Subgroups — #37575779

Order 24448.

560 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259281	C₂	1	1	yes	maximal, characteristic, complemented
2	#27132602	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132599	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132593	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132591	C₂	1	2	yes	maximal, complemented
6	#27132588	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132577	C₂	1	2	yes	maximal
8	C₄×Dic₁₉₁⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊Dic₁₉₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	#20259281	—	2	4	no	—
13	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
23	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
24	C₁₉₁⋊#95	C₂²	1	2	yes	—
25	C₁₉₁⋊#95	C₂²	1	2	yes	—
26	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
27	C₁₉₁⋊#95	C₂²	1	2	yes	—
28	C₁₉₁⋊#95	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₉₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
41	C₄⋊Dic₁₉₁	—	2	2	no	—
42	C₄⋊Dic₁₉₁	—	2	2	no	—
43	Dic₁₉₁⋊C₄	—	2	4	no	—
44	Dic₁₉₁⋊C₄	—	2	4	no	—
45	C₂²⋊Dic₁₉₁	—	2	4	no	—
46	C₂²⋊Dic₁₉₁	—	2	4	no	—
47	C₂²⋊Dic₁₉₁	—	2	4	no	—
48	C₂²⋊Dic₁₉₁	—	2	4	no	—
49	C₄×Dic₁₉₁	—	2	4	no	—
50	C₄×Dic₁₉₁	—	2	4	no	—
51	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
52	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₉₁	—	2	2	no	—
54	C₂×D₄×C₁₉₁	—	2	2	no	—
55	C₂×D₄×C₁₉₁	—	2	2	no	—
56	C₂²×C₄×C₁₉₁	—	2	4	no	—
57	C₂²×C₄×C₁₉₁	—	2	2	no	—
58	C₂²×C₄×C₁₉₁	—	2	4	no	—
59	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
60	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉₁	—	2	2	no	—
66	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
67	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
68	C₄²×C₁₉₁	—	2	2	no	—
69	C₂×Dic₁₉₁	—	2	2	no	—
70	C₂×Dic₁₉₁	—	2	2	no	—
71	C₂×Dic₁₉₁	—	2	2	no	—
72	C₂×Dic₁₉₁	—	2	2	no	—
73	C₂×Dic₁₉₁	—	2	2	no	—
74	C₂×Dic₁₉₁	—	2	2	no	—
75	C₂×Dic₁₉₁	—	2	2	no	—
76	C₂×Dic₁₉₁	—	2	2	no	—
77	C₂×Dic₁₉₁	—	2	2	no	—
78	C₂×Dic₁₉₁	—	2	2	no	—
79	C₂×Dic₁₉₁	—	2	2	no	—
80	C₂×Dic₁₉₁	#41	1	4	yes	—
81	C₂³×C₁₉₁	—	2	1	no	—
82	C₂³×C₁₉₁	—	2	2	no	—
83	C₂³×C₁₉₁	—	2	2	no	—
84	C₂³×C₁₉₁	—	2	1	no	—
85	C₂³×C₁₉₁	—	2	1	no	—
86	C₂³×C₁₉₁	—	2	1	no	—
87	C₂³×C₁₉₁	—	2	2	no	—
88	C₂³×C₁₉₁	—	2	2	no	—
89	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
90	C₂³×C₁₉₁	#41	1	2	yes	—
91	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₁₉₁	—	2	8	no	—
93	C₂×C₄×C₁₉₁	—	2	1	no	—
94	C₂×C₄×C₁₉₁	—	2	2	no	—
95	C₂×C₄×C₁₉₁	—	2	1	no	—
96	C₂×C₄×C₁₉₁	—	2	8	no	—
97	C₂×C₄×C₁₉₁	—	2	1	no	—
98	C₂×C₄×C₁₉₁	—	2	1	no	—
99	C₂×C₄×C₁₉₁	—	2	2	no	—
100	C₂×C₄×C₁₉₁	—	2	2	no	—
101	C₂×C₄×C₁₉₁	—	4	4	no	—
102	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₉₁	#41	1	4	yes	—
105	C₂×C₄×C₁₉₁	#41	1	2	yes	—
106	C₂²×C₁₉₁	—	2	2	no	—
107	C₂²×C₁₉₁	—	2	2	no	—
108	C₂²×C₁₉₁	—	2	2	no	—
109	C₂²×C₁₉₁	—	2	4	no	—
110	C₂²×C₁₉₁	—	2	2	no	—
111	C₂²×C₁₉₁	—	4	1	no	—
112	C₂²×C₁₉₁	—	4	2	no	—
113	C₂²×C₁₉₁	—	4	2	no	—
114	C₂²×C₁₉₁	—	4	2	no	—
115	C₂²×C₁₉₁	—	4	1	no	—
116	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₉₁	C₂×#41	1	2	yes	—
118	C₂²×C₁₉₁	C₂×#41	1	2	yes	—
119	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
121	C₄×C₁₉₁	—	2	2	no	—
122	C₄×C₁₉₁	—	2	2	no	—
123	C₄×C₁₉₁	—	4	1	no	—
124	C₄×C₁₉₁	—	4	1	no	—
125	C₄×C₁₉₁	#123	1	4	yes	—
126	Dic₁₉₁	—	2	4	no	—
127	Dic₁₉₁	—	4	2	no	—
128	Dic₁₉₁	—	4	2	no	—
129	Dic₁₉₁	—	4	2	no	—
130	C₂×C₁₉₁	—	4	2	no	—
131	C₂×C₁₉₁	—	4	2	no	—
132	C₂×C₁₉₁	C₂×#123	1	1	yes	characteristic
133	C₂×C₁₉₁	#525	1	1	yes	characteristic
134	C₂×C₁₉₁	#529	1	2	yes	—
135	C₂×C₁₉₁	#529	1	2	yes	—
136	C₂×C₁₉₁	#532	1	1	yes	characteristic
137	C₁₉₁	#2395	1	1	yes	characteristic, complemented, Sylow
138	#2395	—	191	1	no	maximal, Sylow
139	C₂×C₄×D₄	D₁₉₁	1	1	yes	characteristic
140	C₂×#117	—	191	1	no	—
141	#400	—	191	1	no	—
142	#399	—	191	1	no	—
143	#398	—	191	1	no	—
144	#397	—	191	1	no	—
145	#397	—	191	2	no	—
146	#394	—	191	2	no	—
147	C₄×C₄⋊C₄	—	191	2	no	—
148	C₄×C₂²⋊C₄	—	191	2	no	—
149	C₄×C₂²⋊C₄	—	191	1	no	—
150	C₂²×D₄	C₂×D₁₉₁	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
152	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
153	C₄×D₄	—	2	4	no	—
154	#117	—	382	4	no	—
155	C₂×C₄⋊C₄	—	191	1	no	—
156	C₂×C₄⋊C₄	—	191	1	no	—
157	C₂×C₄⋊C₄	—	191	2	no	—
158	C₂×C₄⋊C₄	—	191	2	no	—
159	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	191	2	no	—
161	C₂×C₂²⋊C₄	—	191	2	no	—
162	C₂×C₂²⋊C₄	—	191	2	no	—
163	C₂×C₂²⋊C₄	—	191	2	no	—
164	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
166	C₂×C₄²	—	191	2	no	—
167	C₂×C₄²	—	191	2	no	—
168	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
169	#95	—	191	2	no	—
170	#95	—	191	1	no	—
171	#95	—	191	2	no	—
172	#95	—	191	1	no	—
173	#95	—	191	2	no	—
174	#95	—	191	2	no	—
175	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
176	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	191	2	no	—
184	C₂²×C₄	—	191	2	no	—
185	C₂²×C₄	—	191	2	no	—
186	C₂²×C₄	—	191	2	no	—
187	C₂²×C₄	C₂²⋊D₁₉₁	1	4	yes	—
188	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	382	4	no	—
195	C₄⋊C₄	—	382	2	no	—
196	C₄⋊C₄	—	382	2	no	—
197	C₄⋊C₄	—	382	4	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	382	4	no	—
201	C₂²⋊C₄	—	382	4	no	—
202	C₂²⋊C₄	—	382	4	no	—
203	C₂²⋊C₄	—	382	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	382	4	no	—
206	C₄²	—	382	4	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×C₂²⋊D₁₉₁	1	2	yes	—
216	C₂³	#15690287	1	2	yes	—
217	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
218	D₄	—	2	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	382	2	no	—
229	C₂×C₄	—	382	2	no	—
230	C₂×C₄	—	382	2	no	—
231	C₂×C₄	—	382	2	no	—
232	C₂×C₄	—	382	2	no	—
233	C₂×C₄	—	382	2	no	—
234	C₂×C₄	—	382	2	no	—
235	C₂×C₄	—	382	2	no	—
236	C₂×C₄	—	382	2	no	—
237	C₂×C₄	—	382	2	no	—
238	C₂×C₄	—	382	2	no	—
239	C₂×C₄	—	191	4	no	—
240	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
242	C₂×C₄	#15690286	1	4	yes	—
243	C₂×C₄	#15690287	1	2	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
255	C₂²	#41×D₁₉₁	1	2	yes	—
256	C₂²	C₂×#15690286	1	2	yes	—
257	C₂²	C₂×#15690287	1	1	yes	characteristic
258	C₂²	C₂×#15690287	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	382	4	no	—
264	C₄	—	764	2	no	—
265	C₄	—	764	2	no	—
266	C₄	—	764	2	no	—
267	C₄	#20259303	1	4	yes	—
268	C₂	—	4	2	no	—
269	C₂	—	4	2	no	—
270	C₂	C₂×#20259303	1	1	yes	characteristic
271	C₂	#27133174	1	1	yes	characteristic
272	C₂	#27133193	1	2	yes	—
273	C₂	#27133195	1	2	yes	—
274	C₂	#27133209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Frattini	C₂³	C₂³×D₁₉₁
Fitting	C₂×C₄×D₄×C₁₉₁	C₂
prime radical	C₂²×D₄×C₁₉₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₉₁
hypocenter	C₁₉₁	#2395
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20259281	C₂	semidirect
2. #27132602	C₂	semidirect
3. #27132599	C₂	semidirect
4. #27132593	C₂	semidirect
5. #27132591	C₂	semidirect
6. #27132588	C₂	semidirect
8. C₄×Dic₁₉₁⋊C₄	C₂	semidirect
9. C₄×C₂²⋊Dic₁₉₁	C₂	semidirect
10. C₄×C₂²⋊Dic₁₉₁	C₂	semidirect
137. C₁₉₁	#2395	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259281	2
#27132602	2
#27132599	2
#27132593	2
#27132591	2
#27132588	2
#27132577	2
C₄×Dic₁₉₁⋊C₄	2
C₄×C₂²⋊Dic₁₉₁	2
C₂×C₄×D₄×C₁₉₁	2
#2395	191