Subgroups — #26199539

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254733	C₂	1	2	yes	maximal
2	#20253902	C₂	1	1	yes	maximal, characteristic
3	#20252322	C₂	1	1	yes	maximal, characteristic
4	#20242917	C₂	1	2	yes	maximal, complemented
5	#20242907	C₂	1	2	yes	maximal, complemented
6	#20242771	C₂	1	4	yes	maximal, complemented
7	#20242076	C₂	1	2	yes	maximal
8	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689680	C₂²	1	1	yes	characteristic
11	#15689528	C₂²	1	1	yes	characteristic
12	#15689528	C₂²	1	1	yes	characteristic
13	#15689525	—	2	1	no	—
14	#15689525	—	2	1	no	—
15	#15689521	C₂²	1	2	yes	—
16	#15689419	—	2	2	no	—
17	#15689419	C₂²	1	4	yes	—
18	#15689364	—	2	4	no	—
19	#15689334	—	2	2	no	—
20	#15689334	C₂²	1	4	yes	—
21	#15689253	—	2	1	no	—
22	#15689253	C₂²	1	2	yes	—
23	C₈×Dic₉₄	—	2	1	no	—
24	C₈×Dic₉₄	C₂²	1	2	yes	—
25	#15689161	—	2	1	no	—
26	#15689161	—	2	1	no	—
27	#15689149	—	2	1	no	—
28	#15689147	C₂²	1	1	yes	characteristic
29	#15689078	—	2	1	no	—
30	#15688871	C₂²	1	4	yes	—
31	#15688870	C₂²	1	4	yes	—
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#334	C₂²	1	2	yes	—
35	C₄₇×#334	C₂²	1	2	yes	—
36	C₂×Dic₁₈₈	—	2	4	no	—
37	C₂×#10625329	—	2	1	no	—
38	#13094024	D₄	1	1	yes	characteristic
39	#13094024	D₄	1	1	yes	characteristic
40	#13094024	C₂³	1	2	yes	—
41	#13094024	C₂³	1	2	yes	—
42	#13094016	—	2	1	no	—
43	#13094016	—	2	2	no	—
44	#13094003	—	2	1	no	—
45	#13094003	—	2	2	no	—
46	#13094003	—	2	2	no	—
47	#13094003	D₄	1	2	yes	—
48	C₄×Dic₉₄	—	2	1	no	—
49	C₄×Dic₉₄	—	2	2	no	—
50	C₄×Dic₉₄	—	2	2	no	—
51	C₄×Dic₉₄	C₂³	1	2	yes	—
52	#13093983	—	2	1	no	—
53	#13093983	—	2	2	no	—
54	#13093958	—	2	1	no	—
55	#13093958	—	2	1	no	—
56	#13093958	—	2	1	no	—
57	#13093958	—	2	1	no	—
58	#13093958	—	2	4	no	—
59	#13093956	—	2	2	no	—
60	#13093956	—	2	2	no	—
61	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
62	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₇⋊C₈	—	2	2	no	—
64	Dic₄₇⋊C₈	—	2	2	no	—
65	#13093949	—	2	4	no	—
66	#13093949	—	2	4	no	—
67	#13093934	—	2	2	no	—
68	C₈×Dic₄₇	—	2	2	no	—
69	C₂²×C₈×C₄₇	—	2	2	no	—
70	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₇	—	2	2	no	—
74	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
82	C₂×Dic₉₄	—	2	1	no	—
83	C₂×Dic₉₄	—	2	2	no	—
84	C₂×Dic₉₄	—	2	2	no	—
85	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	8	2	no	—
89	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₄₇	—	2	2	no	—
99	C₂²⋊Dic₄₇	—	4	2	no	—
100	C₄⋊Dic₄₇	—	2	1	no	—
101	C₄⋊Dic₄₇	—	2	1	no	—
102	C₄⋊Dic₄₇	—	2	1	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
108	Dic₄₇⋊C₄	—	2	2	no	—
109	Dic₄₇⋊C₄	—	2	2	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	Dic₄₇⋊C₄	—	4	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	4	2	no	—
114	#10625329	—	4	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	Dic₁₈₈	—	8	4	no	—
117	C₂³×C₄₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	1	no	—
120	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₄₇	#137	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	Dic₉₄	—	2	4	no	—
133	Dic₉₄	—	4	2	no	—
134	Dic₉₄	—	4	2	no	—
135	Dic₉₄	—	4	2	no	—
136	Dic₉₄	—	8	2	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	—	2	1	no	—
139	C₈×C₄₇	—	4	4	no	—
140	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
141	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
142	C₄₇⋊C₈	—	4	2	no	—
143	#19210	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	#462	1	2	yes	—
147	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	#462	1	2	yes	—
152	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
154	C₄×C₄₇	#577	1	2	yes	—
155	Dic₄₇	—	4	2	no	—
156	Dic₄₇	—	4	2	no	—
157	Dic₄₇	—	8	2	no	—
158	Dic₄₇	—	8	2	no	—
159	#3249	—	47	2	no	—
160	#3160	—	47	1	no	—
161	#2960	—	47	1	no	—
162	#1696	—	47	2	no	—
163	#1694	—	47	2	no	—
164	#1672	—	47	4	no	—
165	C₂×#334	D₄₇	1	1	yes	characteristic
166	#1550	—	47	2	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
169	C₂×C₄₇	#3060	1	1	yes	characteristic
170	C₂×C₄₇	#3067	1	1	yes	characteristic
171	#533	—	47	1	no	—
172	#533	—	47	2	no	—
173	#500	—	47	1	no	—
174	#500	—	47	1	no	—
175	#499	—	94	1	no	—
176	#499	—	94	1	no	—
177	#498	—	47	2	no	—
178	#477	—	47	4	no	—
179	#477	—	94	2	no	—
180	#467	—	94	4	no	—
181	#462	—	47	4	no	—
182	#462	—	94	2	no	—
183	#446	—	47	2	no	—
184	#446	—	94	1	no	—
185	C₈×Q₈	—	47	2	no	—
186	C₈×Q₈	—	94	1	no	—
187	#427	—	94	1	no	—
188	#427	—	94	1	no	—
189	#424	—	94	1	no	—
190	#423	—	47	1	no	—
191	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
192	#405	—	94	1	no	—
193	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
194	#334	C₂×D₄₇	1	2	yes	—
195	#334	C₂×D₄₇	1	2	yes	—
196	#333	—	47	4	no	—
197	#332	—	47	4	no	—
198	C₄₇	#19210	1	1	yes	characteristic, complemented, Sylow
199	C₂×Q₁₆	—	94	4	no	—
200	C₂×M₁₆	—	94	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
203	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
204	C₄⋊Q₈	—	47	1	no	—
205	C₄⋊Q₈	—	47	1	no	—
206	C₄⋊Q₈	—	47	2	no	—
207	C₄⋊Q₈	—	47	2	no	—
208	#125	—	94	2	no	—
209	#125	—	94	1	no	—
210	C₂²⋊Q₈	—	47	2	no	—
211	C₂²⋊Q₈	—	94	1	no	—
212	C₂²⋊Q₈	—	94	2	no	—
213	C₂²⋊Q₈	—	94	2	no	—
214	C₄×Q₈	—	47	2	no	—
215	C₄×Q₈	—	94	1	no	—
216	C₄×Q₈	—	94	2	no	—
217	C₄×Q₈	—	94	2	no	—
218	#117	—	94	1	no	—
219	#117	—	94	2	no	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	#107	—	94	4	no	—
222	#107	—	94	1	no	—
223	#107	—	94	1	no	—
224	#107	—	94	1	no	—
225	#107	—	94	1	no	—
226	#106	—	94	2	no	—
227	#106	—	94	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	47	2	no	—
230	C₄⋊C₈	—	94	2	no	—
231	C₄⋊C₈	—	94	2	no	—
232	C₄⋊C₈	—	94	1	no	—
233	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
234	#103	—	94	4	no	—
235	#103	—	94	4	no	—
236	#97	—	94	2	no	—
237	C₄×C₈	—	94	2	no	—
238	C₄×C₈	D₁₈₈	1	1	yes	characteristic
239	C₄×C₈	D₁₈₈	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×Q₈	—	47	2	no	—
243	C₂×Q₈	—	94	1	no	—
244	C₂×Q₈	—	94	2	no	—
245	C₂×Q₈	—	94	2	no	—
246	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	#10625354	1	1	yes	characteristic
249	Q₁₆	—	376	4	no	—
250	M₁₆	—	188	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	8	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	Dic₁₈₈	1	4	yes	complemented
256	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
257	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
258	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
259	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
260	C₄⋊C₄	—	47	2	no	—
261	C₄⋊C₄	—	94	2	no	—
262	C₄⋊C₄	—	94	1	no	—
263	C₄⋊C₄	—	94	1	no	—
264	C₄⋊C₄	—	94	1	no	—
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₄⋊C₄	—	94	2	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₂²⋊C₄	—	94	2	no	—
273	C₂²⋊C₄	—	188	2	no	—
274	C₄²	—	94	2	no	—
275	C₄²	—	188	2	no	—
276	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
277	C₄²	D₄×D₄₇	1	1	yes	characteristic
278	C₄²	#10625354	1	1	yes	characteristic
279	C₄²	C₂³×D₄₇	1	1	yes	characteristic
280	C₂³	#13093999	1	1	yes	characteristic
281	Q₈	—	94	4	no	—
282	Q₈	—	376	2	no	—
283	Q₈	—	188	2	no	—
284	Q₈	—	188	2	no	—
285	Q₈	—	188	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	#13093999	1	1	yes	characteristic
293	C₂×C₄	#13093999	1	1	yes	characteristic
294	C₂×C₄	#13093999	1	1	yes	characteristic
295	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
296	C₂×C₄	#13094057	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	4	no	—
303	C₈	—	188	2	no	—
304	C₈	C₂×Dic₁₈₈	1	2	yes	—
305	C₈	C₂×Dic₁₈₈	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#15689334	1	2	yes	—
309	C₂²	C₂×#13093999	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	376	2	no	—
314	C₄	—	376	2	no	—
315	C₄	—	188	2	no	—
316	C₄	—	188	2	no	—
317	C₄	#15689334	1	2	yes	—
318	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
319	C₄	C₂×#13094057	1	1	yes	characteristic
320	C₄	#15689929	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	C₂×#15689334	1	1	yes	characteristic
323	C₂	#20253051	1	1	yes	characteristic
324	C₂	#20253109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂³×C₄₇	#121
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#19210
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #20242917	C₂	semidirect
5. #20242907	C₂	semidirect
6. #20242771	C₂	semidirect
89. C₂×C₈×C₄₇	Q₁₆	semidirect
198. C₄₇	#19210	semidirect
255. C₂×C₈	Dic₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
#20254733	2
#20253902	2
#20252322	2
#20242917	2
#20242907	2
#20242771	2
#20242076	2
C₂×C₄₇×#334	2
#19210	47