Subgroups — #26188038

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic
2	#20252134	C₂	1	2	yes	maximal, complemented
3	#20242274	C₂	1	2	yes	maximal, complemented
4	#20242219	C₂	1	4	yes	maximal, complemented
5	C₈×D₃₇₆	C₂	1	2	yes	maximal, complemented
6	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241788	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094032	—	2	2	no	—
10	#15689673	C₂²	1	2	yes	—
11	#15689528	C₄	1	1	yes	characteristic
12	#15689500	C₄	1	1	yes	characteristic
13	#15689470	C₄	1	2	yes	—
14	#15689359	C₄	1	2	yes	—
15	#15689359	C₄	1	2	yes	—
16	C₄×D₃₇₆	C₂²	1	2	yes	—
17	C₄×D₃₇₆	C₂²	1	2	yes	—
18	#15689202	C₂²	1	4	yes	—
19	#15689196	—	2	4	no	—
20	C₈×D₁₈₈	C₂²	1	4	yes	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
22	#15689091	—	2	2	no	—
23	#15688876	C₂²	1	1	yes	characteristic
24	#15688876	C₂²	1	2	yes	—
25	#15688876	C₂²	1	1	yes	characteristic
26	#15688852	C₂²	1	4	yes	—
27	#15688852	C₂²	1	4	yes	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₈²×C₄₇	C₂²	1	2	yes	—
33	C₂×#10625350	—	2	2	no	—
34	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
35	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
36	#13094032	—	4	8	no	—
37	C₂×#10625324	—	2	4	no	—
38	C₂²×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₈×D₄₇	—	2	4	no	—
40	#13094024	C₂×C₄	1	2	yes	—
41	#13094021	C₂×C₄	1	2	yes	—
42	#13094012	—	2	2	no	—
43	#13093998	—	2	2	no	—
44	#13093998	C₂×C₄	1	4	yes	—
45	C₄×D₁₈₈	—	2	2	no	—
46	C₄×D₁₈₈	C₂³	1	4	yes	—
47	#13093958	C₂×C₄	1	2	yes	—
48	#13093958	C₂×C₄	1	2	yes	—
49	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
52	#13093946	C₂×C₄	1	4	yes	—
53	#13093946	C₂×C₄	1	4	yes	—
54	(C₂×D₄₇)⋊C₈	—	2	4	no	—
55	(C₂×D₄₇)⋊C₈	—	2	4	no	—
56	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₂²⋊C₈×C₄₇	—	2	1	no	—
63	#97×C₄₇	D₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₄₇	C₂³	1	2	yes	—
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	—	2	4	no	—
74	#10625350	—	4	4	no	—
75	#10625350	—	4	4	no	—
76	C₂×D₁₈₈	—	2	2	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₄₇	—	2	4	no	—
80	C₂×Dic₉₄	—	2	2	no	—
81	M₁₆×C₄₇	—	2	2	no	—
82	M₁₆×C₄₇	D₈	1	4	yes	complemented
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	#41	1	1	yes	characteristic
95	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
96	(C₂×D₄₇)⋊C₄	—	2	4	no	—
97	(C₂×D₄₇)⋊C₄	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	4	4	no	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	—	4	2	no	—
105	#10625324	—	4	8	no	—
106	C₈×D₄₇	—	4	8	no	—
107	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂²×D₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₇	#135	1	2	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂²⋊D₄₇	—	4	8	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	D₁₈₈	—	2	4	no	—
123	D₁₈₈	—	2	4	no	—
124	D₁₈₈	—	4	4	no	—
125	D₁₈₈	—	8	2	no	—
126	C₄×D₄₇	—	4	4	no	—
127	C₄×D₄₇	—	4	4	no	—
128	Dic₉₄	—	4	4	no	—
129	C₈×C₄₇	—	2	1	no	—
130	C₈×C₄₇	—	2	1	no	—
131	C₈×C₄₇	—	2	1	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	4	2	no	—
134	C₈×C₄₇	#131	1	2	yes	complemented
135	C₈×C₄₇	C₂×D₈	1	4	yes	—
136	C₄₇⋊C₈	—	4	2	no	—
137	C₄₇⋊C₈	—	4	2	no	—
138	#18066	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	C₄×D₈	1	2	yes	—
142	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×D₄₇	—	4	4	no	—
144	C₂×D₄₇	—	4	4	no	—
145	C₂×D₄₇	—	4	4	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	C₄×D₈	1	2	yes	—
150	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
151	C₄×C₄₇	#568	1	1	yes	characteristic
152	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
154	Dic₄₇	—	4	4	no	—
155	#3169	—	47	1	no	—
156	#2935	—	47	2	no	—
157	#1589	—	47	2	no	—
158	#1581	—	47	4	no	—
159	C₈×D₈	—	47	2	no	—
160	C₂×#335	—	47	1	no	—
161	#1480	—	47	2	no	—
162	C₈×M₁₆	D₄₇	1	1	yes	characteristic, complemented
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	#2935	1	1	yes	characteristic
165	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₄₇	#2957	1	1	yes	characteristic
167	D₄₇	—	8	4	no	—
168	C₂×#131	—	94	2	no	—
169	#532	—	47	2	no	—
170	#500	—	47	1	no	—
171	#493	—	47	1	no	—
172	#486	—	47	2	no	—
173	#466	—	47	2	no	—
174	#466	—	47	2	no	—
175	C₄×D₈	—	47	2	no	—
176	C₄×D₈	—	47	2	no	—
177	#436	—	47	4	no	—
178	#435	—	94	4	no	—
179	C₈×D₄	—	47	4	no	—
180	#431	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄⋊C₈	—	47	2	no	—
182	#408	—	94	2	no	—
183	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#335	—	47	1	no	—
186	#335	—	47	2	no	—
187	#335	—	47	1	no	—
188	D₄⋊C₈	—	47	4	no	—
189	D₄⋊C₈	—	47	4	no	—
190	#322	C₂×D₄₇	1	2	yes	—
191	C₈²	C₂×D₄₇	1	2	yes	—
192	C₄₇	#18066	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	94	2	no	—
194	C₂×D₈	—	47	2	no	—
195	C₂×D₈	—	47	2	no	—
196	#131	—	188	8	no	—
197	C₂×M₁₆	—	94	4	no	—
198	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
199	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
200	C₂²×C₈	—	94	2	no	—
201	C₂²×C₈	—	94	4	no	—
202	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
204	C₄⋊Q₈	—	47	2	no	—
205	#127	—	47	2	no	—
206	#124	—	94	2	no	—
207	#121	—	47	4	no	—
208	#121	—	94	2	no	—
209	C₄×D₄	—	47	4	no	—
210	C₄×D₄	—	94	2	no	—
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	#107	—	47	2	no	—
213	#107	—	47	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	47	2	no	—
216	C₄⋊C₈	—	47	2	no	—
217	C₄⋊C₈	—	94	2	no	—
218	#102	—	47	4	no	—
219	#102	—	47	4	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	94	4	no	—
222	C₂²⋊C₈	—	94	4	no	—
223	#97	D₁₈₈	1	2	yes	—
224	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
225	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₄₇	1	2	yes	—
227	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₄₇	1	2	yes	—
230	#41	—	188	4	no	—
231	#41	—	188	4	no	—
232	C₂×Q₈	—	94	2	no	—
233	C₂×D₄	—	47	4	no	—
234	C₂×D₄	—	94	4	no	—
235	C₂×D₄	—	94	2	no	—
236	C₂×D₄	—	94	2	no	—
237	C₂²×C₄	—	94	4	no	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	#10625350	1	1	yes	characteristic
241	D₈	—	94	4	no	—
242	D₈	—	188	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	188	8	no	—
245	M₁₆	D₃₇₆	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	—	94	2	no	—
251	C₂×C₈	—	188	4	no	—
252	C₂×C₈	—	188	8	no	—
253	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
258	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
259	C₄⋊C₄	—	47	4	no	—
260	C₄⋊C₄	—	94	2	no	—
261	C₂²⋊C₄	—	94	4	no	—
262	C₂²⋊C₄	—	94	4	no	—
263	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
265	C₄²	#10625350	1	1	yes	characteristic
266	C₄²	C₂³×D₄₇	1	1	yes	characteristic
267	C₂³	—	94	4	no	—
268	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
269	Q₈	—	188	4	no	—
270	D₄	—	94	4	no	—
271	D₄	—	94	4	no	—
272	D₄	—	376	2	no	—
273	D₄	—	188	8	no	—
274	D₄	—	188	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
282	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
283	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
284	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
285	C₂×C₄	#13094046	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	188	2	no	—
295	C₈	—	188	2	no	—
296	C₈	#13094032	1	2	yes	complemented
297	C₈	C₂×D₃₇₆	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	188	4	no	—
301	C₂²	—	188	4	no	—
302	C₂²	—	188	4	no	—
303	C₂²	C₄×D₃₇₆	1	2	yes	—
304	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	188	4	no	—
309	C₄	C₄×D₃₇₆	1	2	yes	—
310	C₄	C₂×#13094032	1	1	yes	characteristic
311	C₄	#15689889	1	1	yes	characteristic
312	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
313	C₄	C₂×#13094046	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	376	4	no	—
316	C₂	#20252134	1	1	yes	characteristic
317	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
318	C₂	#20252306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253967	C₂
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#18066
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20252134	C₂	semidirect
3. #20242274	C₂	semidirect
4. #20242219	C₂	semidirect
5. C₈×D₃₇₆	C₂	semidirect
6. C₂×#15688876	C₂	semidirect
7. #20241788	C₂	semidirect
8. C₈×M₁₆×C₄₇	C₂	semidirect
82. M₁₆×C₄₇	D₈	semidirect
134. C₈×C₄₇	#131	semidirect
162. C₈×M₁₆	D₄₇	semidirect
192. C₄₇	#18066	semidirect
245. M₁₆	D₃₇₆	semidirect
296. C₈	#13094032	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20252134	2
#20242274	2
#20242219	2
C₈×D₃₇₆	2
C₂×#15688876	2
#20241788	2
C₈×M₁₆×C₄₇	2
#18066	47