Subgroups — #26368894

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689531	C₂	1	2	yes	maximal
2	C₂×#15689527	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689524	C₂	1	1	yes	maximal, characteristic
4	#20249510	C₂	1	2	yes	maximal
5	#20243555	C₂	1	2	yes	maximal
6	#20243554	C₂	1	2	yes	maximal
7	#20243552	C₂	1	1	yes	maximal, characteristic
8	#20243551	C₂	1	1	yes	maximal, characteristic
9	#20243546	C₂	1	1	yes	maximal, characteristic
10	#20243074	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094023	C₂²	1	2	yes	—
13	C₂×#13094015	C₂²	1	2	yes	—
14	#15689531	—	2	8	no	—
15	#15689527	—	2	4	no	—
16	#15689524	—	2	4	no	—
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093957	C₂²	1	2	yes	—
20	C₂×#13093957	C₂²	1	2	yes	—
21	C₂×#13093956	C₂²	1	2	yes	—
22	C₂×#13093955	C₂²	1	2	yes	—
23	C₂×#13093955	C₂²	1	2	yes	—
24	C₂×#13093935	C₂²	1	1	yes	characteristic
25	C₂×#13093934	C₂²	1	1	yes	characteristic
26	C₂×#13093934	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15689056	—	2	4	no	—
30	#15689043	—	2	4	no	—
31	#15688992	C₂²	1	2	yes	—
32	#15688989	C₂²	1	2	yes	—
33	#15688988	C₂²	1	2	yes	—
34	#15688969	C₂²	1	1	yes	characteristic
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#107×C₄₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#384	C₂²	1	2	yes	—
40	C₂²×Dic₉₄	—	2	2	no	—
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	#13094023	—	2	4	no	—
44	#13094023	—	2	4	no	—
45	#13094015	—	2	4	no	—
46	#13094015	—	2	4	no	—
47	C₂×C₄⋊Dic₄₇	—	2	2	no	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×Dic₄₇⋊C₄	—	2	2	no	—
52	C₂×Dic₄₇⋊C₄	—	2	4	no	—
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	#13093958	—	2	2	no	—
56	#13093958	—	2	2	no	—
57	#13093957	—	2	4	no	—
58	#13093957	—	2	4	no	—
59	#13093956	—	2	4	no	—
60	#13093955	—	2	4	no	—
61	#13093955	—	2	4	no	—
62	#13093935	Q₈	1	4	yes	complemented
63	#13093934	D₄	1	4	yes	—
64	#13093934	Q₈	1	4	yes	complemented
65	C₈×Dic₄₇	D₄	1	4	yes	—
66	C₈×Dic₄₇	Q₈	1	4	yes	complemented
67	C₄₇⋊#95	—	2	4	no	—
68	C₄₇⋊#95	—	2	4	no	—
69	C₄₇⋊#95	D₄	1	2	yes	—
70	C₄₇⋊#95	D₄	1	2	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	#107×C₄₇	—	2	2	no	—
78	#107×C₄₇	—	2	2	no	—
79	#107×C₄₇	—	2	4	no	—
80	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	2	no	—
87	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
90	C₂×Dic₉₄	—	4	4	no	—
91	C₂×Dic₉₄	—	8	4	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	8	4	no	—
103	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	4	4	no	—
110	Dic₄₇⋊C₄	—	4	4	no	—
111	Dic₄₇⋊C₄	—	4	4	no	—
112	Dic₄₇⋊C₄	—	8	4	no	—
113	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
114	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
115	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
116	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
121	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
123	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	#127	1	2	yes	—
146	C₂×Dic₄₇	#127	1	2	yes	—
147	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
148	Dic₉₄	—	8	8	no	—
149	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
150	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
151	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
152	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
153	#33319	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄₇	#576	1	2	yes	—
158	C₂²×C₄₇	#578	1	2	yes	—
159	C₄×C₄₇	—	2	4	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	—	8	2	no	—
162	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
164	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	2	2	no	—
168	Dic₄₇	—	2	4	no	—
169	Dic₄₇	—	8	2	no	—
170	Dic₄₇	—	8	2	no	—
171	C₂×#501	—	47	2	no	—
172	C₂×#500	—	47	1	no	—
173	C₂×#499	—	47	1	no	—
174	#2660	—	47	2	no	—
175	#1847	—	47	2	no	—
176	#1847	—	47	2	no	—
177	#1846	—	47	1	no	—
178	#1846	—	47	1	no	—
179	#1845	—	47	1	no	—
180	#1845	D₄₇	1	1	yes	characteristic
181	#1726	—	47	1	no	—
182	C₂×C₄₇	#2868	1	1	yes	characteristic
183	C₂×C₄₇	#3155	1	1	yes	characteristic
184	C₂×C₄₇	#3156	1	1	yes	characteristic
185	C₂×C₄₇	#3164	1	2	yes	—
186	C₂×C₄₇	#3164	1	2	yes	—
187	C₂×C₄⋊Q₈	—	47	2	no	—
188	C₂×#125	—	47	2	no	—
189	#501	—	94	8	no	—
190	#500	—	94	4	no	—
191	#499	—	94	4	no	—
192	C₂×#107	—	47	1	no	—
193	C₂×#107	—	47	1	no	—
194	C₂×#107	—	47	2	no	—
195	C₂×#107	—	47	2	no	—
196	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
197	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#107	C₂×D₄₇	1	2	yes	—
199	C₂×#106	—	47	2	no	—
200	C₂×#106	—	47	2	no	—
201	C₂×#106	—	47	2	no	—
202	C₂×#97	—	47	1	no	—
203	C₂×#97	—	47	1	no	—
204	C₂×#97	—	47	1	no	—
205	C₂×C₄×C₈	—	47	1	no	—
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
208	#398	—	94	4	no	—
209	#395	—	94	4	no	—
210	#384	—	47	2	no	—
211	#384	C₂×D₄₇	1	2	yes	—
212	#382	—	47	2	no	—
213	#382	—	47	2	no	—
214	#376	—	47	1	no	—
215	C₄₇	#33319	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	94	2	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	47	1	no	—
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	94	4	no	—
222	C₄⋊Q₈	—	94	4	no	—
223	#125	—	94	4	no	—
224	#125	—	94	4	no	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	94	4	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₄⋊C₄	—	94	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
234	C₂×C₄²	—	47	1	no	—
235	C₂×C₄²	—	47	1	no	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	#107	—	2	2	no	—
238	#107	—	2	2	no	—
239	#107	—	2	4	no	—
240	#107	—	94	4	no	—
241	#107	—	94	4	no	—
242	#107	—	94	2	no	—
243	#107	—	94	2	no	—
244	#106	—	94	4	no	—
245	#106	—	94	4	no	—
246	#106	—	94	4	no	—
247	#97	—	47	4	no	—
248	#97	—	47	4	no	—
249	#97	—	47	4	no	—
250	C₄×C₈	—	47	4	no	—
251	C₄×C₈	—	47	4	no	—
252	C₄×C₈	Dic₉₄	1	4	yes	complemented
253	#95	—	47	2	no	—
254	#95	—	47	2	no	—
255	#95	—	94	4	no	—
256	#95	—	94	4	no	—
257	C₂×Q₈	—	376	4	no	—
258	C₂×Q₈	—	188	4	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	47	1	no	—
263	C₂²×C₄	—	47	1	no	—
264	C₂²×C₄	—	94	2	no	—
265	C₂²×C₄	—	94	2	no	—
266	C₂²×C₄	D₄×D₄₇	1	2	yes	—
267	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	47	2	no	—
271	C₂×C₈	—	47	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	—	47	2	no	—
274	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
275	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
276	C₂×C₈	D₄×D₄₇	1	2	yes	—
277	C₂×C₈	D₄×D₄₇	1	2	yes	—
278	C₂×C₈	Q₈×D₄₇	1	2	yes	—
279	C₂×C₈	Q₈×D₄₇	1	2	yes	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	8	4	no	—
285	C₄⋊C₄	—	94	2	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₄⋊C₄	—	94	2	no	—
288	C₄⋊C₄	—	94	2	no	—
289	C₄⋊C₄	—	376	4	no	—
290	C₄⋊C₄	—	188	4	no	—
291	C₄⋊C₄	—	188	4	no	—
292	C₄⋊C₄	—	188	4	no	—
293	C₄²	—	47	2	no	—
294	C₄²	—	47	2	no	—
295	C₄²	—	47	2	no	—
296	C₄²	—	47	2	no	—
297	C₄²	C₂×Dic₉₄	1	2	yes	—
298	C₄²	Q₈×D₄₇	1	2	yes	—
299	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
300	Q₈	—	376	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	47	2	no	—
308	C₂×C₄	—	47	2	no	—
309	C₂×C₄	—	47	4	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	4	no	—
317	C₂×C₄	—	188	4	no	—
318	C₂×C₄	#13094023	1	4	yes	—
319	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
325	C₈	—	47	4	no	—
326	C₈	—	47	4	no	—
327	C₈	#13094023	1	4	yes	—
328	C₈	#13094023	1	4	yes	—
329	C₂²	#127×D₄₇	1	1	yes	characteristic
330	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
331	C₂²	C₂×#13094023	1	1	yes	characteristic
332	C₂²	#15689924	1	2	yes	—
333	C₂²	#15689936	1	2	yes	—
334	C₄	—	2	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	94	2	no	—
338	C₄	—	94	2	no	—
339	C₄	—	94	4	no	—
340	C₄	—	376	2	no	—
341	C₄	—	376	2	no	—
342	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
343	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
344	C₄	C₂×#13094023	1	1	yes	characteristic
345	C₄	C₂×#13094023	1	1	yes	characteristic
346	C₂	#20251785	1	1	yes	characteristic
347	C₂	#20253868	1	1	yes	characteristic
348	C₂	#20253876	1	1	yes	characteristic
349	C₂	#20253921	1	2	yes	—
350	C₂	#20253924	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#33319
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
62. #13093935	Q₈	semidirect
64. #13093934	Q₈	semidirect
66. C₈×Dic₄₇	Q₈	semidirect
80. C₄×C₈×C₄₇	Q₈	semidirect
215. C₄₇	#33319	semidirect
252. C₄×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689531	2
C₂×#15689527	2
C₂×#15689524	2
#20249510	2
#20243555	2
#20243554	2
#20243552	2
#20243551	2
#20243546	2
#20243074	2
C₄₇×#1845	2
#33319	47