Subgroups — C₄₇×#24422 #26064120

Order 12032.

1300 conjugacy classes of subgroups.

The subgroup lattice has 590 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×#3068	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3021	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3010	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2889	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2049	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2019	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1914	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1881	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1858	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1791	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#585	C₂²	1	1	yes	characteristic
13	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#578	—	2	4	no	—
15	C₄₇×#577	—	2	4	no	—
16	C₄₇×#575	—	2	2	no	—
17	C₄₇×#575	C₂²	1	2	yes	—
18	C₂×#137×C₄₇	—	2	2	no	—
19	C₂×#137×C₄₇	C₂²	1	2	yes	—
20	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	—
21	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	1	yes	characteristic, complemented
22	C₂²×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
23	C₄₇×#568	C₂²	1	1	yes	characteristic
24	C₄₇×#568	C₂²	1	2	yes	complemented
25	C₄₇×#535	C₂²	1	1	yes	characteristic
26	C₄₇×#472	—	2	4	no	—
27	C₄₇×#471	—	2	4	no	—
28	C₄₇×#454	—	2	2	no	—
29	C₄₇×#453	—	2	2	no	—
30	C₄₇×#449	—	2	2	no	—
31	C₄₇×#448	—	2	2	no	—
32	C₄₇×#430	C₂²	1	2	yes	complemented
33	C₄₇×#421	C₂²	1	2	yes	—
34	C₄₇×#419	C₂²	1	1	yes	characteristic, complemented
35	C₄₇×#416	C₂²	1	1	yes	characteristic, complemented
36	C₄₇×#413	—	2	2	no	—
37	C₄₇×#413	C₂²	1	2	yes	—
38	C₄₇×#413	C₂²	1	1	yes	characteristic
39	C₄₇×#413	C₂²	1	1	yes	characteristic
40	C₂×#101×C₄₇	—	2	2	no	—
41	C₂×#101×C₄₇	C₂²	1	2	yes	complemented
42	C₂×#100×C₄₇	C₂²	1	2	yes	complemented
43	C₄₇×#408	C₂²	1	1	yes	characteristic
44	C₄₇×#408	C₂²	1	2	yes	—
45	C₄₇×#344	C₂²	1	2	yes	complemented
46	C₄₇×#341	C₂²	1	2	yes	complemented
47	2₋¹⁺⁴×C₄₇	—	2	4	no	—
48	2₋¹⁺⁴×C₄₇	—	2	2	no	—
49	2₋¹⁺⁴×C₄₇	—	2	1	no	—
50	2₋¹⁺⁴×C₄₇	—	4	2	no	—
51	H(5,𝔽₂)×C₄₇	—	2	2	no	—
52	H(5,𝔽₂)×C₄₇	—	2	2	no	—
53	H(5,𝔽₂)×C₄₇	—	2	1	no	—
54	C₂×#41×C₄₇	—	2	1	no	—
55	C₂×#41×C₄₇	—	2	2	no	—
56	C₂×#41×C₄₇	—	2	2	no	—
57	C₂×#41×C₄₇	—	4	2	no	—
58	C₂×#41×C₄₇	D₄	1	2	yes	—
59	C₂×#41×C₄₇	C₂³	1	2	yes	—
60	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₄₇	—	2	1	no	—
63	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
64	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
66	#137×C₄₇	—	2	8	no	—
67	#137×C₄₇	—	2	4	no	—
68	#137×C₄₇	—	2	2	no	—
69	#137×C₄₇	—	2	4	no	—
70	#137×C₄₇	—	2	4	no	—
71	#137×C₄₇	D₄	1	4	yes	complemented
72	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
73	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
74	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
75	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
76	AGL(1,ℤ/8)×C₄₇	—	4	2	no	—
77	AGL(1,ℤ/8)×C₄₇	D₄	1	4	yes	complemented
78	#135×C₄₇	—	2	8	no	—
79	#135×C₄₇	—	2	4	no	—
80	C₂×Q₁₆×C₄₇	—	2	4	no	—
81	C₂×Q₁₆×C₄₇	—	2	2	no	—
82	C₂×SD₁₆×C₄₇	—	2	4	no	—
83	C₂×SD₁₆×C₄₇	—	2	2	no	—
84	C₂×SD₁₆×C₄₇	—	2	2	no	—
85	C₂×SD₁₆×C₄₇	—	2	2	no	—
86	C₂×SD₁₆×C₄₇	—	2	1	no	—
87	C₂×SD₁₆×C₄₇	—	4	2	no	—
88	C₂×D₈×C₄₇	—	2	1	no	—
89	C₂×D₈×C₄₇	—	2	2	no	—
90	#131×C₄₇	—	2	4	no	—
91	#131×C₄₇	—	2	4	no	—
92	#131×C₄₇	—	4	2	no	—
93	C₂×M₁₆×C₄₇	—	2	2	no	—
94	C₂×M₁₆×C₄₇	—	2	4	no	—
95	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
96	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
97	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
98	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
99	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
100	#127×C₄₇	—	2	1	no	—
101	#124×C₄₇	—	2	1	no	—
102	#121×C₄₇	—	2	2	no	—
103	#120×C₄₇	—	2	2	no	—
104	#117×C₄₇	C₂³	1	1	yes	characteristic
105	#108×C₄₇	—	2	4	no	—
106	#104×C₄₇	—	2	4	no	—
107	#103×C₄₇	—	2	2	no	—
108	#103×C₄₇	—	2	1	no	—
109	#102×C₄₇	—	2	1	no	—
110	#101×C₄₇	—	2	2	no	—
111	#101×C₄₇	—	2	1	no	—
112	#101×C₄₇	—	2	4	no	—
113	#101×C₄₇	D₄	1	2	yes	—
114	#100×C₄₇	—	2	2	no	—
115	#100×C₄₇	—	2	1	no	—
116	#100×C₄₇	—	2	4	no	—
117	#100×C₄₇	D₄	1	2	yes	—
118	C₂²⋊C₈×C₄₇	—	2	2	no	—
119	C₂²⋊C₈×C₄₇	—	4	2	no	—
120	C₂⁴×C₄₇	—	2	1	no	—
121	#41×C₄₇	—	2	2	no	—
122	#41×C₄₇	—	2	2	no	—
123	#41×C₄₇	—	2	2	no	—
124	#41×C₄₇	—	2	4	no	—
125	#41×C₄₇	—	2	4	no	—
126	#41×C₄₇	—	2	2	no	—
127	#41×C₄₇	—	2	2	no	—
128	#41×C₄₇	—	2	2	no	—
129	#41×C₄₇	—	2	2	no	—
130	#41×C₄₇	—	4	2	no	—
131	#41×C₄₇	—	4	4	no	—
132	#41×C₄₇	—	4	4	no	—
133	#41×C₄₇	—	4	2	no	—
134	#41×C₄₇	—	4	8	no	—
135	#41×C₄₇	—	4	2	no	—
136	#41×C₄₇	—	4	2	no	—
137	#41×C₄₇	—	4	2	no	—
138	#41×C₄₇	C₂×D₄	1	4	yes	—
139	C₂×Q₈×C₄₇	—	2	2	no	—
140	C₂×Q₈×C₄₇	—	2	1	no	—
141	C₂×Q₈×C₄₇	—	2	2	no	—
142	C₂×Q₈×C₄₇	—	2	4	no	—
143	C₂×Q₈×C₄₇	—	2	2	no	—
144	C₂×Q₈×C₄₇	—	2	2	no	—
145	C₂×Q₈×C₄₇	—	4	2	no	—
146	C₂×Q₈×C₄₇	—	4	4	no	—
147	C₂×Q₈×C₄₇	—	4	2	no	—
148	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
150	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
151	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₄₇	—	2	1	no	—
153	C₂×D₄×C₄₇	—	2	2	no	—
154	C₂×D₄×C₄₇	—	2	2	no	—
155	C₂×D₄×C₄₇	—	2	2	no	—
156	C₂×D₄×C₄₇	—	2	1	no	—
157	C₂×D₄×C₄₇	—	2	1	no	—
158	C₂×D₄×C₄₇	—	2	2	no	—
159	C₂×D₄×C₄₇	—	2	2	no	—
160	C₂×D₄×C₄₇	—	4	2	no	—
161	C₂×D₄×C₄₇	—	4	4	no	—
162	C₂×D₄×C₄₇	—	4	2	no	—
163	C₂×D₄×C₄₇	—	4	2	no	—
164	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
166	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₄₇	#41	1	2	yes	—
168	C₂²×C₄×C₄₇	—	2	1	no	—
169	C₂²×C₄×C₄₇	—	2	2	no	—
170	C₂²×C₄×C₄₇	—	2	1	no	—
171	C₂²×C₄×C₄₇	—	2	2	no	—
172	C₂²×C₄×C₄₇	—	2	2	no	—
173	C₂²×C₄×C₄₇	—	4	2	no	—
174	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₄₇	—	2	8	no	—
177	Q₁₆×C₄₇	—	2	4	no	—
178	SD₁₆×C₄₇	—	2	8	no	—
179	SD₁₆×C₄₇	—	2	4	no	—
180	SD₁₆×C₄₇	—	2	4	no	—
181	SD₁₆×C₄₇	—	4	4	no	—
182	SD₁₆×C₄₇	—	4	4	no	—
183	SD₁₆×C₄₇	—	4	2	no	—
184	D₈×C₄₇	—	2	4	no	—
185	D₈×C₄₇	—	4	4	no	—
186	D₈×C₄₇	—	4	2	no	—
187	M₁₆×C₄₇	—	2	4	no	—
188	M₁₆×C₄₇	—	2	2	no	—
189	M₁₆×C₄₇	—	2	2	no	—
190	M₁₆×C₄₇	—	2	2	no	—
191	M₁₆×C₄₇	—	2	2	no	—
192	M₁₆×C₄₇	—	2	2	no	—
193	M₁₆×C₄₇	—	4	4	no	—
194	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
195	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
196	C₂×C₈×C₄₇	—	2	4	no	—
197	C₂×C₈×C₄₇	—	2	2	no	—
198	C₂×C₈×C₄₇	—	2	2	no	—
199	C₂×C₈×C₄₇	—	2	1	no	—
200	C₂×C₈×C₄₇	—	2	1	no	—
201	C₂×C₈×C₄₇	—	2	2	no	—
202	C₂×C₈×C₄₇	—	2	2	no	—
203	C₂×C₈×C₄₇	—	4	4	no	—
204	C₄⋊C₄×C₄₇	—	2	1	no	—
205	C₂²⋊C₄×C₄₇	—	2	1	no	—
206	C₂²⋊C₄×C₄₇	—	4	2	no	—
207	C₄²×C₄₇	—	2	1	no	—
208	C₂³×C₄₇	—	2	1	no	—
209	C₂³×C₄₇	—	2	2	no	—
210	C₂³×C₄₇	—	4	2	no	—
211	C₂³×C₄₇	—	4	1	no	—
212	C₂³×C₄₇	—	4	2	no	—
213	C₂³×C₄₇	—	16	1	no	—
214	C₂³×C₄₇	#121	1	2	yes	—
215	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
216	Q₈×C₄₇	—	2	1	no	—
217	Q₈×C₄₇	—	2	2	no	—
218	Q₈×C₄₇	—	2	2	no	—
219	Q₈×C₄₇	—	2	2	no	—
220	Q₈×C₄₇	—	2	2	no	—
221	Q₈×C₄₇	—	2	2	no	—
222	Q₈×C₄₇	—	4	4	no	—
223	Q₈×C₄₇	—	4	2	no	—
224	Q₈×C₄₇	—	4	2	no	—
225	Q₈×C₄₇	#120	1	2	yes	—
226	D₄×C₄₇	—	2	4	no	—
227	D₄×C₄₇	—	2	4	no	—
228	D₄×C₄₇	—	2	2	no	—
229	D₄×C₄₇	—	2	1	no	—
230	D₄×C₄₇	—	2	2	no	—
231	D₄×C₄₇	—	2	2	no	—
232	D₄×C₄₇	—	4	2	no	—
233	D₄×C₄₇	—	4	4	no	—
234	D₄×C₄₇	—	4	4	no	—
235	D₄×C₄₇	—	4	2	no	—
236	D₄×C₄₇	—	4	2	no	—
237	D₄×C₄₇	—	4	1	no	—
238	D₄×C₄₇	—	4	1	no	—
239	D₄×C₄₇	—	4	2	no	—
240	D₄×C₄₇	—	16	2	no	—
241	D₄×C₄₇	#120	1	2	yes	—
242	C₂×C₄×C₄₇	—	2	1	no	—
243	C₂×C₄×C₄₇	—	2	2	no	—
244	C₂×C₄×C₄₇	—	2	2	no	—
245	C₂×C₄×C₄₇	—	2	4	no	—
246	C₂×C₄×C₄₇	—	2	2	no	—
247	C₂×C₄×C₄₇	—	2	2	no	—
248	C₂×C₄×C₄₇	—	4	4	no	—
249	C₂×C₄×C₄₇	—	4	4	no	—
250	C₂×C₄×C₄₇	—	4	4	no	—
251	C₂×C₄×C₄₇	—	4	2	no	—
252	C₂×C₄×C₄₇	—	4	2	no	—
253	C₂×C₄×C₄₇	—	4	2	no	—
254	C₂×C₄×C₄₇	—	4	2	no	—
255	C₂×C₄×C₄₇	—	4	1	no	—
256	C₂×C₄×C₄₇	#121	1	2	yes	—
257	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
261	C₈×C₄₇	—	2	4	no	—
262	C₈×C₄₇	—	2	4	no	—
263	C₈×C₄₇	—	4	1	no	—
264	C₈×C₄₇	—	4	1	no	—
265	C₈×C₄₇	—	4	2	no	—
266	#24422	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₂²×C₄₇	—	2	2	no	—
268	C₂²×C₄₇	—	2	2	no	—
269	C₂²×C₄₇	—	4	1	no	—
270	C₂²×C₄₇	—	4	2	no	—
271	C₂²×C₄₇	—	4	2	no	—
272	C₂²×C₄₇	—	8	1	no	—
273	C₂²×C₄₇	—	8	2	no	—
274	C₂²×C₄₇	—	8	2	no	—
275	C₂²×C₄₇	—	8	1	no	—
276	C₂²×C₄₇	—	16	2	no	—
277	C₂²×C₄₇	—	16	2	no	—
278	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
279	C₂²×C₄₇	#546	1	2	yes	—
280	C₄×C₄₇	—	2	2	no	—
281	C₄×C₄₇	—	2	2	no	—
282	C₄×C₄₇	—	4	2	no	—
283	C₄×C₄₇	—	4	2	no	—
284	C₄×C₄₇	—	8	1	no	—
285	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
286	C₄×C₄₇	#534	1	1	yes	characteristic
287	C₄×C₄₇	D₄²	1	2	yes	—
288	#3586	C₂×C₄₇	1	1	yes	characteristic, complemented
289	#3068	C₂×C₄₇	1	2	yes	complemented
290	#3021	C₂×C₄₇	1	1	yes	characteristic, complemented
291	#3010	C₂×C₄₇	1	1	yes	characteristic, complemented
292	#2889	C₂×C₄₇	1	1	yes	characteristic, complemented
293	#2049	C₂×C₄₇	1	2	yes	complemented
294	#2019	C₂×C₄₇	1	2	yes	complemented
295	#1914	C₂×C₄₇	1	1	yes	characteristic, complemented
296	#1881	C₂×C₄₇	1	2	yes	complemented
297	#1858	C₂×C₄₇	1	1	yes	characteristic, complemented
298	#1791	C₂×C₄₇	1	1	yes	characteristic, complemented
299	C₂×C₄₇	—	2	1	no	—
300	C₂×C₄₇	—	2	2	no	—
301	C₂×C₄₇	—	4	2	no	—
302	C₂×C₄₇	—	4	2	no	—
303	C₂×C₄₇	—	8	2	no	—
304	C₂×C₄₇	—	16	1	no	—
305	C₂×C₄₇	#2410	1	1	yes	characteristic
306	#585	C₂²×C₄₇	1	1	yes	characteristic
307	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	1	yes	characteristic
308	#578	—	2	4	no	—
309	#577	—	2	4	no	—
310	#575	—	2	2	no	—
311	#575	C₂²×C₄₇	1	2	yes	—
312	C₂×#137	—	2	2	no	—
313	C₂×#137	C₂²×C₄₇	1	2	yes	—
314	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	—
315	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	1	yes	characteristic, complemented
316	C₂²×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
317	#568	C₂²×C₄₇	1	1	yes	characteristic
318	#568	C₂²×C₄₇	1	2	yes	complemented
319	#535	C₂²×C₄₇	1	1	yes	characteristic
320	#472	—	2	4	no	—
321	#471	—	2	4	no	—
322	#454	—	2	2	no	—
323	#453	—	2	2	no	—
324	#449	—	2	2	no	—
325	#448	—	2	2	no	—
326	#430	C₂²×C₄₇	1	2	yes	complemented
327	#421	C₂²×C₄₇	1	2	yes	—
328	#419	C₂²×C₄₇	1	1	yes	characteristic, complemented
329	#416	C₂²×C₄₇	1	1	yes	characteristic, complemented
330	#413	—	2	2	no	—
331	#413	C₂²×C₄₇	1	2	yes	—
332	#413	C₂²×C₄₇	1	1	yes	characteristic
333	#413	C₂²×C₄₇	1	1	yes	characteristic
334	C₂×#101	—	2	2	no	—
335	C₂×#101	C₂²×C₄₇	1	2	yes	complemented
336	C₂×#100	C₂²×C₄₇	1	2	yes	complemented
337	#408	C₂²×C₄₇	1	1	yes	characteristic
338	#408	C₂²×C₄₇	1	2	yes	—
339	#344	C₂²×C₄₇	1	2	yes	complemented
340	#341	C₂²×C₄₇	1	2	yes	complemented
341	C₄₇	#24422	1	1	yes	characteristic, direct product factor, Sylow
342	2₋¹⁺⁴	—	2	4	no	—
343	2₋¹⁺⁴	—	2	2	no	—
344	2₋¹⁺⁴	—	2	1	no	—
345	2₋¹⁺⁴	—	4	2	no	—
346	H(5,𝔽₂)	—	2	2	no	—
347	H(5,𝔽₂)	—	2	2	no	—
348	H(5,𝔽₂)	—	2	1	no	—
349	C₂×#41	—	2	1	no	—
350	C₂×#41	—	2	2	no	—
351	C₂×#41	—	2	2	no	—
352	C₂×#41	—	4	2	no	—
353	C₂×#41	D₄×C₄₇	1	2	yes	—
354	C₂×#41	C₂³×C₄₇	1	2	yes	—
355	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
356	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
357	C₂²×Q₈	—	2	1	no	—
358	C₂²×Q₈	D₄×C₄₇	1	2	yes	—
359	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
360	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
361	#137	—	2	8	no	—
362	#137	—	2	4	no	—
363	#137	—	2	2	no	—
364	#137	—	2	4	no	—
365	#137	—	2	4	no	—
366	#137	D₄×C₄₇	1	4	yes	complemented
367	AGL(1,ℤ/8)	—	2	4	no	—
368	AGL(1,ℤ/8)	—	2	2	no	—
369	AGL(1,ℤ/8)	—	2	4	no	—
370	AGL(1,ℤ/8)	—	2	4	no	—
371	AGL(1,ℤ/8)	—	4	2	no	—
372	AGL(1,ℤ/8)	D₄×C₄₇	1	4	yes	complemented
373	#135	—	2	8	no	—
374	#135	—	2	4	no	—
375	C₂×Q₁₆	—	2	4	no	—
376	C₂×Q₁₆	—	2	2	no	—
377	C₂×SD₁₆	—	2	4	no	—
378	C₂×SD₁₆	—	2	2	no	—
379	C₂×SD₁₆	—	2	2	no	—
380	C₂×SD₁₆	—	2	2	no	—
381	C₂×SD₁₆	—	2	1	no	—
382	C₂×SD₁₆	—	4	2	no	—
383	C₂×D₈	—	2	2	no	—
384	C₂×D₈	—	2	1	no	—
385	#131	—	2	4	no	—
386	#131	—	2	4	no	—
387	#131	—	4	2	no	—
388	C₂×M₁₆	—	2	2	no	—
389	C₂×M₁₆	—	2	4	no	—
390	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
391	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
392	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
393	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
394	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
395	#127	—	2	1	no	—
396	#124	—	2	1	no	—
397	#121	—	2	2	no	—
398	#120	—	2	2	no	—
399	#117	C₂³×C₄₇	1	1	yes	characteristic
400	#108	—	2	4	no	—
401	#104	—	2	4	no	—
402	#103	—	2	1	no	—
403	#103	—	2	2	no	—
404	#102	—	2	1	no	—
405	#101	—	2	1	no	—
406	#101	—	2	2	no	—
407	#101	—	2	4	no	—
408	#101	D₄×C₄₇	1	2	yes	—
409	#100	—	2	1	no	—
410	#100	—	2	2	no	—
411	#100	—	2	4	no	—
412	#100	D₄×C₄₇	1	2	yes	—
413	C₂²⋊C₈	—	2	2	no	—
414	C₂²⋊C₈	—	4	2	no	—
415	C₂⁴	—	2	1	no	—
416	#41	—	2	2	no	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	2	4	no	—
420	#41	—	2	4	no	—
421	#41	—	2	2	no	—
422	#41	—	2	2	no	—
423	#41	—	2	2	no	—
424	#41	—	2	2	no	—
425	#41	—	4	2	no	—
426	#41	—	4	4	no	—
427	#41	—	4	4	no	—
428	#41	—	4	2	no	—
429	#41	—	4	8	no	—
430	#41	—	4	2	no	—
431	#41	—	4	2	no	—
432	#41	—	4	2	no	—
433	#41	C₂×D₄×C₄₇	1	4	yes	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	4	no	—
442	C₂×Q₈	—	4	2	no	—
443	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
444	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
445	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
446	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	4	2	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	2	no	—
459	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
460	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
461	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
462	C₂×D₄	#41×C₄₇	1	2	yes	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
471	Q₁₆	—	2	8	no	—
472	Q₁₆	—	2	4	no	—
473	SD₁₆	—	2	8	no	—
474	SD₁₆	—	2	4	no	—
475	SD₁₆	—	2	4	no	—
476	SD₁₆	—	4	4	no	—
477	SD₁₆	—	4	4	no	—
478	SD₁₆	—	4	2	no	—
479	D₈	—	2	4	no	—
480	D₈	—	4	4	no	—
481	D₈	—	4	2	no	—
482	M₁₆	—	2	4	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	2	2	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	4	4	no	—
489	M₁₆	C₂×D₄×C₄₇	1	4	yes	—
490	M₁₆	C₂×D₄×C₄₇	1	4	yes	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	2	no	—
497	C₂×C₈	—	2	2	no	—
498	C₂×C₈	—	4	4	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	4	2	no	—
502	C₄²	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	16	1	no	—
509	C₂³	#121×C₄₇	1	2	yes	—
510	C₂³	C₂×#41×C₄₇	1	1	yes	characteristic
511	Q₈	—	2	1	no	—
512	Q₈	—	2	2	no	—
513	Q₈	—	2	2	no	—
514	Q₈	—	2	2	no	—
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	4	4	no	—
518	Q₈	—	4	2	no	—
519	Q₈	—	4	2	no	—
520	Q₈	#120×C₄₇	1	2	yes	—
521	D₄	—	2	4	no	—
522	D₄	—	2	4	no	—
523	D₄	—	2	2	no	—
524	D₄	—	2	1	no	—
525	D₄	—	2	2	no	—
526	D₄	—	2	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	4	2	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	2	no	—
535	D₄	—	16	2	no	—
536	D₄	#120×C₄₇	1	2	yes	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	#121×C₄₇	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
555	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
556	C₈	—	2	4	no	—
557	C₈	—	2	4	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	8	2	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	1	no	—
570	C₂²	—	16	2	no	—
571	C₂²	—	16	2	no	—
572	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
573	C₂²	C₄₇×#546	1	2	yes	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	8	1	no	—
579	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
580	C₄	C₄₇×#534	1	1	yes	characteristic
581	C₄	D₄²×C₄₇	1	2	yes	—
582	C₂	—	2	1	no	—
583	C₂	—	2	2	no	—
584	C₂	—	4	2	no	—
585	C₂	—	4	2	no	—
586	C₂	—	8	2	no	—
587	C₂	—	16	1	no	—
588	C₂	C₄₇×#2410	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#2410
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×#3068	C₂	semidirect
3. C₄₇×#3021	C₂	semidirect
4. C₄₇×#3010	C₂	semidirect
5. C₄₇×#2889	C₂	semidirect
6. C₄₇×#2049	C₂	semidirect
7. C₄₇×#2019	C₂	semidirect
8. C₄₇×#1914	C₂	semidirect
9. C₄₇×#1881	C₂	semidirect
10. C₄₇×#1858	C₂	semidirect
11. C₄₇×#1791	C₂	semidirect
21. C₂×AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
22. C₂²×SD₁₆×C₄₇	C₂²	semidirect
24. C₄₇×#568	C₂²	semidirect
32. C₄₇×#430	C₂²	semidirect
34. C₄₇×#419	C₂²	semidirect
35. C₄₇×#416	C₂²	semidirect
41. C₂×#101×C₄₇	C₂²	semidirect
42. C₂×#100×C₄₇	C₂²	semidirect
45. C₄₇×#344	C₂²	semidirect
46. C₄₇×#341	C₂²	semidirect
71. #137×C₄₇	D₄	semidirect
77. AGL(1,ℤ/8)×C₄₇	D₄	semidirect
266. #24422	C₄₇	direct (N × H)
288. #3586	C₂×C₄₇	semidirect
289. #3068	C₂×C₄₇	semidirect
290. #3021	C₂×C₄₇	semidirect
291. #3010	C₂×C₄₇	semidirect
292. #2889	C₂×C₄₇	semidirect
293. #2049	C₂×C₄₇	semidirect
294. #2019	C₂×C₄₇	semidirect
295. #1914	C₂×C₄₇	semidirect
296. #1881	C₂×C₄₇	semidirect
297. #1858	C₂×C₄₇	semidirect
298. #1791	C₂×C₄₇	semidirect
315. C₂×AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
316. C₂²×SD₁₆	C₂²×C₄₇	semidirect
318. #568	C₂²×C₄₇	semidirect
326. #430	C₂²×C₄₇	semidirect
328. #419	C₂²×C₄₇	semidirect
329. #416	C₂²×C₄₇	semidirect
335. C₂×#101	C₂²×C₄₇	semidirect
336. C₂×#100	C₂²×C₄₇	semidirect
339. #344	C₂²×C₄₇	semidirect
340. #341	C₂²×C₄₇	semidirect
341. C₄₇	#24422	direct (N × H)
366. #137	D₄×C₄₇	semidirect
372. AGL(1,ℤ/8)	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×#3068	2
C₄₇×#3021	2
C₄₇×#3010	2
C₄₇×#2889	2
C₄₇×#2049	2
C₄₇×#2019	2
C₄₇×#1914	2
C₄₇×#1881	2
C₄₇×#1858	2
C₄₇×#1791	2
#24422	47

Subgroups — C47×#24422 #26064120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#24422 #26064120