Subgroups — #37581637

Order 24448.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27132794	C₂	1	2	yes	maximal, complemented
2	#27132741	C₂	1	2	yes	maximal, complemented
3	#27132735	C₂	1	4	yes	maximal, complemented
4	#27132720	C₂	1	2	yes	maximal, complemented
5	#27132716	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132687	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₁₉₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#15690271	C₂²	1	4	yes	—
10	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
12	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
13	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	—
15	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
17	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	—
18	#20259233	C₂²	1	4	yes	—
19	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
20	C₂×#41×C₁₉₁	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₉₁	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₉₁	C₄	1	2	yes	—
24	C₄×D₄×C₁₉₁	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₉₁	C₂²	1	2	yes	—
26	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
27	#117×C₁₉₁	C₄	1	2	yes	—
28	#117×C₁₉₁	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	—
31	#15690271	—	4	4	no	—
32	C₂×C₁₉₁⋊C₈	—	4	2	no	—
33	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
36	#41×C₁₉₁	—	2	4	no	—
37	C₂×Q₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₉₁	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
53	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
55	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
56	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
58	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
59	C₄²×C₁₉₁	C₂³	1	2	yes	—
60	C₁₉₁⋊C₈	—	2	4	no	—
61	C₁₉₁⋊C₈	—	2	2	no	—
62	C₁₉₁⋊C₈	—	2	2	no	—
63	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
64	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
65	Q₈×C₁₉₁	—	2	2	no	—
66	D₄×C₁₉₁	—	2	4	no	—
67	D₄×C₁₉₁	—	2	2	no	—
68	C₂×C₄×C₁₉₁	—	2	4	no	—
69	C₂×C₄×C₁₉₁	—	2	1	no	—
70	C₂×C₄×C₁₉₁	—	2	2	no	—
71	C₂×C₄×C₁₉₁	—	2	1	no	—
72	C₂×C₄×C₁₉₁	—	2	2	no	—
73	C₂×C₄×C₁₉₁	—	4	1	no	—
74	C₂×C₄×C₁₉₁	—	4	2	no	—
75	C₂×C₄×C₁₉₁	M₁₆	1	4	yes	complemented
76	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₉₁	—	2	1	no	—
86	C₂²×C₁₉₁	—	2	1	no	—
87	C₂²×C₁₉₁	—	2	2	no	—
88	C₂²×C₁₉₁	—	2	2	no	—
89	C₂²×C₁₉₁	—	2	2	no	—
90	C₂²×C₁₉₁	C₂×M₁₆	1	2	yes	—
91	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₁₉₁	—	2	1	no	—
93	C₄×C₁₉₁	—	2	2	no	—
94	C₄×C₁₉₁	—	2	2	no	—
95	C₄×C₁₉₁	—	2	1	no	—
96	C₄×C₁₉₁	—	2	1	no	—
97	C₄×C₁₉₁	—	2	2	no	—
98	C₄×C₁₉₁	C₂×M₁₆	1	2	yes	—
99	C₄×C₁₉₁	C₂×M₁₆	1	2	yes	—
100	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₁₉₁	—	2	2	no	—
103	C₂×C₁₉₁	—	4	2	no	—
104	C₂×C₁₉₁	#520	1	1	yes	characteristic
105	C₂×C₁₉₁	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₁₉₁	#568	1	1	yes	characteristic
107	C₁₉₁	#2989	1	1	yes	characteristic, complemented, Sylow
108	#2989	—	191	1	no	maximal, Sylow
109	C₄×#41	D₁₉₁	1	1	yes	characteristic
110	#446	—	191	2	no	—
111	#436	—	191	2	no	—
112	#435	—	191	4	no	—
113	#432	—	191	2	no	—
114	#431	—	191	1	no	—
115	#423	—	191	2	no	—
116	C₂×C₄⋊C₈	—	191	1	no	—
117	C₂×#41	Dic₁₉₁	1	1	yes	characteristic
118	C₂×M₁₆	—	191	4	no	—
119	C₂²×C₈	—	191	2	no	—
120	C₄×Q₈	Dic₁₉₁	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
122	C₄×D₄	Dic₁₉₁	1	2	yes	—
123	C₄×D₄	Dic₁₉₁	1	1	yes	characteristic
124	C₄×D₄	C₂×D₁₉₁	1	2	yes	—
125	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
126	#117	Dic₁₉₁	1	2	yes	—
127	#117	Dic₁₉₁	1	1	yes	characteristic
128	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₁₉₁	1	2	yes	—
130	C₄⋊C₈	—	191	2	no	—
131	C₄⋊C₈	—	191	2	no	—
132	C₄⋊C₈	—	191	1	no	—
133	C₄⋊C₈	—	191	4	no	—
134	C₄⋊C₈	—	191	1	no	—
135	C₂²⋊C₈	—	191	2	no	—
136	C₂²⋊C₈	—	191	4	no	—
137	#97	—	191	4	no	—
138	C₄×C₈	—	191	2	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×Dic₁₉₁	1	1	yes	characteristic
141	C₂×D₄	C₂×Dic₁₉₁	1	2	yes	—
142	C₂×D₄	C₂×Dic₁₉₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
149	M₁₆	—	764	4	no	—
150	C₂×C₈	—	191	2	no	—
151	C₂×C₈	—	191	2	no	—
152	C₂×C₈	—	191	4	no	—
153	C₂×C₈	—	764	2	no	—
154	C₄⋊C₄	C₂×Dic₁₉₁	1	2	yes	—
155	C₄⋊C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
156	C₄⋊C₄	C₂×Dic₁₉₁	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×Dic₁₉₁	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₁₉₁	1	2	yes	—
160	C₂²⋊C₄	C₂×Dic₁₉₁	1	2	yes	—
161	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
162	C₄²	C₂×Dic₁₉₁	1	2	yes	—
163	C₄²	C₂×Dic₁₉₁	1	2	yes	—
164	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
165	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
166	C₄²	C₂²×D₁₉₁	1	2	yes	—
167	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
168	C₂³	C₂²×Dic₁₉₁	1	2	yes	—
169	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	#15690271	1	4	yes	complemented
181	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
184	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
190	C₈	—	382	4	no	—
191	C₈	—	382	2	no	—
192	C₈	—	382	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂×#15690271	1	2	yes	—
199	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂×#15690271	1	2	yes	—
207	C₄	C₂×#15690271	1	2	yes	—
208	C₄	#20259368	1	1	yes	characteristic
209	C₄	#20259369	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	4	2	no	—
212	C₂	#27133146	1	1	yes	characteristic
213	C₂	C₂²×#15690271	1	1	yes	characteristic
214	C₂	#27133430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₄×#41×C₁₉₁	C₂
prime radical	C₂×#41×C₁₉₁	C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₄×#41	D₁₉₁
hypocenter	C₁₉₁	#2989
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #27132794	C₂	semidirect
2. #27132741	C₂	semidirect
3. #27132735	C₂	semidirect
4. #27132720	C₂	semidirect
5. #27132716	C₂	semidirect
6. #27132687	C₂	semidirect
7. C₂×C₄⋊(C₁₉₁⋊C₈)	C₂	semidirect
75. C₂×C₄×C₁₉₁	M₁₆	semidirect
107. C₁₉₁	#2989	semidirect
180. C₂×C₄	#15690271	semidirect

Maximal subgroups

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

subgroup	index
#27132794	2
#27132741	2
#27132735	2
#27132720	2
#27132716	2
#27132687	2
C₂×C₄⋊(C₁₉₁⋊C₈)	2
C₄×#41×C₁₉₁	2
#2989	191