Subgroups — #26371003

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253924	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252262	C₂	1	1	yes	maximal, characteristic
3	#20245555	C₂	1	2	yes	maximal, complemented
4	#20245544	C₂	1	2	yes	maximal, complemented
5	#20245534	C₂	1	2	yes	maximal, complemented
6	#20245529	C₂	1	2	yes	maximal, complemented
7	#20245174	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245146	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245098	C₂	1	1	yes	maximal, characteristic
10	#20245096	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689754	—	2	1	no	—
14	#15689604	—	2	1	no	—
15	C₂×#13093964	C₂²	1	1	yes	characteristic
16	#15689531	—	2	2	no	—
17	#15689527	C₂²	1	2	yes	—
18	#15689524	C₂²	1	2	yes	—
19	#15689222	—	2	2	no	—
20	#15689217	C₂²	1	2	yes	—
21	#15689205	C₂²	1	2	yes	—
22	#15689163	—	2	1	no	—
23	#15689161	C₂²	1	1	yes	characteristic
24	#15689122	—	2	1	no	—
25	#15689078	C₂²	1	1	yes	characteristic
26	#15688950	C₂²	1	2	yes	—
27	#15688948	C₂²	1	2	yes	—
28	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
29	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
30	#15688928	C₂²	1	1	yes	characteristic
31	#15688928	C₂²	1	1	yes	characteristic
32	#15688926	C₂²	1	2	yes	—
33	#15688925	C₂²	1	1	yes	characteristic
34	#15688925	C₂²	1	1	yes	characteristic
35	#15688923	C₂²	1	2	yes	—
36	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#367	C₂²	1	2	yes	—
40	C₄₇×#367	C₂²	1	2	yes	—
41	C₂×#10625350	—	2	1	no	—
42	#13094046	—	2	2	no	—
43	#13094046	Q₈	1	4	yes	complemented
44	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
45	C₂×#10625325	—	2	1	no	—
46	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
47	C₂×#10625329	D₄	1	1	yes	characteristic
48	#13094023	—	2	2	no	—
49	#13094023	—	4	1	no	—
50	#13094015	—	2	2	no	—
51	#13094015	—	4	1	no	—
52	#13094003	—	4	2	no	—
53	#13093989	—	2	2	no	—
54	#13093986	—	2	2	no	—
55	C₄×C₂²⋊D₄₇	—	4	2	no	—
56	#13093983	—	2	1	no	—
57	#13093983	D₄	1	1	yes	characteristic
58	#13093982	—	4	1	no	—
59	C₄⋊C₄×D₄₇	—	4	1	no	—
60	#13093964	—	2	2	no	—
61	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
62	#13093958	C₂³	1	1	yes	characteristic
63	#13093958	C₂³	1	1	yes	characteristic
64	#13093957	—	2	2	no	—
65	#13093956	—	2	1	no	—
66	#13093955	—	2	2	no	—
67	#13093947	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	#13093934	D₄	1	2	yes	—
70	C₈×Dic₄₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₂×C₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
77	#107×C₄₇	—	2	1	no	—
78	#107×C₄₇	—	2	1	no	—
79	#107×C₄₇	C₂³	1	2	yes	—
80	#107×C₄₇	C₂³	1	2	yes	—
81	C₂²×C₄×C₄₇	—	4	1	no	—
82	C₂²×C₄×C₄₇	—	4	1	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₇	—	4	1	no	—
85	#10625350	—	2	4	no	—
86	#10625350	—	4	2	no	—
87	C₂×D₁₈₈	—	2	1	no	—
88	C₂×C₄×D₄₇	—	4	1	no	—
89	C₂×C₄×D₄₇	—	4	2	no	—
90	C₂×Dic₉₄	—	2	1	no	—
91	C₂×Dic₉₄	—	4	2	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₄₇	#41	1	2	yes	—
95	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	4	1	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₂²⋊Dic₄₇	—	4	1	no	—
101	C₂²⋊Dic₄₇	D₈	1	2	yes	—
102	(C₂×D₄₇)⋊C₄	—	4	2	no	—
103	C₄⋊Dic₄₇	—	2	1	no	—
104	C₄⋊Dic₄₇	—	2	1	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
107	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
108	Dic₄₇⋊C₄	—	4	2	no	—
109	Dic₄₇⋊C₄	—	4	2	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	C₄×Dic₄₇	—	4	1	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
114	#10625329	—	2	2	no	—
115	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
116	Dic₁₈₈	—	2	1	no	—
117	Dic₁₈₈	C₂×Q₈	1	2	yes	—
118	D₃₇₆	—	2	1	no	—
119	D₃₇₆	C₂×Q₈	1	2	yes	—
120	#10625325	—	2	4	no	—
121	C₁₆×C₄₇	—	2	1	no	—
122	C₁₆×C₄₇	—	2	1	no	—
123	C₄₇⋊C₁₆	—	2	2	no	—
124	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	16	1	no	—
129	C₂×C₄×C₄₇	—	16	1	no	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	1	no	—
136	C₂×Dic₄₇	—	4	1	no	—
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
139	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
140	D₁₈₈	—	2	2	no	—
141	D₁₈₈	—	4	1	no	—
142	C₄×D₄₇	—	4	4	no	—
143	C₄×D₄₇	—	16	2	no	—
144	Dic₉₄	—	2	2	no	—
145	Dic₉₄	—	4	1	no	—
146	Dic₉₄	—	16	2	no	—
147	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
148	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₄₇⋊C₈	—	2	2	no	—
151	#33634	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	#474	1	2	yes	—
155	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₇	—	4	2	no	—
157	C₂×D₄₇	—	8	1	no	—
158	C₂×D₄₇	—	8	1	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	#474	1	2	yes	—
162	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
164	Dic₄₇	—	2	2	no	—
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	—	4	2	no	—
167	Dic₄₇	—	8	1	no	—
168	Dic₄₇	—	8	1	no	—
169	#3164	—	47	1	no	—
170	#2951	—	47	1	no	—
171	#2229	—	47	2	no	—
172	#2228	—	47	2	no	—
173	#2226	—	47	2	no	—
174	#2225	—	47	2	no	—
175	#2159	—	47	1	no	—
176	C₂×#367	D₄₇	1	1	yes	characteristic, complemented
177	#2151	—	47	1	no	—
178	#2139	—	47	1	no	—
179	#2138	—	47	1	no	—
180	C₂×C₄₇	—	2	2	no	—
181	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
182	C₂×C₄₇	#3416	1	1	yes	characteristic
183	C₂×C₄₇	#3416	1	1	yes	characteristic
184	D₄₇	—	8	2	no	—
185	C₂×#135	—	47	1	no	—
186	#544	—	94	1	no	—
187	#520	—	94	1	no	—
188	C₂×M₃₂	—	47	1	no	—
189	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
190	#501	—	94	2	no	—
191	#500	—	47	2	no	—
192	#499	—	47	2	no	—
193	#440	—	94	2	no	—
194	C₄×Q₁₆	—	47	2	no	—
195	C₄×D₈	—	47	2	no	—
196	#428	—	94	1	no	—
197	#427	—	47	1	no	—
198	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
200	#417	—	94	1	no	—
201	#405	—	47	1	no	—
202	#367	—	47	2	no	—
203	#367	C₂×D₄₇	1	2	yes	—
204	#367	C₂×D₄₇	1	2	yes	—
205	#366	—	47	2	no	—
206	C₄⋊C₁₆	—	47	2	no	—
207	C₄⋊C₁₆	—	47	2	no	—
208	#358	—	47	1	no	—
209	#358	—	47	1	no	—
210	#358	—	47	2	no	—
211	#357	—	47	1	no	—
212	#357	—	47	1	no	—
213	#357	—	47	2	no	—
214	C₄₇	#33634	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	94	1	no	—
216	#135	—	47	4	no	—
217	#135	—	94	2	no	—
218	C₂×Q₁₆	—	47	1	no	—
219	C₂×SD₁₆	—	94	1	no	—
220	C₂×D₈	—	47	1	no	—
221	C₂×M₁₆	—	47	1	no	—
222	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊Q₈	—	94	2	no	—
224	C₄⋊Q₈	—	188	1	no	—
225	#125	—	94	2	no	—
226	#125	—	188	1	no	—
227	C₂²⋊Q₈	—	188	2	no	—
228	C₄×Q₈	—	94	2	no	—
229	C₄×D₄	—	94	2	no	—
230	C₄×D₄	—	188	2	no	—
231	#117	—	47	1	no	—
232	#117	—	94	1	no	—
233	#117	—	188	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	188	1	no	—
237	M₃₂	—	94	2	no	—
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	—	47	2	no	—
240	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	#107	—	47	1	no	—
245	#107	—	47	1	no	—
246	#107	—	94	2	no	—
247	#107	C₂²×D₄₇	1	2	yes	—
248	#107	C₂²×D₄₇	1	2	yes	—
249	#106	—	94	2	no	—
250	#106	—	94	1	no	—
251	#103	—	94	2	no	—
252	#102	—	94	2	no	—
253	#97	—	47	2	no	—
254	C₄×C₈	—	47	2	no	—
255	#41	—	94	4	no	—
256	#41	—	188	2	no	—
257	C₂×Q₈	—	94	1	no	—
258	C₂×Q₈	—	188	2	no	—
259	C₂×D₄	—	94	1	no	—
260	C₂×D₄	—	188	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	188	2	no	—
264	C₂²×C₄	—	188	1	no	—
265	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
266	Q₁₆	—	47	2	no	—
267	Q₁₆	—	94	1	no	—
268	SD₁₆	—	94	4	no	—
269	D₈	—	47	2	no	—
270	D₈	—	94	1	no	—
271	M₁₆	—	94	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	#10625350	1	2	yes	—
274	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
275	C₂×C₈	Q₈×D₄₇	1	2	yes	—
276	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	47	1	no	—
282	C₄⋊C₄	—	47	1	no	—
283	C₄⋊C₄	—	94	1	no	—
284	C₄⋊C₄	—	94	1	no	—
285	C₄⋊C₄	—	188	2	no	—
286	C₄⋊C₄	—	188	2	no	—
287	C₄⋊C₄	—	188	2	no	—
288	C₄⋊C₄	—	188	2	no	—
289	C₂²⋊C₄	—	47	2	no	—
290	C₂²⋊C₄	—	188	2	no	—
291	C₂²⋊C₄	—	188	1	no	—
292	C₄²	—	47	2	no	—
293	C₄²	—	188	2	no	—
294	C₄²	—	188	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	94	2	no	—
298	C₂³	—	188	1	no	—
299	C₂³	D₈×D₄₇	1	1	yes	characteristic
300	Q₈	—	94	2	no	—
301	Q₈	—	188	1	no	—
302	Q₈	—	752	2	no	—
303	D₄	—	94	2	no	—
304	D₄	—	188	4	no	—
305	D₄	—	188	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	16	1	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	47	2	no	—
312	C₂×C₄	—	188	1	no	—
313	C₂×C₄	—	188	1	no	—
314	C₂×C₄	—	188	1	no	—
315	C₂×C₄	—	188	4	no	—
316	C₂×C₄	—	752	2	no	—
317	C₂×C₄	#13094002	1	2	yes	—
318	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
319	C₂×C₄	#13094056	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
321	C₈	—	94	2	no	—
322	C₈	#13094002	1	2	yes	—
323	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
324	C₈	C₂×#10625350	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	376	1	no	—
328	C₂²	—	376	1	no	—
329	C₂²	—	188	2	no	—
330	C₂²	#15689402	1	2	yes	—
331	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	—	94	2	no	—
335	C₄	—	94	2	no	—
336	C₄	—	376	1	no	—
337	C₄	—	376	1	no	—
338	C₄	—	188	2	no	—
339	C₄	#15689402	1	2	yes	—
340	C₄	C₂×#13094002	1	1	yes	characteristic
341	C₄	C₂×#13094056	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	376	2	no	—
344	C₂	C₂×#15689402	1	1	yes	characteristic
345	C₂	#20256530	1	1	yes	characteristic
346	C₂	#20256530	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#367	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#13094046	C₂²
hypercenter	C₂×#367	D₄₇
hypocenter	C₄₇	#33634
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
1. #20253924	C₂	semidirect
3. #20245555	C₂	semidirect
4. #20245544	C₂	semidirect
5. #20245534	C₂	semidirect
6. #20245529	C₂	semidirect
7. #20245174	C₂	semidirect
8. #20245146	C₂	semidirect
11. C₂×C₄₇×#367	C₂	semidirect
43. #13094046	Q₈	semidirect
176. C₂×#367	D₄₇	semidirect
214. C₄₇	#33634	semidirect

Maximal subgroups

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

subgroup	index
#20253924	2
#20252262	2
#20245555	2
#20245544	2
#20245534	2
#20245529	2
#20245174	2
#20245146	2
#20245098	2
#20245096	2
C₂×C₄₇×#367	2
#33634	47