Finite Group Databasebeta

Subgroups — #37574369

Order 24448.

162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂×#20259265	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132745	C₂	1	1	yes	maximal, characteristic
4	#27132490	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132465	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₅₂₈	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₉₁	—	2	1	no	—
10	#20259284	—	2	1	no	—
11	#20259265	C₄	1	4	yes	complemented
12	C₂×C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#20259256	C₂²	1	1	yes	characteristic
14	#20259246	—	2	1	no	—
15	#20259242	—	2	1	no	—
16	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
19	#107×C₁₉₁	C₂²	1	1	yes	characteristic
20	C₂×D₇₆₄	—	2	1	no	—
21	C₂×C₄×D₁₉₁	—	4	1	no	—
22	D₁₅₂₈	—	2	1	no	—
23	D₁₅₂₈	C₂×C₄	1	2	yes	—
24	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₁₉₁	—	2	1	no	—
26	C₄⋊Dic₁₉₁	—	4	1	no	—
27	(C₂×D₁₉₁)⋊C₄	—	4	1	no	—
28	C₄×Dic₁₉₁	—	4	1	no	—
29	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
30	C₁₉₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₁₉₁	—	2	1	no	—
32	Q₁₆×C₁₉₁	—	2	1	no	—
33	Q₁₆×C₁₉₁	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₁	—	2	1	no	—
36	C₁₆×C₁₉₁	—	2	1	no	—
37	C₂²×D₁₉₁	—	4	1	no	—
38	D₇₆₄	—	2	2	no	—
39	D₇₆₄	—	4	1	no	—
40	C₂×Dic₁₉₁	—	4	1	no	—
41	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
42	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
43	C₄×D₁₉₁	—	8	2	no	—
44	C₁₉₁⋊C₈	—	2	1	no	—
45	Q₈×C₁₉₁	—	2	2	no	—
46	Q₈×C₁₉₁	—	4	1	no	—
47	C₂×C₄×C₁₉₁	—	4	1	no	—
48	C₂×C₄×C₁₉₁	—	4	1	no	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₉₁	#41	1	1	yes	characteristic
52	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₉₁	—	4	2	no	—
54	C₂×D₁₉₁	—	8	1	no	—
55	C₂×D₁₉₁	—	8	1	no	—
56	C₄×C₁₉₁	—	4	2	no	—
57	C₄×C₁₉₁	—	8	1	no	—
58	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₉₁	#135	1	1	yes	characteristic
60	Dic₁₉₁	—	2	1	no	—
61	Dic₁₉₁	—	8	1	no	—
62	Dic₁₉₁	SD₃₂	1	2	yes	complemented
63	C₂×C₁₉₁	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₉₁	C₂×SD₃₂	1	1	yes	characteristic
65	C₂×C₁₉₁	#508	1	1	yes	characteristic
66	D₁₉₁	—	8	2	no	—
67	C₁₉₁	C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₄×SD₃₂	—	191	1	no	maximal, Sylow
69	C₂×SD₃₂	—	191	1	no	—
70	C₄×Q₁₆	—	191	1	no	—
71	C₄×D₈	—	191	1	no	—
72	#367	—	191	1	no	—
73	#358	D₁₉₁	1	1	yes	characteristic, complemented
74	#357	—	191	1	no	—
75	C₄×C₁₆	—	191	1	no	—
76	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
77	C₂×D₈	—	191	1	no	—
78	C₄×Q₈	—	382	1	no	—
79	C₄×D₄	—	382	1	no	—
80	SD₃₂	—	191	4	no	—
81	C₂×C₁₆	—	191	1	no	—
82	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
83	#107	—	191	1	no	—
84	#107	C₂×D₁₉₁	1	1	yes	characteristic
85	#103	—	382	1	no	—
86	#102	—	382	1	no	—
87	C₄×C₈	—	191	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	382	1	no	—
90	C₂²×C₄	—	764	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₁₉₁	1	2	yes	complemented
93	D₈	—	382	1	no	—
94	D₈	—	191	2	no	—
95	C₂×C₈	—	191	1	no	—
96	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	382	1	no	—
99	C₄⋊C₄	—	764	1	no	—
100	C₂²⋊C₄	—	764	1	no	—
101	C₄²	—	191	1	no	—
102	C₄²	—	764	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	191	2	no	—
105	C₂³	—	764	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	382	2	no	—
109	D₄	—	764	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	191	1	no	—
113	C₂×C₄	—	191	1	no	—
114	C₂×C₄	—	1528	2	no	—
115	C₂×C₄	—	764	1	no	—
116	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
117	C₈	—	382	1	no	—
118	C₈	C₂×C₄×D₁₉₁	1	1	yes	characteristic
119	C₈	#15690286	1	1	yes	characteristic
120	C₂²	—	1528	1	no	—
121	C₂²	—	1528	1	no	—
122	C₂²	—	764	2	no	—
123	C₂²	D₈×D₁₉₁	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	382	1	no	—
127	C₄	—	191	2	no	—
128	C₄	—	1528	1	no	—
129	C₄	(C₂²⋊D₁₉₁)⋊C₄	1	1	yes	characteristic
130	C₄	#20259343	1	1	yes	characteristic
131	C₂	—	1528	2	no	—
132	C₂	#27132743	1	1	yes	characteristic
133	C₂	SD₃₂×D₁₉₁	1	1	yes	characteristic
134	C₂	#27133096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₉₁
commutator	C₈×C₁₉₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#358×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₈
prime radical	C₂×D₁₅₂₈	C₂²
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#358	D₁₉₁
hypocenter	C₁₉₁	C₄×SD₃₂
LCS	G ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₉₁	C₂	semidirect
4. #27132490	C₂	semidirect
6. C₁₆×Dic₁₉₁	C₂	semidirect
7. #358×C₁₉₁	C₂	semidirect
11. #20259265	C₄	semidirect
30. C₁₉₁⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₁₉₁	C₂×C₄	semidirect
62. Dic₁₉₁	SD₃₂	semidirect
67. C₁₉₁	C₄×SD₃₂	semidirect
73. #358	D₁₉₁	semidirect
92. Q₁₆	C₄×D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259265	2
Q₁₆×Dic₁₉₁	2
#27132745	2
#27132490	2
#27132465	2
C₁₆×Dic₁₉₁	2
#358×C₁₉₁	2
C₄×SD₃₂	191