Subgroups — #26371708

Order 12032.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689561	C₂	1	4	yes	maximal, complemented
2	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689500	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245654	C₂	1	4	yes	maximal, complemented
5	#20245644	C₂	1	2	yes	maximal, complemented
6	#20245638	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	—	2	1	no	—
9	C₂×#13094046	C₂²	1	2	yes	—
10	C₂²×D₃₇₆	—	2	1	no	—
11	C₂²×D₃₇₆	C₂²	1	2	yes	—
12	#15689673	—	2	1	no	—
13	C₂×#13094021	—	2	1	no	—
14	#15689561	—	2	16	no	—
15	C₂×#13093969	C₂²	1	8	yes	complemented
16	C₂×D₇₅₂	C₂²	1	8	yes	complemented
17	#15689507	—	2	1	no	—
18	#15689507	C₂²	1	2	yes	complemented
19	#15689503	C₂²	1	1	yes	characteristic, complemented
20	#15689500	—	2	1	no	—
21	#15689500	—	2	1	no	—
22	#15689500	C₂²	1	2	yes	—
23	#15689500	C₂²	1	2	yes	—
24	#15689500	C₂²	1	1	yes	characteristic, complemented
25	#15689496	—	2	1	no	—
26	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
30	C₂×#10625350	—	2	2	no	—
31	C₂²×D₁₈₈	—	2	2	no	—
32	C₂²×D₁₈₈	—	4	2	no	—
33	#13094046	—	2	8	no	—
34	#13094046	—	4	4	no	—
35	C₂×Dic₁₈₈	—	2	1	no	—
36	C₂×Dic₁₈₈	C₂³	1	2	yes	—
37	C₂×#10625325	—	2	2	no	—
38	C₂×#10625325	—	2	2	no	—
39	C₂×D₃₇₆	—	2	8	no	—
40	C₂×D₃₇₆	—	2	2	no	—
41	C₂×D₃₇₆	—	2	2	no	—
42	C₂×D₃₇₆	—	2	2	no	—
43	C₂×D₃₇₆	—	2	1	no	—
44	C₂×D₃₇₆	—	4	4	no	—
45	C₂×D₃₇₆	C₂³	1	4	yes	—
46	C₂×D₃₇₆	C₂³	1	2	yes	—
47	#13094024	—	2	1	no	—
48	#13094021	—	2	1	no	—
49	#13094021	—	2	2	no	—
50	#13094021	—	2	2	no	—
51	#13094021	—	4	2	no	—
52	#13094012	—	2	2	no	—
53	#13093998	—	4	2	no	—
54	C₄×D₁₈₈	—	4	2	no	—
55	#13093969	—	2	16	no	—
56	D₇₅₂	—	2	16	no	—
57	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	M₃₂×C₄₇	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
62	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂³×D₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	D₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	2	no	—
70	#10625350	—	4	4	no	—
71	#10625350	—	4	4	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	2	4	no	—
74	C₂×D₁₈₈	—	4	4	no	—
75	C₂×D₁₈₈	—	4	4	no	—
76	C₂×D₁₈₈	—	4	2	no	—
77	C₂×D₁₈₈	—	4	4	no	—
78	C₂×D₁₈₈	—	4	4	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	8	8	no	—
81	C₂×C₄×D₄₇	—	4	2	no	—
82	C₂×Dic₉₄	—	2	2	no	—
83	C₂×C₈×C₄₇	D₈	1	4	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄²×C₄₇	D₈	1	2	yes	—
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	4	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	Dic₁₈₈	—	2	4	no	—
95	Dic₁₈₈	—	4	2	no	—
96	D₃₇₆	—	2	4	no	—
97	D₃₇₆	—	2	8	no	—
98	D₃₇₆	—	4	4	no	—
99	D₃₇₆	—	4	4	no	—
100	D₃₇₆	—	4	4	no	—
101	D₃₇₆	—	4	2	no	—
102	#10625325	—	4	4	no	—
103	#10625325	—	4	4	no	—
104	C₁₆×C₄₇	C₂×D₄	1	8	yes	complemented
105	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
106	C₂²×D₄₇	—	4	2	no	—
107	C₂²×D₄₇	—	4	4	no	—
108	C₂²×D₄₇	—	8	8	no	—
109	C₂²×D₄₇	—	8	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	#127	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₄₇	—	8	4	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	D₁₈₈	—	4	2	no	—
120	D₁₈₈	—	4	4	no	—
121	D₁₈₈	—	4	2	no	—
122	D₁₈₈	—	4	4	no	—
123	D₁₈₈	—	8	8	no	—
124	D₁₈₈	—	8	4	no	—
125	D₁₈₈	—	8	4	no	—
126	D₁₈₈	—	8	8	no	—
127	C₄×D₄₇	—	8	4	no	—
128	Dic₉₄	—	4	2	no	—
129	Dic₉₄	—	4	2	no	—
130	C₈×C₄₇	#127	1	2	yes	—
131	C₈×C₄₇	C₂×D₈	1	2	yes	—
132	C₈×C₄₇	C₂×D₈	1	2	yes	—
133	C₈×C₄₇	C₂²×D₄	1	2	yes	—
134	#33717	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	#493	1	2	yes	—
137	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₄₇	—	8	2	no	—
139	C₂×D₄₇	—	8	4	no	—
140	C₂×D₄₇	—	8	8	no	—
141	C₂×D₄₇	—	8	4	no	—
142	C₂×D₄₇	—	16	8	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	#493	1	2	yes	—
145	C₄×C₄₇	#509	1	4	yes	complemented
146	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
147	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	Dic₄₇	—	8	2	no	—
149	C₂×#509	—	47	4	no	—
150	#3147	—	47	1	no	—
151	C₂×#493	—	47	1	no	—
152	#2250	—	47	4	no	—
153	#2248	—	47	2	no	—
154	#2246	—	47	2	no	—
155	C₄×M₃₂	D₄₇	1	1	yes	characteristic, complemented
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#509	1	2	yes	—
159	D₄₇	—	16	2	no	—
160	D₄₇	—	16	4	no	—
161	C₂×#135	—	47	2	no	—
162	C₂×#135	—	94	1	no	—
163	C₂²×D₈	—	47	2	no	—
164	C₂²×D₈	—	94	1	no	—
165	#532	—	94	1	no	—
166	C₂×#127	—	94	1	no	—
167	#509	—	94	16	no	—
168	C₂×SD₃₂	—	47	8	no	—
169	C₂×D₁₆	—	47	8	no	—
170	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
171	#495	—	47	2	no	—
172	#495	—	94	1	no	—
173	#494	—	47	1	no	—
174	#493	—	47	2	no	—
175	#493	—	47	2	no	—
176	#493	—	47	1	no	—
177	#493	—	94	1	no	—
178	#493	—	94	1	no	—
179	#492	—	94	1	no	—
180	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
181	#346	C₂×D₄₇	1	2	yes	complemented
182	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
183	C₄₇	#33717	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	94	2	no	—
185	C₂²×D₄	—	94	2	no	—
186	C₂²×D₄	—	188	2	no	—
187	#135	—	94	8	no	—
188	#135	—	188	4	no	—
189	C₂×Q₁₆	—	47	2	no	—
190	C₂×Q₁₆	—	94	1	no	—
191	C₂×SD₁₆	—	94	2	no	—
192	C₂×SD₁₆	—	94	2	no	—
193	C₂×D₈	—	47	4	no	—
194	C₂×D₈	—	47	2	no	—
195	C₂×D₈	—	94	8	no	—
196	C₂×D₈	—	94	2	no	—
197	C₂×D₈	—	94	2	no	—
198	C₂×D₈	—	94	2	no	—
199	C₂×D₈	—	94	1	no	—
200	C₂×D₈	—	188	4	no	—
201	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	94	1	no	—
204	#127	—	94	2	no	—
205	#127	—	94	2	no	—
206	#127	—	94	1	no	—
207	#127	—	188	2	no	—
208	#124	—	94	2	no	—
209	#121	—	188	2	no	—
210	C₄×D₄	—	188	2	no	—
211	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
212	SD₃₂	—	94	16	no	—
213	D₁₆	—	94	16	no	—
214	M₃₂	D₁₈₈	1	8	yes	complemented
215	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
216	C₄×C₈	D₁₈₈	1	1	yes	characteristic
217	C₄×C₈	D₁₈₈	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	C₂⁴	—	188	2	no	—
221	#41	—	188	4	no	—
222	#41	—	188	4	no	—
223	C₂×Q₈	—	94	2	no	—
224	C₂×D₄	—	94	2	no	—
225	C₂×D₄	—	94	4	no	—
226	C₂×D₄	—	376	8	no	—
227	C₂×D₄	—	188	2	no	—
228	C₂×D₄	—	188	4	no	—
229	C₂×D₄	—	188	2	no	—
230	C₂×D₄	—	188	4	no	—
231	C₂×D₄	—	188	4	no	—
232	C₂×D₄	—	188	2	no	—
233	C₂×D₄	—	188	4	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	D₃₇₆	1	2	yes	—
236	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
237	Q₁₆	—	94	4	no	—
238	Q₁₆	—	188	2	no	—
239	SD₁₆	—	188	4	no	—
240	SD₁₆	—	188	4	no	—
241	D₈	—	94	4	no	—
242	D₈	—	94	8	no	—
243	D₈	—	188	4	no	—
244	D₈	—	188	4	no	—
245	D₈	—	188	4	no	—
246	D₈	—	188	2	no	—
247	C₂×C₈	D₃₇₆	1	4	yes	—
248	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
253	C₄⋊C₄	—	188	2	no	—
254	C₂²⋊C₄	—	188	4	no	—
255	C₄²	D₃₇₆	1	2	yes	—
256	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
257	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
258	C₁₆	C₂×D₁₈₈	1	8	yes	complemented
259	C₂³	—	376	8	no	—
260	C₂³	—	376	4	no	—
261	C₂³	—	188	2	no	—
262	C₂³	—	188	4	no	—
263	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
264	Q₈	—	188	2	no	—
265	Q₈	—	188	2	no	—
266	D₄	—	376	8	no	—
267	D₄	—	376	4	no	—
268	D₄	—	376	4	no	—
269	D₄	—	376	8	no	—
270	D₄	—	376	4	no	—
271	D₄	—	188	4	no	—
272	D₄	—	188	2	no	—
273	D₄	—	188	4	no	—
274	D₄	—	188	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	376	4	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	#13094021	1	2	yes	—
279	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
280	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
281	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
282	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
283	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
284	C₈	#13094021	1	2	yes	—
285	C₈	C₂×D₃₇₆	1	2	yes	—
286	C₈	C₂×D₃₇₆	1	2	yes	—
287	C₈	C₂²×D₁₈₈	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	376	8	no	—
290	C₂²	—	376	4	no	—
291	C₂²	—	376	2	no	—
292	C₂²	—	376	4	no	—
293	C₂²	—	752	8	no	—
294	C₂²	#15689500	1	2	yes	—
295	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	376	2	no	—
298	C₄	#15689500	1	2	yes	—
299	C₄	#15689561	1	4	yes	complemented
300	C₄	C₂×#13094021	1	1	yes	characteristic
301	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	752	4	no	—
304	C₂	—	752	2	no	—
305	C₂	C₂×#15689500	1	1	yes	characteristic
306	C₂	C₂×#15689561	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#33717
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#15689561	C₂	semidirect
2. #20253816	C₂	semidirect
3. C₂×#15689500	C₂	semidirect
4. #20245654	C₂	semidirect
5. #20245644	C₂	semidirect
6. #20245638	C₂	semidirect
7. C₄×M₃₂×C₄₇	C₂	semidirect
15. C₂×#13093969	C₂²	semidirect
16. C₂×D₇₅₂	C₂²	semidirect
18. #15689507	C₂²	semidirect
19. #15689503	C₂²	semidirect
24. #15689500	C₂²	semidirect
28. C₄₇×#346	C₂²	semidirect
29. C₄×C₁₆×C₄₇	C₂²	semidirect
60. M₃₂×C₄₇	D₄	semidirect
104. C₁₆×C₄₇	C₂×D₄	semidirect
145. C₄×C₄₇	#509	semidirect
155. C₄×M₃₂	D₄₇	semidirect
181. #346	C₂×D₄₇	semidirect
182. C₄×C₁₆	C₂×D₄₇	semidirect
183. C₄₇	#33717	semidirect
214. M₃₂	D₁₈₈	semidirect
258. C₁₆	C₂×D₁₈₈	semidirect
299. C₄	#15689561	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689561	2
#20253816	2
C₂×#15689500	2
#20245654	2
#20245644	2
#20245638	2
C₄×M₃₂×C₄₇	2
#33717	47