Subgroups — #26234719

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈₈	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20246308	C₂	1	2	yes	maximal
4	#20244121	C₂	1	2	yes	maximal
5	#20243659	C₂	1	4	yes	maximal
6	#20243552	C₂	1	1	yes	maximal, characteristic
7	#20243404	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
11	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
12	C₄×Dic₁₈₈	—	2	8	no	—
13	C₄×Dic₁₈₈	C₄	1	8	yes	complemented
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093949	C₂²	1	4	yes	—
18	C₂×#13093949	C₂²	1	2	yes	—
19	#15689021	C₂²	1	4	yes	—
20	#15689018	—	2	4	no	—
21	#15688989	C₂²	1	2	yes	—
22	#15688987	—	2	2	no	—
23	#15688974	—	2	2	no	—
24	#15688886	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#336	C₂²	1	2	yes	—
30	C₂²×Dic₉₄	C₂³	1	2	yes	—
31	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
32	C₂×Dic₁₈₈	D₄	1	4	yes	—
33	C₂×Dic₁₈₈	Q₈	1	4	yes	—
34	C₄×Dic₉₄	—	2	4	no	—
35	C₄×Dic₉₄	—	2	8	no	—
36	C₄×Dic₉₄	—	4	4	no	—
37	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×Dic₄₇⋊C₄	—	2	2	no	—
41	C₂×Dic₄₇⋊C₄	—	2	4	no	—
42	C₂×Dic₄₇⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	4	no	—
45	#13093958	—	2	4	no	—
46	#13093958	C₂×C₄	1	4	yes	—
47	#13093956	—	4	2	no	—
48	#13093949	—	2	8	no	—
49	#13093949	C₂×C₄	1	8	yes	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	—	2	2	no	—
60	C₄×C₈×C₄₇	—	2	4	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
62	#95×C₄₇	D₄	1	1	yes	characteristic
63	#95×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	—	2	2	no	—
74	C₂×Dic₉₄	—	2	2	no	—
75	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
76	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
77	C₂×Dic₉₄	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
96	Dic₄₇⋊C₄	—	2	8	no	—
97	Dic₄₇⋊C₄	—	4	8	no	—
98	Dic₄₇⋊C₄	—	4	8	no	—
99	C₄×Dic₄₇	—	2	8	no	—
100	C₄×Dic₄₇	—	4	8	no	—
101	Dic₁₈₈	C₄⋊C₄	1	16	yes	complemented
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	#135	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	Dic₉₄	—	2	4	no	—
128	Dic₉₄	—	2	4	no	—
129	Dic₉₄	—	2	4	no	—
130	Dic₉₄	C₂×C₄⋊C₄	1	8	yes	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	—	4	2	no	—
133	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
134	#22685	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₇	D₄²	1	1	yes	characteristic
137	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#135	1	2	yes	—
139	C₂²×C₄₇	#578	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₇	#520	1	1	yes	characteristic
146	C₄×C₄₇	#547	1	1	yes	characteristic
147	C₄×C₄₇	#554	1	1	yes	characteristic
148	Dic₄₇	—	2	8	no	—
149	Dic₄₇	—	4	4	no	—
150	Dic₄₇	—	8	4	no	—
151	C₂×C₄×Q₁₆	—	47	1	no	—
152	C₂×C₄×Q₁₆	—	47	2	no	—
153	#2351	—	47	2	no	—
154	#1942	—	47	2	no	—
155	#1865	—	47	4	no	—
156	#1846	—	47	1	no	—
157	#1805	—	47	2	no	—
158	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
159	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
161	C₂×C₄₇	#2950	1	1	yes	characteristic
162	C₂×C₄₇	#3285	1	2	yes	—
163	C₂×C₄₇	#3383	1	2	yes	—
164	C₂²×Q₁₆	—	47	1	no	—
165	C₂×C₄×Q₈	—	47	2	no	—
166	C₂×C₄×Q₈	—	47	4	no	—
167	C₄×Q₁₆	—	47	8	no	—
168	C₄×Q₁₆	—	94	8	no	—
169	C₂×#107	—	47	1	no	—
170	C₂×#107	—	47	2	no	—
171	C₂×#106	—	47	2	no	—
172	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₂×#103	—	47	2	no	—
174	C₂×#103	—	47	4	no	—
175	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	#389	—	47	4	no	—
178	#389	—	94	4	no	—
179	#382	—	47	2	no	—
180	#382	—	94	2	no	—
181	C₄×C₄⋊C₄	—	94	2	no	—
182	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
183	#340	—	47	4	no	—
184	#336	C₂×D₄₇	1	2	yes	—
185	C₄₇	#22685	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	—	47	2	no	—
187	C₂×Q₁₆	—	47	4	no	—
188	C₂×Q₁₆	—	47	4	no	—
189	C₂×Q₁₆	—	47	4	no	—
190	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₄×Q₈	—	47	8	no	—
194	C₄×Q₈	—	94	8	no	—
195	C₄×Q₈	—	94	4	no	—
196	C₄×Q₈	—	188	4	no	—
197	C₂×C₄⋊C₄	—	47	2	no	—
198	C₂×C₄⋊C₄	—	47	4	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	94	4	no	—
201	C₂×C₄⋊C₄	—	94	4	no	—
202	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	94	4	no	—
205	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	47	4	no	—
208	#107	—	94	4	no	—
209	#106	—	188	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	#103	—	47	8	no	—
212	#103	—	94	8	no	—
213	C₄×C₈	—	2	4	no	—
214	C₄×C₈	C₄×D₄₇	1	4	yes	—
215	#95	—	94	2	no	—
216	#95	—	94	2	no	—
217	#95	—	94	4	no	—
218	#95	D₁₈₈	1	1	yes	characteristic
219	#95	D₁₈₈	1	1	yes	characteristic
220	C₂×Q₈	—	47	4	no	—
221	C₂×Q₈	—	47	4	no	—
222	C₂×Q₈	—	47	4	no	—
223	C₂×Q₈	—	94	2	no	—
224	C₂×Q₈	—	94	2	no	—
225	C₂×Q₈	—	94	2	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
230	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	Q₁₆	—	47	16	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
242	C₂×C₈	D₄×D₄₇	1	2	yes	—
243	C₂×C₈	Q₈×D₄₇	1	2	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	47	4	no	—
246	C₄⋊C₄	—	94	8	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	94	4	no	—
250	C₄⋊C₄	—	188	8	no	—
251	C₄⋊C₄	—	188	8	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	94	8	no	—
255	C₄²	—	188	8	no	—
256	C₄²	C₂×C₄×D₄₇	1	2	yes	—
257	C₄²	#10625350	1	2	yes	—
258	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
259	Q₈	—	47	8	no	—
260	Q₈	—	94	4	no	—
261	Q₈	—	94	4	no	—
262	Q₈	—	94	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
279	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
280	C₂×C₄	#13094046	1	4	yes	—
281	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
285	C₂×C₄	#13094070	1	1	yes	characteristic
286	C₂×C₄	#13094071	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	C₄⋊C₄×D₄₇	1	4	yes	—
290	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
291	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
292	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
293	C₂²	C₂×#13094046	1	2	yes	—
294	C₂²	#15689940	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	94	8	no	—
300	C₄	—	376	4	no	—
301	C₄	—	188	4	no	—
302	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
303	C₄	#15689603	1	1	yes	characteristic
304	C₄	#15689778	1	1	yes	characteristic
305	C₄	#15689833	1	1	yes	characteristic
306	C₂	C₄⋊C₄×D₁₈₈	1	1	yes	characteristic
307	C₂	C₄×#13094046	1	1	yes	characteristic
308	C₂	#20252257	1	1	yes	characteristic
309	C₂	#20255131	1	2	yes	—
310	C₂	#20256280	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#22685
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₁₈₈	C₄	semidirect
101. Dic₁₈₈	C₄⋊C₄	semidirect
185. C₄₇	#22685	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
#20246308	2
#20244121	2
#20243659	2
#20243552	2
#20243404	2
C₈×C₄⋊C₄×C₄₇	2
#22685	47