Subgroups — #37581670

Order 24448.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133195	C₂	1	1	yes	maximal, characteristic
2	#27132790	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132741	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132738	C₂	1	2	yes	maximal, complemented
6	#27132734	C₂	1	2	yes	maximal, complemented
7	#27132721	C₂	1	2	yes	maximal, complemented
8	#27132714	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132689	C₂	1	2	yes	maximal, complemented
10	#27132685	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15690271	C₂²	1	2	yes	—
13	C₂×#15690270	C₂²	1	2	yes	—
14	#20259315	C₄	1	2	yes	—
15	#20259313	C₄	1	1	yes	characteristic
16	#20259309	C₄	1	1	yes	characteristic
17	#20259304	C₄	1	2	yes	—
18	#20259302	C₄	1	1	yes	characteristic
19	#20259293	C₄	1	1	yes	characteristic
20	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₉₁	C₂²	1	2	yes	—
22	#20259281	C₂²	1	2	yes	—
23	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
26	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
28	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
33	#20259232	C₂²	1	2	yes	—
34	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₉₁	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
37	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₉₁	C₂²	1	2	yes	—
40	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₈₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
46	#15690271	—	4	2	no	—
47	#15690270	—	4	2	no	—
48	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
54	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
59	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₉₁	C₂³	1	2	yes	—
63	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
66	C₂×C₈×C₁₉₁	—	4	2	no	—
67	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
72	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
73	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
74	Dic₃₈₂	—	2	2	no	—
75	C₂²⋊D₁₉₁	—	4	2	no	—
76	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
81	C₄×D₁₉₁	—	2	2	no	—
82	C₄×D₁₉₁	—	2	1	no	—
83	C₄×D₁₉₁	—	2	1	no	—
84	C₄×D₁₉₁	—	4	1	no	—
85	C₁₉₁⋊C₈	—	2	2	no	—
86	C₁₉₁⋊C₈	—	2	1	no	—
87	C₁₉₁⋊C₈	—	2	1	no	—
88	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
89	D₄×C₁₉₁	—	2	2	no	—
90	C₂×C₄×C₁₉₁	—	2	2	no	—
91	C₂×C₄×C₁₉₁	—	2	2	no	—
92	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₉₁	—	2	1	no	—
97	C₈×C₁₉₁	—	2	2	no	—
98	C₈×C₁₉₁	—	2	1	no	—
99	C₂²×C₁₉₁	—	2	2	no	—
100	C₂²×C₁₉₁	—	2	2	no	—
101	C₂²×C₁₉₁	—	2	2	no	—
102	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₉₁	—	2	1	no	—
104	C₂×D₁₉₁	—	2	1	no	—
105	C₂×D₁₉₁	—	2	1	no	—
106	C₄×C₁₉₁	—	2	2	no	—
107	C₄×C₁₉₁	—	2	1	no	—
108	C₄×C₁₉₁	#131	1	2	yes	—
109	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₁₉₁	—	2	1	no	—
112	Dic₁₉₁	—	2	1	no	—
113	Dic₁₉₁	—	2	2	no	—
114	Dic₁₉₁	—	2	1	no	—
115	Dic₁₉₁	—	2	2	no	—
116	C₂×C₁₉₁	—	4	2	no	—
117	C₂×C₁₉₁	#520	1	1	yes	characteristic
118	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₉₁	#568	1	1	yes	characteristic
120	D₁₉₁	—	4	1	no	—
121	C₁₉₁	#2993	1	1	yes	characteristic, complemented, Sylow
122	#2993	—	191	1	no	maximal, Sylow
123	#529	—	191	1	no	—
124	#446	—	191	1	no	—
125	C₈×Q₈	—	191	1	no	—
126	#436	—	191	1	no	—
127	#436	—	191	2	no	—
128	#435	—	191	2	no	—
129	C₈×D₄	D₁₉₁	1	1	yes	characteristic, complemented
130	#433	—	191	2	no	—
131	#431	—	191	1	no	—
132	#424	—	191	2	no	—
133	#423	—	191	1	no	—
134	C₂×M₁₆	—	191	2	no	—
135	C₂×M₁₆	—	191	2	no	—
136	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
137	#126	—	191	2	no	—
138	#125	—	191	1	no	—
139	#124	—	191	1	no	—
140	#123	—	191	2	no	—
141	C₂²⋊Q₈	—	191	1	no	—
142	#121	—	191	1	no	—
143	C₄×Q₈	—	191	1	no	—
144	C₄×D₄	—	191	2	no	—
145	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
146	#117	—	191	2	no	—
147	C₂×C₄²	—	191	1	no	—
148	C₄⋊C₈	—	191	2	no	—
149	C₄⋊C₈	—	191	1	no	—
150	C₄⋊C₈	—	191	2	no	—
151	C₄⋊C₈	—	191	2	no	—
152	C₄⋊C₈	—	191	1	no	—
153	C₄⋊C₈	—	191	1	no	—
154	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
155	C₂²⋊C₈	—	191	2	no	—
156	C₂²⋊C₈	—	191	1	no	—
157	C₂²⋊C₈	—	191	1	no	—
158	C₂²⋊C₈	C₂×D₁₉₁	1	2	yes	—
159	#97	—	191	2	no	—
160	C₄×C₈	—	191	2	no	—
161	C₄×C₈	—	191	1	no	—
162	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
163	C₂×Q₈	—	191	1	no	—
164	C₂×D₄	—	191	2	no	—
165	C₂×D₄	C₄×D₁₉₁	1	1	yes	characteristic
166	C₂²×C₄	—	191	1	no	—
167	C₂²×C₄	—	191	1	no	—
168	C₂²×C₄	—	191	1	no	—
169	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
170	M₁₆	—	764	2	no	—
171	M₁₆	—	764	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	191	1	no	—
174	C₂×C₈	—	191	2	no	—
175	C₂×C₈	—	191	1	no	—
176	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
179	C₄⋊C₄	—	191	2	no	—
180	C₄⋊C₄	—	191	2	no	—
181	C₄⋊C₄	—	191	2	no	—
182	C₄⋊C₄	—	191	2	no	—
183	C₄⋊C₄	—	191	1	no	—
184	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
185	C₂²⋊C₄	—	191	2	no	—
186	C₂²⋊C₄	—	191	2	no	—
187	C₂²⋊C₄	—	191	2	no	—
188	C₂²⋊C₄	—	191	2	no	—
189	C₂²⋊C₄	C₄×D₁₉₁	1	2	yes	—
190	C₄²	—	191	1	no	—
191	C₄²	—	191	1	no	—
192	C₄²	—	191	2	no	—
193	C₄²	—	191	1	no	—
194	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
195	C₂³	—	191	1	no	—
196	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
197	Q₈	—	382	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	764	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	382	2	no	—
203	C₂×C₄	—	382	1	no	—
204	C₂×C₄	—	382	1	no	—
205	C₂×C₄	—	191	1	no	—
206	C₂×C₄	—	191	1	no	—
207	C₂×C₄	—	191	1	no	—
208	C₂×C₄	—	191	2	no	—
209	C₂×C₄	—	191	2	no	—
210	C₂×C₄	—	764	1	no	—
211	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	382	2	no	—
219	C₈	—	382	1	no	—
220	C₈	—	382	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	382	1	no	—
225	C₂²	—	382	1	no	—
226	C₂²	—	382	1	no	—
227	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	382	1	no	—
231	C₄	—	382	1	no	—
232	C₄	—	382	2	no	—
233	C₄	—	382	1	no	—
234	C₄	—	382	2	no	—
235	C₄	#20259328	1	2	yes	—
236	C₄	#20259367	1	1	yes	characteristic
237	C₄	#20259369	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	764	1	no	—
240	C₂	#27133142	1	1	yes	characteristic
241	C₂	C₂×#20259328	1	1	yes	characteristic
242	C₂	#27133428	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₈×D₄×C₁₉₁	C₂
prime radical	#20259293	C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₈×D₄	D₁₉₁
hypocenter	C₁₉₁	#2993
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #27132790	C₂	semidirect
3. C₈×Dic₃₈₂	C₂	semidirect
4. #27132741	C₂	semidirect
5. #27132738	C₂	semidirect
6. #27132734	C₂	semidirect
7. #27132721	C₂	semidirect
8. #27132714	C₂	semidirect
9. #27132689	C₂	semidirect
10. #27132685	C₂	semidirect
11. C₈×D₄×C₁₉₁	C₂	semidirect
121. C₁₉₁	#2993	semidirect
129. C₈×D₄	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133195	2
#27132790	2
C₈×Dic₃₈₂	2
#27132741	2
#27132738	2
#27132734	2
#27132721	2
#27132714	2
#27132689	2
#27132685	2
C₈×D₄×C₁₉₁	2
#2993	191