇	—	4	2	no	—
206	C₂×D₄×C₄₇	—	4	8	no	—
207	C₂×D₄×C₄₇	—	4	4	no	—
208	C₂×D₄×C₄₇	—	4	2	no	—
209	C₂×D₄×C₄₇	—	4	2	no	—
210	C₂×D₄×C₄₇	—	4	2	no	—
211	C₂×D₄×C₄₇	—	4	4	no	—
212	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
213	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
214	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
215	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
216	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₄₇	—	2	1	no	—
218	C₂²×C₄×C₄₇	—	2	2	no	—
219	C₂²×C₄×C₄₇	—	2	2	no	—
220	C₂²×C₄×C₄₇	—	2	1	no	—
221	C₂²×C₄×C₄₇	—	2	2	no	—
222	C₂²×C₄×C₄₇	—	2	1	no	—
223	C₂²×C₄×C₄₇	—	2	2	no	—
224	C₂²×C₄×C₄₇	—	2	1	no	—
225	C₂²×C₄×C₄₇	—	2	2	no	—
226	C₂²×C₄×C₄₇	—	4	2	no	—
227	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
228	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
229	Q₁₆×C₄₇	—	2	8	no	—
230	Q₁₆×C₄₇	—	2	4	no	—
231	Q₁₆×C₄₇	—	4	2	no	—
232	SD₁₆×C₄₇	—	2	8	no	—
233	SD₁₆×C₄₇	—	2	4	no	—
234	SD₁₆×C₄₇	—	2	4	no	—
235	SD₁₆×C₄₇	—	4	2	no	—
236	SD₁₆×C₄₇	—	4	2	no	—
237	D₈×C₄₇	—	2	4	no	—
238	D₈×C₄₇	—	4	2	no	—
239	M₁₆×C₄₇	—	2	4	no	—
240	M₁₆×C₄₇	—	2	2	no	—
241	M₁₆×C₄₇	—	2	2	no	—
242	M₁₆×C₄₇	—	2	2	no	—
243	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
244	C₂×C₈×C₄₇	—	2	4	no	—
245	C₂×C₈×C₄₇	—	2	2	no	—
246	C₂×C₈×C₄₇	—	2	2	no	—
247	C₄⋊C₄×C₄₇	—	2	2	no	—
248	C₄⋊C₄×C₄₇	—	2	4	no	—
249	C₄⋊C₄×C₄₇	—	2	2	no	—
250	C₄⋊C₄×C₄₇	—	2	1	no	—
251	C₄⋊C₄×C₄₇	—	2	2	no	—
252	C₂²⋊C₄×C₄₇	—	2	4	no	—
253	C₂²⋊C₄×C₄₇	—	2	4	no	—
254	C₂²⋊C₄×C₄₇	—	2	4	no	—
255	C₂²⋊C₄×C₄₇	—	2	2	no	—
256	C₂²⋊C₄×C₄₇	—	2	2	no	—
257	C₂²⋊C₄×C₄₇	—	2	2	no	—
258	C₂²⋊C₄×C₄₇	—	2	4	no	—
259	C₂²⋊C₄×C₄₇	—	2	2	no	—
260	C₂²⋊C₄×C₄₇	—	4	4	no	—
261	C₂²⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
262	C₄²×C₄₇	—	2	2	no	—
263	C₄²×C₄₇	—	2	1	no	—
264	C₄²×C₄₇	—	2	1	no	—
265	C₄²×C₄₇	—	2	2	no	—
266	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
268	C₂³×C₄₇	—	2	1	no	—
269	C₂³×C₄₇	—	2	2	no	—
270	C₂³×C₄₇	—	2	1	no	—
271	C₂³×C₄₇	—	2	4	no	—
272	C₂³×C₄₇	—	2	2	no	—
273	C₂³×C₄₇	—	4	2	no	—
274	C₂³×C₄₇	—	4	4	no	—
275	C₂³×C₄₇	—	8	2	no	—
276	C₂³×C₄₇	—	8	4	no	—
277	C₂³×C₄₇	#120	1	2	yes	—
278	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
279	Q₈×C₄₇	—	2	1	no	—
280	Q₈×C₄₇	—	2	2	no	—
281	Q₈×C₄₇	—	2	2	no	—
282	Q₈×C₄₇	—	2	2	no	—
283	Q₈×C₄₇	—	2	2	no	—
284	Q₈×C₄₇	—	2	2	no	—
285	Q₈×C₄₇	—	2	2	no	—
286	Q₈×C₄₇	—	4	4	no	—
287	Q₈×C₄₇	—	4	2	no	—
288	Q₈×C₄₇	—	4	2	no	—
289	Q₈×C₄₇	#120	1	2	yes	—
290	D₄×C₄₇	—	2	4	no	—
291	D₄×C₄₇	—	2	4	no	—
292	D₄×C₄₇	—	2	2	no	—
293	D₄×C₄₇	—	2	1	no	—
294	D₄×C₄₇	—	2	2	no	—
295	D₄×C₄₇	—	2	2	no	—
296	D₄×C₄₇	—	2	2	no	—
297	D₄×C₄₇	—	4	4	no	—
298	D₄×C₄₇	—	4	2	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	—	4	2	no	—
308	D₄×C₄₇	—	8	2	no	—
309	D₄×C₄₇	—	8	4	no	—
310	D₄×C₄₇	—	16	4	no	—
311	D₄×C₄₇	#120	1	2	yes	—
312	C₂×C₄×C₄₇	—	2	1	no	—
313	C₂×C₄×C₄₇	—	2	2	no	—
314	C₂×C₄×C₄₇	—	2	2	no	—
315	C₂×C₄×C₄₇	—	2	4	no	—
316	C₂×C₄×C₄₇	—	2	1	no	—
317	C₂×C₄×C₄₇	—	2	2	no	—
318	C₂×C₄×C₄₇	—	2	1	no	—
319	C₂×C₄×C₄₇	—	2	2	no	—
320	C₂×C₄×C₄₇	—	2	1	no	—
321	C₂×C₄×C₄₇	—	2	2	no	—
322	C₂×C₄×C₄₇	—	4	4	no	—
323	C₂×C₄×C₄₇	—	4	4	no	—
324	C₂×C₄×C₄₇	—	4	4	no	—
325	C₂×C₄×C₄₇	—	4	4	no	—
326	C₂×C₄×C₄₇	—	4	2	no	—
327	C₂×C₄×C₄₇	—	4	2	no	—
328	C₂×C₄×C₄₇	—	4	1	no	—
329	C₂×C₄×C₄₇	—	4	2	no	—
330	C₂×C₄×C₄₇	—	4	1	no	—
331	C₂×C₄×C₄₇	—	4	2	no	—
332	C₂×C₄×C₄₇	—	4	2	no	—
333	C₂×C₄×C₄₇	—	8	1	no	—
334	C₂×C₄×C₄₇	—	8	2	no	—
335	C₂×C₄×C₄₇	#120	1	2	yes	—
336	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
337	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
338	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
339	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
340	C₈×C₄₇	—	2	4	no	—
341	C₈×C₄₇	—	4	2	no	—
342	#23873	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
343	C₂²×C₄₇	—	2	2	no	—
344	C₂²×C₄₇	—	2	2	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	2	no	—
350	C₂²×C₄₇	—	4	2	no	—
351	C₂²×C₄₇	—	4	1	no	—
352	C₂²×C₄₇	—	4	2	no	—
353	C₂²×C₄₇	—	8	2	no	—
354	C₂²×C₄₇	—	8	4	no	—
355	C₂²×C₄₇	—	16	2	no	—
356	C₂²×C₄₇	—	16	4	no	—
357	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
358	C₂²×C₄₇	D₄²	1	2	yes	—
359	C₄×C₄₇	—	2	2	no	—
360	C₄×C₄₇	—	2	2	no	—
361	C₄×C₄₇	—	4	1	no	—
362	C₄×C₄₇	—	4	2	no	—
363	C₄×C₄₇	—	4	1	no	—
364	C₄×C₄₇	—	4	1	no	—
365	C₄×C₄₇	—	4	2	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	#3586	C₂×C₄₇	1	1	yes	characteristic, complemented
372	#3500	C₂×C₄₇	1	1	yes	characteristic, complemented
373	#3027	C₂×C₄₇	1	1	yes	characteristic, complemented
374	#3019	C₂×C₄₇	1	2	yes	complemented
375	#3018	C₂×C₄₇	1	1	yes	characteristic, complemented
376	#2009	C₂×C₄₇	1	2	yes	complemented
377	#2004	C₂×C₄₇	1	2	yes	complemented
378	#1914	C₂×C₄₇	1	1	yes	characteristic, complemented
379	#1906	C₂×C₄₇	1	2	yes	complemented
380	#1884	C₂×C₄₇	1	1	yes	characteristic, complemented
381	#1796	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	2	no	—
386	C₂×C₄₇	—	8	2	no	—
387	C₂×C₄₇	—	8	1	no	—
388	C₂×C₄₇	—	8	2	no	—
389	C₂×C₄₇	#2403	1	1	yes	characteristic
390	#585	C₂²×C₄₇	1	1	yes	characteristic, complemented
391	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	1	yes	characteristic
392	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	1	yes	characteristic, complemented
393	#578	—	2	4	no	—
394	#577	—	2	4	no	—
395	#575	—	2	2	no	—
396	#575	C₂²×C₄₇	1	2	yes	—
397	#575	C₂²×C₄₇	1	2	yes	—
398	C₂×#137	—	2	2	no	—
399	C₂×#137	C₂²×C₄₇	1	2	yes	complemented
400	#568	C₂²×C₄₇	1	1	yes	characteristic, complemented
401	#548	—	2	2	no	—
402	#546	—	2	2	no	—
403	#546	—	2	4	no	—
404	#536	—	2	1	no	—
405	#536	C₂²×C₄₇	1	1	yes	characteristic, complemented
406	#535	—	2	1	no	—
407	#535	C₂²×C₄₇	1	2	yes	—
408	#535	C₂²×C₄₇	1	1	yes	characteristic
409	#532	—	2	1	no	—
410	#532	C₂²×C₄₇	1	1	yes	characteristic
411	C₂×#124	—	2	1	no	—
412	C₂×#124	C₂²×C₄₇	1	1	yes	characteristic, complemented
413	C₄×#41	C₂²×C₄₇	1	1	yes	characteristic, complemented
414	B(4,𝔽₂)	—	2	4	no	—
415	#456	—	2	1	no	—
416	#455	—	2	4	no	—
417	#455	—	2	1	no	—
418	#454	—	2	1	no	—
419	#453	—	2	4	no	—
420	#453	—	2	1	no	—
421	#421	C₂²×C₄₇	1	2	yes	complemented
422	#421	C₂²×C₄₇	1	2	yes	complemented
423	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
424	#419	C₂²×C₄₇	1	2	yes	complemented
425	#413	C₂²×C₄₇	1	1	yes	characteristic, complemented
426	C₂×#100	C₂²×C₄₇	1	2	yes	complemented
427	#410	C₂²×C₄₇	1	1	yes	characteristic, complemented
428	#410	C₂²×C₄₇	1	2	yes	complemented
429	#344	C₂²×C₄₇	1	2	yes	complemented
430	#339	C₂²×C₄₇	1	2	yes	complemented
431	C₄₇	#23873	1	1	yes	characteristic, direct product factor, Sylow
432	2₋¹⁺⁴	—	2	4	no	—
433	2₋¹⁺⁴	—	2	2	no	—
434	2₋¹⁺⁴	—	2	1	no	—
435	2₋¹⁺⁴	—	4	2	no	—
436	H(5,𝔽₂)	—	2	2	no	—
437	H(5,𝔽₂)	—	2	2	no	—
438	H(5,𝔽₂)	—	2	4	no	—
439	H(5,𝔽₂)	—	2	1	no	—
440	H(5,𝔽₂)	—	4	2	no	—
441	C₂×#41	—	2	1	no	—
442	C₂×#41	—	2	1	no	—
443	C₂×#41	—	2	1	no	—
444	C₂×#41	—	2	2	no	—
445	C₂×#41	—	2	2	no	—
446	C₂×#41	—	4	2	no	—
447	C₂×#41	D₄×C₄₇	1	2	yes	complemented
448	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
449	C₂×#41	C₂³×C₄₇	1	2	yes	—
450	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
451	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
452	C₂²×Q₈	—	2	1	no	—
453	C₂²×Q₈	D₄×C₄₇	1	2	yes	complemented
454	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
455	C₂²×D₄	—	2	2	no	—
456	C₂²×D₄	—	2	1	no	—
457	C₂²×D₄	—	2	1	no	—
458	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
459	#137	—	2	8	no	—
460	#137	—	2	4	no	—
461	#137	—	2	2	no	—
462	#137	—	2	4	no	—
463	#137	—	2	2	no	—
464	#137	—	2	2	no	—
465	#137	—	2	4	no	—
466	#137	D₄×C₄₇	1	4	yes	complemented
467	AGL(1,ℤ/8)	—	2	4	no	—
468	AGL(1,ℤ/8)	—	2	2	no	—
469	AGL(1,ℤ/8)	—	2	2	no	—
470	AGL(1,ℤ/8)	—	2	2	no	—
471	AGL(1,ℤ/8)	D₄×C₄₇	1	4	yes	complemented
472	#135	—	2	8	no	—
473	#135	—	2	4	no	—
474	#135	—	4	2	no	—
475	C₂×Q₁₆	—	2	4	no	—
476	C₂×Q₁₆	—	2	2	no	—
477	C₂×SD₁₆	—	2	4	no	—
478	C₂×SD₁₆	—	2	2	no	—
479	#131	—	2	4	no	—
480	C₂×M₁₆	—	2	2	no	—
481	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
482	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
483	C₄⋊Q₈	—	2	1	no	—
484	C₄⋊Q₈	D₄×C₄₇	1	1	yes	characteristic, complemented
485	#127	—	2	1	no	—
486	#127	—	2	2	no	—
487	#127	D₄×C₄₇	1	1	yes	characteristic, complemented
488	#124	—	2	4	no	—
489	#124	—	2	1	no	—
490	#124	—	2	2	no	—
491	#124	—	2	1	no	—
492	#124	—	2	2	no	—
493	#124	—	2	1	no	—
494	#124	—	2	1	no	—
495	#124	D₄×C₄₇	1	1	yes	characteristic, complemented
496	#124	D₄×C₄₇	1	1	yes	characteristic, complemented
497	#123	—	2	4	no	—
498	#123	—	2	2	no	—
499	C₂²⋊Q₈	—	2	4	no	—
500	C₂²⋊Q₈	—	2	2	no	—
501	#121	—	2	4	no	—
502	#121	—	2	4	no	—
503	#121	—	2	2	no	—
504	#121	—	2	2	no	—
505	#121	—	4	2	no	—
506	#120	—	2	4	no	—
507	#120	—	2	4	no	—
508	#120	D₄×C₄₇	1	4	yes	—
509	C₄×Q₈	—	2	1	no	—
510	C₄×D₄	—	2	4	no	—
511	C₄×D₄	—	2	1	no	—
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	—	2	2	no	—
514	#117	—	2	1	no	—
515	#117	C₂³×C₄₇	1	2	yes	—
516	#117	C₂³×C₄₇	1	1	yes	characteristic
517	C₂×C₂²⋊C₄	—	2	4	no	—
518	C₂×C₂²⋊C₄	—	2	2	no	—
519	C₂×C₄²	—	2	1	no	—
520	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
521	#104	—	2	4	no	—
522	#104	—	2	4	no	—
523	#104	—	2	2	no	—
524	#104	—	2	2	no	—
525	#103	—	4	2	no	—
526	#101	—	2	1	no	—
527	#100	—	2	1	no	—
528	#100	—	2	4	no	—
529	#99	—	2	2	no	—
530	#99	—	2	4	no	—
531	C₂⁴	—	2	2	no	—
532	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
533	#41	—	2	2	no	—
534	#41	—	2	2	no	—
535	#41	—	2	2	no	—
536	#41	—	2	4	no	—
537	#41	—	2	4	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	—	2	2	no	—
544	#41	—	4	2	no	—
545	#41	—	4	4	no	—
546	#41	—	4	4	no	—
547	#41	—	4	4	no	—
548	#41	—	4	2	no	—
549	#41	—	4	8	no	—
550	#41	—	4	2	no	—
551	#41	—	4	2	no	—
552	#41	—	4	2	no	—
553	#41	—	4	2	no	—
554	#41	—	4	2	no	—
555	#41	C₂×D₄×C₄₇	1	4	yes	—
556	C₂×Q₈	—	2	2	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	2	no	—
559	C₂×Q₈	—	2	4	no	—
560	C₂×Q₈	—	2	2	no	—
561	C₂×Q₈	—	2	2	no	—
562	C₂×Q₈	—	4	2	no	—
563	C₂×Q₈	—	4	4	no	—
564	C₂×Q₈	—	4	2	no	—
565	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
566	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
567	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
568	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
569	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
570	C₂×D₄	—	2	1	no	—
571	C₂×D₄	—	2	2	no	—
572	C₂×D₄	—	2	2	no	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	2	2	no	—
575	C₂×D₄	—	2	2	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	2	4	no	—
579	C₂×D₄	—	2	2	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	2	2	no	—
583	C₂×D₄	—	4	4	no	—
584	C₂×D₄	—	4	2	no	—
585	C₂×D₄	—	4	8	no	—
586	C₂×D₄	—	4	4	no	—
587	C₂×D₄	—	4	2	no	—
588	C₂×D₄	—	4	2	no	—
589	C₂×D₄	—	4	2	no	—
590	C₂×D₄	—	4	4	no	—
591	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
592	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
593	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
594	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
595	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
596	C₂²×C₄	—	2	1	no	—
597	C₂²×C₄	—	2	2	no	—
598	C₂²×C₄	—	2	2	no	—
599	C₂²×C₄	—	2	1	no	—
600	C₂²×C₄	—	2	2	no	—
601	C₂²×C₄	—	2	1	no	—
602	C₂²×C₄	—	2	2	no	—
603	C₂²×C₄	—	2	1	no	—
604	C₂²×C₄	—	2	2	no	—
605	C₂²×C₄	—	4	2	no	—
606	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
607	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
608	Q₁₆	—	2	8	no	—
609	Q₁₆	—	2	4	no	—
610	Q₁₆	—	4	2	no	—
611	SD₁₆	—	2	8	no	—
612	SD₁₆	—	2	4	no	—
613	SD₁₆	—	2	4	no	—
614	SD₁₆	—	4	2	no	—
615	SD₁₆	—	4	2	no	—
616	D₈	—	2	4	no	—
617	D₈	—	4	2	no	—
618	M₁₆	—	2	4	no	—
619	M₁₆	—	2	2	no	—
620	M₁₆	—	2	2	no	—
621	M₁₆	—	2	2	no	—
622	M₁₆	C₂×D₄×C₄₇	1	4	yes	—
623	C₂×C₈	—	2	4	no	—
624	C₂×C₈	—	2	2	no	—
625	C₂×C₈	—	2	2	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	—	2	2	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₄⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	4	no	—
632	C₂²⋊C₄	—	2	4	no	—
633	C₂²⋊C₄	—	2	4	no	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	4	no	—
638	C₂²⋊C₄	—	2	2	no	—
639	C₂²⋊C₄	—	4	4	no	—
640	C₂²⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
641	C₄²	—	2	2	no	—
642	C₄²	—	2	1	no	—
643	C₄²	—	2	1	no	—
644	C₄²	—	2	2	no	—
645	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
646	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
647	C₂³	—	2	1	no	—
648	C₂³	—	2	2	no	—
649	C₂³	—	2	1	no	—
650	C₂³	—	2	4	no	—
651	C₂³	—	2	2	no	—
652	C₂³	—	4	2	no	—
653	C₂³	—	4	4	no	—
654	C₂³	—	8	2	no	—
655	C₂³	—	8	4	no	—
656	C₂³	#120×C₄₇	1	2	yes	—
657	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
658	Q₈	—	2	1	no	—
659	Q₈	—	2	2	no	—
660	Q₈	—	2	2	no	—
661	Q₈	—	2	2	no	—
662	Q₈	—	2	2	no	—
663	Q₈	—	2	2	no	—
664	Q₈	—	2	2	no	—
665	Q₈	—	4	4	no	—
666	Q₈	—	4	2	no	—
667	Q₈	—	4	2	no	—
668	Q₈	#120×C₄₇	1	2	yes	—
669	D₄	—	2	4	no	—
670	D₄	—	2	4	no	—
671	D₄	—	2	2	no	—
672	D₄	—	2	1	no	—
673	D₄	—	2	2	no	—
674	D₄	—	2	2	no	—
675	D₄	—	2	2	no	—
676	D₄	—	4	4	no	—
677	D₄	—	4	2	no	—
678	D₄	—	4	4	no	—
679	D₄	—	4	4	no	—
680	D₄	—	4	4	no	—
681	D₄	—	4	2	no	—
682	D₄	—	4	2	no	—
683	D₄	—	4	2	no	—
684	D₄	—	4	1	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	2	no	—
687	D₄	—	8	2	no	—
688	D₄	—	8	4	no	—
689	D₄	—	16	4	no	—
690	D₄	#120×C₄₇	1	2	yes	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	4	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	4	4	no	—
702	C₂×C₄	—	4	4	no	—
703	C₂×C₄	—	4	4	no	—
704	C₂×C₄	—	4	4	no	—
705	C₂×C₄	—	4	2	no	—
706	C₂×C₄	—	4	2	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	4	2	no	—
709	C₂×C₄	—	4	1	no	—
710	C₂×C₄	—	4	2	no	—
711	C₂×C₄	—	4	2	no	—
712	C₂×C₄	—	8	1	no	—
713	C₂×C₄	—	8	2	no	—
714	C₂×C₄	#120×C₄₇	1	2	yes	—
715	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
716	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
717	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
718	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
719	C₈	—	2	4	no	—
720	C₈	—	4	2	no	—
721	C₂²	—	2	2	no	—
722	C₂²	—	2	2	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	2	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	4	no	—
727	C₂²	—	4	2	no	—
728	C₂²	—	4	2	no	—
729	C₂²	—	4	1	no	—
730	C₂²	—	4	2	no	—
731	C₂²	—	8	2	no	—
732	C₂²	—	8	4	no	—
733	C₂²	—	16	4	no	—
734	C₂²	—	16	2	no	—
735	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
736	C₂²	D₄²×C₄₇	1	2	yes	—
737	C₄	—	2	2	no	—
738	C₄	—	2	2	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	2	no	—
745	C₄	—	8	2	no	—
746	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
747	C₄	C₄₇×#534	1	1	yes	characteristic
748	C₄	D₄²×C₄₇	1	2	yes	—
749	C₂	—	2	1	no	—
750	C₂	—	2	2	no	—
751	C₂	—	4	2	no	—
752	C₂	—	4	2	no	—
753	C₂	—	8	2	no	—
754	C₂	—	8	1	no	—
755	C₂	—	8	2	no	—
756	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×#3586	C₂	semidirect
2. C₄₇×#3500	C₂	semidirect
3. C₄₇×#3027	C₂	semidirect
4. C₄₇×#3019	C₂	semidirect
5. C₄₇×#3018	C₂	semidirect
6. C₄₇×#2009	C₂	semidirect
7. C₄₇×#2004	C₂	semidirect
8. C₄₇×#1914	C₂	semidirect
9. C₄₇×#1906	C₂	semidirect
10. C₄₇×#1884	C₂	semidirect
11. C₄₇×#1796	C₂	semidirect
12. C₄₇×#585	C₂²	semidirect
14. C₂×H(5,𝔽₂)×C₄₇	C₂²	semidirect
21. C₂×#137×C₄₇	C₂²	semidirect
22. C₄₇×#568	C₂²	semidirect
27. C₄₇×#536	C₂²	semidirect
34. C₂×#124×C₄₇	C₂²	semidirect
35. C₄×#41×C₄₇	C₂²	semidirect
43. C₄₇×#421	C₂²	semidirect
44. C₄₇×#421	C₂²	semidirect
45. C₂×#104×C₄₇	C₂²	semidirect
46. C₄₇×#419	C₂²	semidirect
47. C₄₇×#413	C₂²	semidirect
48. C₂×#100×C₄₇	C₂²	semidirect
49. C₄₇×#410	C₂²	semidirect
50. C₄₇×#410	C₂²	semidirect
51. C₄₇×#344	C₂²	semidirect
52. C₄₇×#339	C₂²	semidirect
68. C₂×#41×C₄₇	D₄	semidirect
74. C₂²×Q₈×C₄₇	D₄	semidirect
87. #137×C₄₇	D₄	semidirect
92. AGL(1,ℤ/8)×C₄₇	D₄	semidirect
105. C₄⋊Q₈×C₄₇	D₄	semidirect
108. #127×C₄₇	D₄	semidirect
116. #124×C₄₇	D₄	semidirect
117. #124×C₄₇	D₄	semidirect
342. #23873	C₄₇	direct (N × H)
371. #3586	C₂×C₄₇	semidirect
372. #3500	C₂×C₄₇	semidirect
373. #3027	C₂×C₄₇	semidirect
374. #3019	C₂×C₄₇	semidirect
375. #3018	C₂×C₄₇	semidirect
376. #2009	C₂×C₄₇	semidirect
377. #2004	C₂×C₄₇	semidirect
378. #1914	C₂×C₄₇	semidirect
379. #1906	C₂×C₄₇	semidirect
380. #1884	C₂×C₄₇	semidirect
381. #1796	C₂×C₄₇	semidirect
390. #585	C₂²×C₄₇	semidirect
392. C₂×H(5,𝔽₂)	C₂²×C₄₇	semidirect
399. C₂×#137	C₂²×C₄₇	semidirect
400. #568	C₂²×C₄₇	semidirect
405. #536	C₂²×C₄₇	semidirect
412. C₂×#124	C₂²×C₄₇	semidirect
413. C₄×#41	C₂²×C₄₇	semidirect
421. #421	C₂²×C₄₇	semidirect
422. #421	C₂²×C₄₇	semidirect
423. C₂×#104	C₂²×C₄₇	semidirect
424. #419	C₂²×C₄₇	semidirect
425. #413	C₂²×C₄₇	semidirect
426. C₂×#100	C₂²×C₄₇	semidirect
427. #410	C₂²×C₄₇	semidirect
428. #410	C₂²×C₄₇	semidirect
429. #344	C₂²×C₄₇	semidirect
430. #339	C₂²×C₄₇	semidirect
431. C₄₇	#23873	direct (N × H)
447. C₂×#41	D₄×C₄₇	semidirect
453. C₂²×Q₈	D₄×C₄₇	semidirect
466. #137	D₄×C₄₇	semidirect
471. AGL(1,ℤ/8)	D₄×C₄₇	semidirect
484. C₄⋊Q₈	D₄×C₄₇	semidirect
487. #127	D₄×C₄₇	semidirect
495. #124	D₄×C₄₇	semidirect
496. #124	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×#3586	2
C₄₇×#3500	2
C₄₇×#3027	2
C₄₇×#3019	2
C₄₇×#3018	2
C₄₇×#2009	2
C₄₇×#2004	2
C₄₇×#1914	2
C₄₇×#1906	2
C₄₇×#1884	2
C₄₇×#1796	2
#23873	47

Subgroups — C47×#23873 #26063571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23873 #26063571