Subgroups — #26188271

Order 12032.

572 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253974	C₂	1	1	yes	maximal, characteristic
2	#20252489	C₂	1	2	yes	maximal
3	#20242286	C₂	1	2	yes	maximal, complemented
4	#20242261	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₁₈₈	C₂	1	2	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic
7	#20241810	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	C₄	1	1	yes	characteristic
11	#15689503	C₄	1	1	yes	characteristic
12	#15689474	C₄	1	2	yes	—
13	#15689364	C₄	1	2	yes	—
14	#15689364	C₄	1	2	yes	—
15	#15689253	C₂²	1	4	yes	—
16	#15689250	—	2	2	no	—
17	C₈×Dic₉₄	C₂²	1	4	yes	—
18	#15689247	—	2	2	no	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	—
20	C₄×Dic₁₈₈	C₂²	1	2	yes	—
21	#15689181	—	2	4	no	—
22	#15689176	C₂²	1	2	yes	—
23	#15688876	C₂²	1	1	yes	characteristic
24	#15688876	C₂²	1	1	yes	characteristic
25	#15688874	C₂²	1	2	yes	—
26	#15688855	C₂²	1	4	yes	—
27	#15688855	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×Dic₁₈₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
35	#13094024	C₂×C₄	1	2	yes	—
36	#13094024	C₂×C₄	1	2	yes	—
37	#13094016	—	2	2	no	—
38	#13094003	—	2	2	no	—
39	#13094003	C₂×C₄	1	4	yes	—
40	C₄×Dic₉₄	—	2	2	no	—
41	C₄×Dic₉₄	C₂³	1	4	yes	—
42	#13093983	—	2	2	no	—
43	#13093958	C₂×C₄	1	2	yes	—
44	#13093958	C₂×C₄	1	2	yes	—
45	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
47	Dic₄₇⋊C₈	—	2	4	no	—
48	Dic₄₇⋊C₈	—	2	4	no	—
49	#13093949	C₂×C₄	1	4	yes	—
50	#13093949	C₂×C₄	1	4	yes	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
52	#13093934	—	2	4	no	—
53	C₄×C₄₇⋊C₈	—	2	2	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
61	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
62	#97×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
74	M₁₆×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	4	1	no	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	#41	1	1	yes	characteristic
87	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
93	Dic₄₇⋊C₄	—	2	4	no	—
94	Dic₄₇⋊C₄	—	2	4	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	Dic₁₈₈	—	2	4	no	—
99	Dic₁₈₈	—	4	2	no	—
100	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	Dic₉₄	—	2	4	no	—
115	Dic₉₄	—	2	4	no	—
116	Dic₉₄	—	4	4	no	—
117	C₈×C₄₇	—	2	4	no	—
118	C₈×C₄₇	—	2	1	no	—
119	C₈×C₄₇	—	2	1	no	—
120	C₈×C₄₇	—	2	1	no	—
121	C₈×C₄₇	—	2	2	no	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	#131	1	2	yes	—
124	C₄₇⋊C₈	—	4	2	no	—
125	C₄₇⋊C₈	—	4	2	no	—
126	#18086	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	#443	1	2	yes	—
135	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
136	C₄×C₄₇	#568	1	1	yes	characteristic
137	C₄×C₄₇	#578	1	1	yes	characteristic
138	C₄×C₄₇	#578	1	1	yes	characteristic
139	Dic₄₇	—	4	4	no	—
140	Dic₄₇	—	4	4	no	—
141	#3170	—	47	1	no	—
142	#2986	—	47	2	no	—
143	#1591	—	47	2	no	—
144	#1587	—	47	4	no	—
145	C₈×Q₁₆	—	47	2	no	—
146	#1564	—	47	1	no	—
147	#1485	—	47	2	no	—
148	C₈×M₁₆	D₄₇	1	1	yes	characteristic
149	C₂×C₄₇	—	4	1	no	—
150	C₂×C₄₇	#2935	1	1	yes	characteristic
151	C₂×C₄₇	#2950	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
153	#533	—	47	2	no	—
154	#500	—	47	1	no	—
155	#494	—	47	1	no	—
156	#487	—	47	2	no	—
157	#467	—	47	2	no	—
158	#467	—	47	2	no	—
159	#446	—	47	4	no	—
160	#446	—	94	2	no	—
161	C₈×Q₈	—	47	4	no	—
162	C₈×Q₈	—	94	2	no	—
163	C₄×Q₁₆	—	47	2	no	—
164	C₄×Q₁₆	—	47	2	no	—
165	#433	—	94	4	no	—
166	#431	—	47	2	no	—
167	#431	C₂×D₄₇	1	1	yes	characteristic
168	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
170	#335	—	47	1	no	—
171	#335	—	47	1	no	—
172	#334	—	47	2	no	—
173	Q₈⋊C₈	—	47	4	no	—
174	Q₈⋊C₈	—	47	4	no	—
175	#322	C₂×D₄₇	1	2	yes	—
176	C₈²	C₂×D₄₇	1	2	yes	—
177	C₄₇	#18086	1	1	yes	characteristic, complemented, Sylow
178	C₂×Q₁₆	—	47	2	no	—
179	C₂×Q₁₆	—	47	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
183	C₄⋊Q₈	—	47	2	no	—
184	C₄⋊Q₈	—	47	2	no	—
185	#125	—	94	2	no	—
186	C₂²⋊Q₈	—	47	4	no	—
187	C₂²⋊Q₈	—	94	2	no	—
188	C₄×Q₈	—	47	4	no	—
189	C₄×Q₈	—	94	2	no	—
190	#117	—	94	2	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#107	—	47	2	no	—
193	#107	—	47	2	no	—
194	C₄⋊C₈	—	47	2	no	—
195	C₄⋊C₈	—	47	2	no	—
196	C₄⋊C₈	—	94	4	no	—
197	C₄⋊C₈	—	94	4	no	—
198	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
199	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
200	#103	—	47	4	no	—
201	#103	—	47	4	no	—
202	C₂²⋊C₈	—	94	2	no	—
203	C₂²⋊C₈	D₁₈₈	1	2	yes	—
204	#97	—	2	1	no	—
205	#97	—	94	4	no	—
206	C₄×C₈	—	94	4	no	—
207	C₄×C₈	—	94	2	no	—
208	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
209	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₄₇	1	2	yes	—
211	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₄₇	1	2	yes	—
214	C₂×Q₈	—	47	4	no	—
215	C₂×Q₈	—	94	2	no	—
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	#10625350	1	1	yes	characteristic
219	Q₁₆	—	94	4	no	—
220	Q₁₆	—	188	2	no	—
221	M₁₆	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
232	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
233	C₄⋊C₄	—	47	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	94	2	no	—
236	C₄⋊C₄	—	94	2	no	—
237	C₄⋊C₄	—	94	2	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₂²⋊C₄	—	94	4	no	—
240	C₄²	—	94	4	no	—
241	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
243	C₄²	#10625350	1	1	yes	characteristic
244	C₄²	C₂³×D₄₇	1	1	yes	characteristic
245	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
246	Q₈	—	94	4	no	—
247	Q₈	—	94	4	no	—
248	Q₈	—	188	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
257	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
258	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	188	2	no	—
269	C₈	—	188	2	no	—
270	C₈	#13094032	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	188	4	no	—
279	C₄	—	188	4	no	—
280	C₄	#15689241	1	2	yes	—
281	C₄	C₂×#13094032	1	1	yes	characteristic
282	C₄	#15689889	1	1	yes	characteristic
283	C₄	#15689940	1	1	yes	characteristic
284	C₄	#15689940	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	#20252134	1	1	yes	characteristic
287	C₂	#20252257	1	1	yes	characteristic
288	C₂	C₂×#15689241	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	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#18086
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20242286	C₂	semidirect
4. #20242261	C₂	semidirect
5. C₈×Dic₁₈₈	C₂	semidirect
177. C₄₇	#18086	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20252489	2
#20242286	2
#20242261	2
C₈×Dic₁₈₈	2
#20242136	2
#20241810	2
C₈×M₁₆×C₄₇	2
#18086	47