Subgroups — #26204289

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254314	C₂	1	2	yes	maximal, complemented
2	#20253968	C₂	1	1	yes	maximal, characteristic
3	#500×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253783	C₂	1	2	yes	maximal, complemented
5	#20243043	C₂	1	2	yes	maximal, complemented
6	#20243042	C₂	1	2	yes	maximal, complemented
7	#20242997	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242981	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242980	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689674	C₂²	1	2	yes	—
13	C₄⋊Q₈×D₄₇	C₂²	1	2	yes	—
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689527	—	2	2	no	—
16	#15689526	C₂²	1	2	yes	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689497	C₂²	1	2	yes	—
19	#15689496	C₂²	1	2	yes	—
20	#15689484	—	2	4	no	—
21	#15689475	C₂²	1	2	yes	—
22	#15689473	C₂²	1	2	yes	—
23	#15689471	C₂²	1	2	yes	—
24	#15689471	C₂²	1	2	yes	—
25	#15689470	C₂²	1	2	yes	—
26	#15689360	—	2	2	no	—
27	#15689327	—	2	4	no	—
28	#15689263	—	2	4	no	—
29	#15689205	—	2	2	no	—
30	#15689174	C₂²	1	1	yes	characteristic
31	#107×D₄₇	—	2	2	no	—
32	#15689123	—	2	4	no	—
33	C₄×#10625324	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄₇×#500	C₂²	1	1	yes	characteristic
40	C₄₇×#487	C₂²	1	2	yes	—
41	C₄₇×#487	C₂²	1	2	yes	—
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625354	—	2	2	no	—
44	C₂×Q₈×D₄₇	—	2	2	no	—
45	C₂×#10625325	—	2	4	no	—
46	C₂×#10625336	—	2	4	no	—
47	C₂×D₃₇₆	—	2	2	no	—
48	C₂×#10625324	D₄	1	2	yes	—
49	C₂×C₈×D₄₇	D₄	1	2	yes	—
50	#13094024	C₂³	1	2	yes	—
51	#13094023	—	2	2	no	—
52	#13094023	—	2	2	no	—
53	#13094022	—	2	2	no	—
54	#13094021	C₂³	1	2	yes	—
55	#13094013	—	2	2	no	—
56	#13093999	—	2	2	no	—
57	#13093999	—	2	2	no	—
58	#13093986	—	2	2	no	—
59	#13093986	—	2	2	no	—
60	C₄⋊C₄×D₄₇	—	2	2	no	—
61	C₄⋊C₄×D₄₇	—	2	2	no	—
62	C₄²×D₄₇	C₂³	1	1	yes	characteristic
63	#13093958	—	2	2	no	—
64	#13093957	—	2	4	no	—
65	Dic₄₇⋊C₈	D₄	1	2	yes	—
66	#13093948	—	2	4	no	—
67	#13093947	—	2	4	no	—
68	#13093946	—	2	4	no	—
69	#13093944	—	2	4	no	—
70	#13093944	—	2	4	no	—
71	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
72	#13093934	D₄	1	2	yes	—
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₇	D₄	1	2	yes	—
77	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
79	#107×C₄₇	—	2	2	no	—
80	#107×C₄₇	—	2	2	no	—
81	#103×C₄₇	—	2	4	no	—
82	#103×C₄₇	—	2	4	no	—
83	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₄₇	—	2	2	no	—
87	C₂×Q₈×C₄₇	—	2	2	no	—
88	#10625354	—	4	8	no	—
89	Q₈×D₄₇	—	8	4	no	—
90	C₂×D₁₈₈	—	2	2	no	—
91	C₂×D₁₈₈	—	2	2	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	4	2	no	—
94	C₂×C₄×D₄₇	—	4	2	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	4	2	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
109	#10625336	—	8	4	no	—
110	(C₂×D₄₇)⋊C₄	—	4	2	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	Dic₄₇⋊C₄	—	4	2	no	—
115	Dic₄₇⋊C₄	—	4	2	no	—
116	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
120	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
121	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
122	D₃₇₆	—	4	4	no	—
123	#10625325	—	8	4	no	—
124	#10625324	—	2	4	no	—
125	C₈×D₄₇	—	2	4	no	—
126	C₂²×D₄₇	—	4	2	no	—
127	C₂²×D₄₇	#127	1	1	yes	characteristic
128	Q₈×C₄₇	—	4	4	no	—
129	Q₈×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	#127	1	1	yes	characteristic
137	C₂×Dic₄₇	#127	1	1	yes	characteristic
138	C₂×Dic₄₇	#127	1	1	yes	characteristic
139	D₁₈₈	—	4	4	no	—
140	D₁₈₈	—	4	4	no	—
141	D₁₈₈	—	4	4	no	—
142	D₁₈₈	—	8	4	no	—
143	C₄×D₄₇	—	2	2	no	—
144	C₄×D₄₇	—	2	2	no	—
145	C₄×D₄₇	—	8	4	no	—
146	C₄×D₄₇	—	8	4	no	—
147	C₄×D₄₇	#135	1	4	yes	—
148	Dic₉₄	—	4	4	no	—
149	Dic₉₄	—	8	4	no	—
150	C₈×C₄₇	—	2	2	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	#137	1	4	yes	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₄₇⋊C₈	—	2	2	no	—
155	C₄₇⋊C₈	—	2	2	no	—
156	#19638	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
158	C₂×D₄₇	—	2	1	no	—
159	C₂×D₄₇	—	2	1	no	—
160	C₂×D₄₇	—	8	2	no	—
161	C₂×D₄₇	—	8	2	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	#495	1	2	yes	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	C₂×#135	1	2	yes	—
167	C₄×C₄₇	C₂×#137	1	2	yes	—
168	C₄×C₄₇	#576	1	2	yes	—
169	Dic₄₇	—	2	1	no	—
170	Dic₄₇	—	2	1	no	—
171	Dic₄₇	—	8	2	no	—
172	Dic₄₇	#495	1	2	yes	—
173	#3206	—	47	2	no	—
174	#3169	—	47	1	no	—
175	C₂×#500	—	47	1	no	—
176	#3141	—	47	2	no	—
177	#1719	—	47	2	no	—
178	#1719	—	47	2	no	—
179	#1710	—	47	1	no	—
180	#1710	D₄₇	1	1	yes	characteristic, complemented
181	#1706	—	47	1	no	—
182	#1706	—	47	1	no	—
183	C₈×M₁₆	—	47	1	no	—
184	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
185	C₂×C₄₇	#3152	1	1	yes	characteristic
186	C₂×C₄₇	#3155	1	1	yes	characteristic
187	D₄₇	—	2	2	no	—
188	D₄₇	—	16	2	no	—
189	#532	—	47	2	no	—
190	C₂×C₄⋊Q₈	—	47	2	no	—
191	#500	—	47	2	no	—
192	#500	—	47	1	no	—
193	#500	—	94	2	no	—
194	#500	C₂×D₄₇	1	1	yes	characteristic
195	#500	C₂×D₄₇	1	1	yes	characteristic
196	#493	—	47	1	no	—
197	#492	—	47	2	no	—
198	#492	—	47	2	no	—
199	#489	—	94	4	no	—
200	#487	—	47	2	no	—
201	#487	—	47	2	no	—
202	#487	C₂×D₄₇	1	2	yes	—
203	#487	C₂×D₄₇	1	2	yes	—
204	#486	—	47	2	no	—
205	#486	—	47	2	no	—
206	#486	—	47	2	no	—
207	#466	—	94	2	no	—
208	#461	—	94	4	no	—
209	#448	—	94	4	no	—
210	C₄×D₈	—	94	2	no	—
211	#431	—	47	1	no	—
212	C₂×#107	—	94	2	no	—
213	#417	—	94	4	no	—
214	C₄×M₁₆	—	47	1	no	—
215	C₂×C₄×C₈	—	47	1	no	—
216	#322	—	47	1	no	—
217	#322	—	47	1	no	—
218	C₈²	—	47	1	no	—
219	C₈²	C₂×D₄₇	1	1	yes	characteristic
220	C₄₇	#19638	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	94	2	no	—
222	C₂²×Q₈	—	94	2	no	—
223	C₂×SD₁₆	—	94	4	no	—
224	C₂×SD₁₆	—	94	4	no	—
225	C₂×D₈	—	94	2	no	—
226	C₂×M₁₆	—	47	2	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₄⋊Q₈	—	47	2	no	—
229	C₄⋊Q₈	—	94	2	no	—
230	C₄⋊Q₈	—	94	2	no	—
231	C₄⋊Q₈	—	94	2	no	—
232	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
233	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
234	#127	—	47	2	no	—
235	#124	—	94	2	no	—
236	#121	—	94	2	no	—
237	#121	—	94	2	no	—
238	C₄×D₄	—	94	2	no	—
239	C₄×D₄	—	94	2	no	—
240	C₂×C₄⋊C₄	—	94	2	no	—
241	C₂×C₄⋊C₄	—	94	2	no	—
242	C₂×C₄²	—	47	1	no	—
243	#107	—	2	2	no	—
244	#107	—	2	2	no	—
245	#107	—	94	4	no	—
246	#107	—	94	2	no	—
247	C₄⋊C₈	—	47	2	no	—
248	#103	—	2	4	no	—
249	#103	—	2	4	no	—
250	#103	—	94	4	no	—
251	#103	—	94	4	no	—
252	#102	—	94	4	no	—
253	#102	—	94	4	no	—
254	#102	—	94	4	no	—
255	C₂²⋊C₈	—	47	2	no	—
256	#97	—	47	2	no	—
257	C₄×C₈	—	47	2	no	—
258	C₄×C₈	—	47	1	no	—
259	C₄×C₈	—	47	1	no	—
260	C₄×C₈	—	47	1	no	—
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
264	#41	—	188	8	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	94	2	no	—
268	C₂×Q₈	—	376	4	no	—
269	C₂×Q₈	—	188	2	no	—
270	C₂×D₄	—	94	2	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂×D₄	—	94	2	no	—
273	C₂×D₄	—	188	2	no	—
274	C₂²×C₄	—	47	1	no	—
275	C₂²×C₄	—	47	1	no	—
276	C₂²×C₄	—	47	1	no	—
277	C₂²×C₄	—	188	2	no	—
278	C₂²×C₄	—	188	2	no	—
279	SD₁₆	—	376	4	no	—
280	SD₁₆	—	376	4	no	—
281	D₈	—	188	4	no	—
282	M₁₆	—	94	4	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	94	4	no	—
287	C₂×C₈	D₄×D₄₇	1	2	yes	—
288	C₂×C₈	D₄×D₄₇	1	2	yes	—
289	C₂×C₈	D₄×D₄₇	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	188	2	no	—
297	C₄⋊C₄	—	188	2	no	—
298	C₂²⋊C₄	—	188	2	no	—
299	C₂²⋊C₄	—	188	2	no	—
300	C₄²	—	47	1	no	—
301	C₄²	—	47	1	no	—
302	C₄²	—	47	1	no	—
303	C₄²	C₂³×D₄₇	1	1	yes	characteristic
304	C₂³	—	47	1	no	—
305	C₂³	—	188	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	4	4	no	—
308	Q₈	—	376	4	no	—
309	Q₈	—	188	4	no	—
310	D₄	—	376	4	no	—
311	D₄	—	188	4	no	—
312	D₄	—	188	4	no	—
313	D₄	—	188	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	47	1	no	—
317	C₂×C₄	—	47	4	no	—
318	C₂×C₄	—	47	1	no	—
319	C₂×C₄	—	47	1	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	376	4	no	—
323	C₂×C₄	—	376	4	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	94	2	no	—
331	C₈	—	94	2	no	—
332	C₈	—	94	2	no	—
333	C₈	#13094056	1	4	yes	—
334	C₂²	—	47	2	no	—
335	C₂²	—	94	1	no	—
336	C₂²	—	94	1	no	—
337	C₂²	—	376	2	no	—
338	C₂²	—	376	2	no	—
339	C₂²	—	376	2	no	—
340	C₂²	#127×D₄₇	1	1	yes	characteristic
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	47	2	no	—
344	C₄	—	94	1	no	—
345	C₄	—	94	1	no	—
346	C₄	—	376	2	no	—
347	C₄	#135×D₄₇	1	2	yes	—
348	C₄	C₂×#13094056	1	2	yes	—
349	C₄	#15689925	1	2	yes	—
350	C₂	—	94	2	no	—
351	C₂	—	752	2	no	—
352	C₂	#495×D₄₇	1	1	yes	characteristic
353	C₂	#20253846	1	1	yes	characteristic
354	C₂	#20253870	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1710	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253968	C₂
hypercenter	#1710	D₄₇
hypocenter	C₄₇	#19638
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

Complements

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

N	H ≅ G/N	type
1. #20254314	C₂	semidirect
3. #500×D₄₇	C₂	semidirect
4. #20253783	C₂	semidirect
5. #20243043	C₂	semidirect
6. #20243042	C₂	semidirect
7. #20242997	C₂	semidirect
8. #20242981	C₂	semidirect
9. #20242980	C₂	semidirect
10. C₈×#10625324	C₂	semidirect
11. C₄₇×#1710	C₂	semidirect
180. #1710	D₄₇	semidirect
220. C₄₇	#19638	semidirect

Maximal subgroups

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

subgroup	index
#20254314	2
#20253968	2
#500×D₄₇	2
#20253783	2
#20243043	2
#20243042	2
#20242997	2
#20242981	2
#20242980	2
C₈×#10625324	2
C₄₇×#1710	2
#19638	47