Finite Group Databasebeta

Subgroups — #37574403

Order 24448.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133093	C₂	1	1	yes	maximal, characteristic
2	#27132780	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132779	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132492	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132471	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132438	C₂	1	1	yes	maximal, characteristic, complemented
7	#359×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20259344	C₂²	1	1	yes	characteristic
9	#20259330	—	2	1	no	—
10	#20259329	—	2	1	no	—
11	#20259268	C₄	1	1	yes	characteristic, complemented
12	#20259266	C₄	1	1	yes	characteristic, complemented
13	#20259265	C₄	1	1	yes	characteristic, complemented
14	#20259263	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#20259258	C₂²	1	1	yes	characteristic
17	#20259249	—	2	1	no	—
18	#20259248	—	2	1	no	—
19	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
20	#135×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
22	#108×C₁₉₁	C₂²	1	1	yes	characteristic
23	#15690287	—	2	1	no	—
24	Dic₇₆₄	C₂×C₄	1	1	yes	characteristic
25	#15690276	—	2	1	no	—
26	D₁₅₂₈	C₂×C₄	1	1	yes	characteristic
27	#15690271	—	2	1	no	—
28	#15690271	—	4	1	no	—
29	#15690270	—	4	1	no	—
30	C₂×C₁₉₁⋊C₈	—	4	1	no	—
31	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
32	C₈×D₁₉₁	—	4	1	no	—
33	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
34	C₁₉₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₁₉₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
36	#41×C₁₉₁	—	2	1	no	—
37	Q₁₆×C₁₉₁	C₂×C₄	1	1	yes	characteristic
38	SD₁₆×C₁₉₁	—	2	1	no	—
39	D₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₁₉₁	—	2	1	no	—
41	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
42	C₁₆×C₁₉₁	—	2	1	no	—
43	Dic₃₈₂	—	2	1	no	—
44	D₇₆₄	—	2	1	no	—
45	C₂²⋊D₁₉₁	—	4	1	no	—
46	C₂×Dic₁₉₁	—	2	1	no	—
47	C₄×D₁₉₁	—	4	1	no	—
48	C₁₉₁⋊C₈	—	4	1	no	—
49	C₁₉₁⋊C₈	D₈	1	1	yes	characteristic
50	C₁₉₁⋊C₈	D₈	1	1	yes	characteristic
51	Q₈×C₁₉₁	—	2	1	no	—
52	D₄×C₁₉₁	—	2	1	no	—
53	D₄×C₁₉₁	—	4	1	no	—
54	C₂×C₄×C₁₉₁	—	4	1	no	—
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₉₁	—	4	1	no	—
57	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₉₁	#41	1	1	yes	characteristic
59	C₂²×C₁₉₁	—	4	1	no	—
60	C₂²×C₁₉₁	#135	1	1	yes	characteristic
61	C₂×D₁₉₁	—	4	1	no	—
62	C₄×C₁₉₁	—	4	1	no	—
63	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₉₁	C₂×D₈	1	1	yes	characteristic
65	Dic₁₉₁	—	4	1	no	—
66	Dic₁₉₁	—	4	1	no	—
67	C₂×C₁₉₁	—	2	1	no	—
68	C₂×C₁₉₁	—	8	1	no	—
69	C₂×C₁₉₁	C₄×D₈	1	1	yes	characteristic
70	D₁₉₁	—	8	1	no	—
71	C₁₉₁	#2179	1	1	yes	characteristic, complemented, Sylow
72	#2179	—	191	1	no	maximal, Sylow
73	#508	—	191	1	no	—
74	#443	—	191	1	no	—
75	#443	—	191	1	no	—
76	#368	—	191	1	no	—
77	#359	—	191	1	no	—
78	#359	D₁₉₁	1	1	yes	characteristic, complemented
79	#346	—	191	1	no	—
80	#135	—	191	1	no	—
81	#135	C₂×D₁₉₁	1	1	yes	characteristic
82	#131	—	382	1	no	—
83	#131	—	382	1	no	—
84	Q₃₂	—	191	1	no	—
85	SD₃₂	—	191	1	no	—
86	SD₃₂	—	191	1	no	—
87	D₁₆	—	191	1	no	—
88	C₂×C₁₆	—	191	1	no	—
89	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
90	#108	—	191	1	no	—
91	#108	C₂×D₁₉₁	1	1	yes	characteristic
92	#104	—	382	1	no	—
93	#104	—	382	1	no	—
94	C₄×C₈	—	191	1	no	—
95	#41	—	2	1	no	—
96	#41	—	382	1	no	—
97	Q₁₆	—	191	1	no	—
98	Q₁₆	C₄×D₁₉₁	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	382	1	no	—
101	D₈	—	191	1	no	—
102	D₈	C₄×D₁₉₁	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	382	1	no	—
105	M₁₆	—	764	1	no	—
106	M₁₆	—	764	1	no	—
107	C₂×C₈	—	191	1	no	—
108	C₂×C₈	—	764	1	no	—
109	C₂×C₈	—	764	1	no	—
110	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
111	C₄²	—	191	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	191	1	no	—
114	C₁₆	—	191	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	382	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	382	1	no	—
120	D₄	—	764	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	382	1	no	—
123	C₂×C₄	—	764	1	no	—
124	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	191	1	no	—
127	C₈	—	191	1	no	—
128	C₈	—	764	1	no	—
129	C₈	C₂×C₄×D₁₉₁	1	1	yes	characteristic
130	C₈	#15690286	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	764	1	no	—
133	C₂²	#20259343	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	764	1	no	—
136	C₄	—	764	1	no	—
137	C₄	(C₂²⋊D₁₉₁)⋊C₄	1	1	yes	characteristic
138	C₄	D₈×D₁₉₁	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	1528	1	no	—
142	C₂	#27132743	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₉₁
commutator	C₈×C₁₉₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#359×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₄×D₈
prime radical	#27133093	C₂
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#359	D₁₉₁
hypocenter	C₁₉₁	#2179
LCS	G ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
2. #27132780	C₂	semidirect
3. #27132779	C₂	semidirect
4. #27132492	C₂	semidirect
5. #27132471	C₂	semidirect
6. #27132438	C₂	semidirect
7. #359×C₁₉₁	C₂	semidirect
11. #20259268	C₄	semidirect
12. #20259266	C₄	semidirect
13. #20259265	C₄	semidirect
14. #20259263	C₄	semidirect
71. C₁₉₁	#2179	semidirect
78. #359	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133093	2
#27132780	2
#27132779	2
#27132492	2
#27132471	2
#27132438	2
#359×C₁₉₁	2
#2179	191