Subgroups — #26252530

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257198	C₂	1	1	yes	maximal, characteristic
2	#20253193	C₂	1	2	yes	maximal
3	#20244733	C₂	1	2	yes	maximal
4	#20244549	C₂	1	2	yes	maximal
5	#20244365	C₂	1	2	yes	maximal
6	#20243782	C₂	1	2	yes	maximal
7	#20243602	C₂	1	2	yes	maximal
8	#20243384	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	Q₈×Dic₉₄	—	2	2	no	—
11	#15689852	—	2	2	no	—
12	#15689849	—	2	4	no	—
13	#15689754	—	2	2	no	—
14	#15689754	C₂²	1	2	yes	—
15	#15689680	—	2	2	no	—
16	#15689680	C₂²	1	2	yes	—
17	#15689601	C₂²	1	2	yes	—
18	#15689430	—	2	4	no	—
19	#15689418	—	2	2	no	—
20	#15689408	—	2	2	no	—
21	#15689163	C₂²	1	4	yes	—
22	#15689147	C₂²	1	2	yes	—
23	#15689132	C₂²	1	4	yes	—
24	C₂×#13093943	C₂²	1	2	yes	—
25	#15688892	C₂²	1	2	yes	—
26	#15688883	C₂²	1	4	yes	—
27	#15688883	C₂²	1	4	yes	—
28	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#391	C₂²	1	1	yes	characteristic
32	C₄₇×#391	C₂²	1	1	yes	characteristic
33	C₄₇×#389	—	2	2	no	—
34	C₄₇×#384	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄³×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#10625329	C₂³	1	4	yes	—
38	#13094024	—	2	2	no	—
39	#13094024	—	2	2	no	—
40	#13094024	D₄	1	2	yes	—
41	#13094024	D₄	1	2	yes	—
42	#13094023	—	2	8	no	—
43	#13094023	—	2	4	no	—
44	#13094016	—	2	4	no	—
45	#13094016	D₄	1	4	yes	—
46	#13094015	—	2	8	no	—
47	#13094015	—	2	4	no	—
48	#13094003	—	2	2	no	—
49	#13094003	—	2	4	no	—
50	#13094003	—	2	2	no	—
51	Q₈×Dic₄₇	—	2	4	no	—
52	Q₈×Dic₄₇	—	2	4	no	—
53	C₄×Dic₉₄	—	2	4	no	—
54	C₄×Dic₉₄	—	2	4	no	—
55	C₄×Dic₉₄	—	2	2	no	—
56	C₄×Dic₉₄	—	2	2	no	—
57	#13093983	—	2	4	no	—
58	#13093983	C₂³	1	4	yes	—
59	#13093957	—	2	4	no	—
60	#13093955	—	2	4	no	—
61	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
63	#13093948	—	2	4	no	—
64	#13093948	—	2	4	no	—
65	#13093943	—	2	4	no	—
66	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₄₇	—	2	2	no	—
69	C₄⋊Q₈×C₄₇	Q₈	1	4	yes	complemented
70	C₄×Q₈×C₄₇	—	2	2	no	—
71	C₄×Q₈×C₄₇	—	2	2	no	—
72	C₄×Q₈×C₄₇	—	2	2	no	—
73	C₄×Q₈×C₄₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
78	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₄₇	—	2	1	no	—
81	C₂×C₄²×C₄₇	—	2	1	no	—
82	C₂×C₄²×C₄₇	—	2	1	no	—
83	C₂×C₄²×C₄₇	D₄	1	2	yes	—
84	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
87	#95×C₄₇	—	2	2	no	—
88	#95×C₄₇	—	2	2	no	—
89	C₂×Q₈×C₄₇	—	2	2	no	—
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×Q₈×C₄₇	—	2	2	no	—
92	C₂×Q₈×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	2	2	no	—
94	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂×Q₈×C₄₇	#41	1	2	yes	—
97	C₂²×C₄×C₄₇	—	2	1	no	—
98	C₂²×C₄×C₄₇	—	2	1	no	—
99	C₂²×C₄×C₄₇	—	2	1	no	—
100	C₂²×C₄×C₄₇	—	2	1	no	—
101	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₉₄	—	2	2	no	—
107	C₂×Dic₉₄	—	2	4	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	4	no	—
110	C₄⋊C₄×C₄₇	—	2	8	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	—	4	4	no	—
114	C₄⋊C₄×C₄₇	—	4	2	no	—
115	C₄²×C₄₇	—	2	4	no	—
116	C₄²×C₄₇	—	2	2	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	—	2	1	no	—
119	C₄²×C₄₇	—	2	2	no	—
120	C₄²×C₄₇	—	4	2	no	—
121	C₄²×C₄₇	—	4	1	no	—
122	C₄²×C₄₇	—	4	1	no	—
123	C₄²×C₄₇	C₂×D₄	1	2	yes	—
124	C₄²×C₄₇	C₂×D₄	1	2	yes	—
125	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
126	C₂²⋊Dic₄₇	—	2	4	no	—
127	C₄⋊Dic₄₇	—	2	4	no	—
128	C₄⋊Dic₄₇	—	2	4	no	—
129	C₄⋊Dic₄₇	—	2	4	no	—
130	C₄⋊Dic₄₇	—	2	2	no	—
131	C₄⋊Dic₄₇	—	2	2	no	—
132	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
133	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
134	Dic₄₇⋊C₄	—	2	8	no	—
135	Dic₄₇⋊C₄	—	2	4	no	—
136	Dic₄₇⋊C₄	—	2	4	no	—
137	C₄×Dic₄₇	—	2	8	no	—
138	C₄×Dic₄₇	—	2	4	no	—
139	#10625329	—	2	4	no	—
140	#10625329	—	2	4	no	—
141	C₂×C₄₇⋊C₈	—	2	4	no	—
142	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	Q₈×C₄₇	—	2	4	no	—
144	Q₈×C₄₇	—	2	4	no	—
145	Q₈×C₄₇	—	2	4	no	—
146	Q₈×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	—	2	2	no	—
148	C₂×C₄×C₄₇	—	2	2	no	—
149	C₂×C₄×C₄₇	—	2	2	no	—
150	C₂×C₄×C₄₇	—	2	2	no	—
151	C₂×C₄×C₄₇	—	2	2	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	2	no	—
154	C₂×C₄×C₄₇	—	2	2	no	—
155	C₂×C₄×C₄₇	—	2	2	no	—
156	C₂×C₄×C₄₇	—	4	1	no	—
157	C₂×C₄×C₄₇	—	4	1	no	—
158	C₂×C₄×C₄₇	—	4	1	no	—
159	C₂×C₄×C₄₇	—	4	1	no	—
160	C₂×C₄×C₄₇	—	4	1	no	—
161	C₂×C₄×C₄₇	—	4	2	no	—
162	C₂×C₄×C₄₇	—	4	2	no	—
163	C₂×C₄×C₄₇	—	4	1	no	—
164	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
165	C₂×C₄×C₄₇	#120	1	2	yes	—
166	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
167	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
168	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
169	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
172	C₂×Dic₄₇	—	2	4	no	—
173	C₂×Dic₄₇	—	2	4	no	—
174	Dic₉₄	—	4	4	no	—
175	Dic₉₄	—	8	2	no	—
176	Dic₉₄	—	8	4	no	—
177	C₄₇⋊C₈	—	4	4	no	—
178	#24315	—	47	1	no	maximal, Sylow
179	C₂²×C₄₇	—	2	1	no	—
180	C₂²×C₄₇	—	2	1	no	—
181	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
182	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₄₇	#536	1	1	yes	characteristic
184	C₄×C₄₇	—	2	4	no	—
185	C₄×C₄₇	—	2	4	no	—
186	C₄×C₄₇	—	2	1	no	—
187	C₄×C₄₇	—	2	2	no	—
188	C₄×C₄₇	—	4	2	no	—
189	C₄×C₄₇	—	4	2	no	—
190	C₄×C₄₇	—	4	2	no	—
191	C₄×C₄₇	—	8	1	no	—
192	C₄×C₄₇	#456	1	2	yes	—
193	C₄×C₄₇	#548	1	2	yes	—
194	C₄×C₄₇	D₄×Q₈	1	2	yes	—
195	Dic₄₇	—	4	4	no	—
196	Dic₄₇	—	4	4	no	—
197	#3502	—	47	1	no	—
198	#3078	—	47	2	no	—
199	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
200	#2060	—	47	2	no	—
201	#2028	—	47	2	no	—
202	#1994	—	47	2	no	—
203	#1886	—	47	2	no	—
204	#1857	—	47	2	no	—
205	#1801	—	47	1	no	—
206	C₂×C₄₇	—	2	1	no	—
207	C₂×C₄₇	—	2	1	no	—
208	C₂×C₄₇	#2409	1	1	yes	characteristic
209	C₂×C₄₇	C₂×#456	1	1	yes	characteristic
210	C₂×C₄₇	#3018	1	1	yes	characteristic
211	Q₈²	—	94	2	no	—
212	#557	—	94	4	no	—
213	#557	—	94	2	no	—
214	#544	—	47	2	no	—
215	#544	—	94	2	no	—
216	#533	—	47	2	no	—
217	#533	—	94	2	no	—
218	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
219	#519	—	47	2	no	—
220	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
221	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
222	#479	—	94	4	no	—
223	#477	—	94	2	no	—
224	#475	—	94	2	no	—
225	#428	—	47	4	no	—
226	#423	—	47	2	no	—
227	#419	—	47	4	no	—
228	C₂×#101	—	47	2	no	—
229	#391	C₂×D₄₇	1	1	yes	characteristic
230	#391	C₂×D₄₇	1	1	yes	characteristic
231	#389	—	2	2	no	—
232	#384	C₂×D₄₇	1	1	yes	characteristic
233	C₄×C₄⋊C₄	—	2	2	no	—
234	C₄³	C₂×D₄₇	1	1	yes	characteristic
235	#343	—	47	2	no	—
236	#339	—	47	4	no	—
237	#339	—	47	4	no	—
238	C₄₇	#24315	1	1	yes	characteristic, complemented, Sylow
239	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
240	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×M₁₆	—	47	4	no	—
242	C₄⋊Q₈	—	2	2	no	—
243	C₄⋊Q₈	—	47	2	no	—
244	C₄⋊Q₈	—	47	2	no	—
245	C₄⋊Q₈	—	94	2	no	—
246	C₄⋊Q₈	—	94	2	no	—
247	C₄⋊Q₈	—	94	4	no	—
248	C₄⋊Q₈	—	94	8	no	—
249	C₄⋊Q₈	Dic₉₄	1	4	yes	complemented
250	#125	—	47	4	no	—
251	#125	—	94	4	no	—
252	#125	—	94	4	no	—
253	#125	—	94	8	no	—
254	C₂²⋊Q₈	—	94	2	no	—
255	C₂²⋊Q₈	—	94	2	no	—
256	C₂²⋊Q₈	—	94	4	no	—
257	C₄×Q₈	—	2	2	no	—
258	C₄×Q₈	—	2	2	no	—
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	—	94	4	no	—
262	C₄×Q₈	—	94	4	no	—
263	C₄×Q₈	—	94	4	no	—
264	C₄×Q₈	—	94	4	no	—
265	C₄×Q₈	—	94	2	no	—
266	C₄×Q₈	—	94	2	no	—
267	#117	—	47	4	no	—
268	#117	—	94	4	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
279	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
280	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
281	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
282	#107	—	94	4	no	—
283	#106	—	94	4	no	—
284	C₄⋊C₈	—	94	2	no	—
285	C₄⋊C₈	—	94	2	no	—
286	#103	—	94	4	no	—
287	#103	—	94	4	no	—
288	#101	—	94	4	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	94	4	no	—
297	C₂×Q₈	—	94	2	no	—
298	C₂×Q₈	—	94	2	no	—
299	C₂×Q₈	C₂×Dic₉₄	1	2	yes	—
300	C₂×Q₈	C₂×Dic₉₄	1	2	yes	—
301	C₂×Q₈	#10625350	1	2	yes	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
308	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
309	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
311	M₁₆	—	94	4	no	—
312	M₁₆	—	94	4	no	—
313	C₂×C₈	—	94	4	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	8	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	4	4	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	47	4	no	—
321	C₄⋊C₄	—	47	4	no	—
322	C₄⋊C₄	—	94	8	no	—
323	C₄⋊C₄	—	94	4	no	—
324	C₄⋊C₄	—	94	4	no	—
325	C₄⋊C₄	—	94	4	no	—
326	C₄⋊C₄	—	94	4	no	—
327	C₄⋊C₄	—	94	4	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	—	94	2	no	—
330	C₂²⋊C₄	—	94	4	no	—
331	C₄²	—	2	4	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	2	no	—
337	C₄²	—	4	1	no	—
338	C₄²	—	4	1	no	—
339	C₄²	—	94	8	no	—
340	C₄²	—	94	4	no	—
341	C₄²	C₂×Dic₉₄	1	2	yes	—
342	C₄²	D₄×D₄₇	1	2	yes	—
343	C₄²	D₄×D₄₇	1	2	yes	—
344	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
345	Q₈	—	2	4	no	—
346	Q₈	—	2	4	no	—
347	Q₈	—	2	4	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	376	2	no	—
350	Q₈	—	376	4	no	—
351	Q₈	—	188	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	94	4	no	—
370	C₂×C₄	—	94	4	no	—
371	C₂×C₄	#13093992	1	1	yes	characteristic
372	C₂×C₄	#13093992	1	2	yes	—
373	C₂×C₄	#13093992	1	1	yes	characteristic
374	C₂×C₄	#13094003	1	4	yes	—
375	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
376	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
377	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
378	C₂×C₄	#13094071	1	2	yes	—
379	C₈	—	188	4	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	C₂×#13093992	1	1	yes	characteristic
383	C₂²	C₂×#13094003	1	1	yes	characteristic
384	C₂²	#15689694	1	1	yes	characteristic
385	C₄	—	2	4	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	4	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	8	1	no	—
393	C₄	—	188	4	no	—
394	C₄	—	188	4	no	—
395	C₄	#15689305	1	2	yes	—
396	C₄	#15689782	1	2	yes	—
397	C₄	D₄×Dic₉₄	1	2	yes	—
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	#20246753	1	1	yes	characteristic
401	C₂	C₂×#15689305	1	1	yes	characteristic
402	C₂	#20252736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093992
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#120
prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#24315
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
69. C₄⋊Q₈×C₄₇	Q₈	semidirect
238. C₄₇	#24315	semidirect
249. C₄⋊Q₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20257198	2
#20253193	2
#20244733	2
#20244549	2
#20244365	2
#20243782	2
#20243602	2
#20243384	2
C₄×C₄⋊Q₈×C₄₇	2
#24315	47