Group #27150677 — subgroups

Subgroups — #27150677

Order 12416.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260866	C₂	1	8	yes	maximal, complemented
2	#20260762	C₂	1	2	yes	maximal
3	#20260759	C₂	1	1	yes	maximal, characteristic
4	#20260691	C₂	1	2	yes	maximal
5	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
6	#533×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13095495	C₂²	1	2	yes	—
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	C₂²×#11715895	C₂²	1	1	yes	characteristic
10	C₄⋊#11715895	C₂²	1	4	yes	—
11	C₄⋊#11715895	C₂²	1	8	yes	—
12	#15690721	C₂²	1	4	yes	—
13	#15690721	C₂²	1	4	yes	—
14	C₄×#11715895	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₉₇	C₄	1	2	yes	—
16	#125×C₉₇	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₉₇	C₄	1	4	yes	—
18	C₄×Q₈×C₉₇	C₂²	1	4	yes	—
19	#117×C₉₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
21	#13095495	—	2	4	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	Q₈	1	4	yes	complemented
24	C₂×#11715895	C₂³	1	2	yes	—
25	C₂×#11715895	C₂³	1	4	yes	—
26	C₂×#11715895	C₂³	1	2	yes	—
27	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₉₇	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
34	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₉₇	C₂³	1	2	yes	—
37	C₄²×C₉₇	C₂³	1	4	yes	—
38	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₉₇	—	4	4	no	—
40	C₂×C₄×C₉₇	—	4	2	no	—
41	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
48	#11715895	—	2	2	no	—
49	#11715895	—	2	4	no	—
50	#11715895	C₂×Q₈	1	4	yes	—
51	C₂²×C₉₇	—	2	2	no	—
52	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₉₇	—	2	2	no	—
55	C₄×C₉₇	—	2	2	no	—
56	C₄×C₉₇	—	2	8	no	—
57	C₄×C₉₇	C₄×Q₈	1	2	yes	—
58	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
60	C₂×C₉₇	—	2	2	no	—
61	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₉₇	#568	1	2	yes	—
63	#2961	—	97	1	no	maximal, Sylow
64	C₉₇	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	D₉₇	1	1	yes	characteristic
66	#446	—	97	8	no	—
67	#424	—	97	2	no	—
68	#423	—	97	1	no	—
69	#405	—	97	2	no	—
70	C₂×#97	—	97	1	no	—
71	C₂×M₁₆	—	97	2	no	—
72	C₂²×C₈	—	97	1	no	—
73	C₂²×C₈	—	97	1	no	—
74	C₄⋊Q₈	Dic₉₇	1	2	yes	—
75	#125	Dic₉₇	1	2	yes	—
76	C₂²⋊Q₈	Dic₉₇	1	4	yes	—
77	C₄×Q₈	C₂×D₉₇	1	4	yes	—
78	#117	C₂×D₉₇	1	2	yes	—
79	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
80	C₄⋊C₈	—	97	4	no	—
81	C₄⋊C₈	—	97	8	no	—
82	#97	—	97	4	no	—
83	#97	—	97	4	no	—
84	C₄×C₈	—	97	4	no	—
85	C₂×Q₈	C₂×Dic₉₇	1	4	yes	—
86	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	M₁₆	—	194	4	no	—
89	C₂×C₈	—	97	4	no	—
90	C₂×C₈	—	97	2	no	—
91	C₂×C₈	—	97	4	no	—
92	C₂×C₈	—	97	2	no	—
93	C₂×C₈	—	194	2	no	—
94	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₉₇	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₉₇	1	4	yes	—
98	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
100	C₄²	C₂²×D₉₇	1	4	yes	—
101	C₄²	C₂²×D₉₇	1	2	yes	—
102	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
107	C₂×C₄	C₂²×Dic₉₇	1	8	yes	—
108	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₉₇	1	2	yes	—
110	C₂×C₄	#13095512	1	2	yes	—
111	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
112	C₈	—	97	4	no	—
113	C₈	—	194	2	no	—
114	C₈	—	194	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₉₇	1	2	yes	—
117	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₉₇	1	2	yes	—
122	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
123	C₄	C₂×#13095512	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₉₇	1	1	yes	characteristic
126	C₂	#20261502	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260866

C₂

yes

maximal, complemented

#20260762

C₂

yes

maximal

#20260759

C₂

yes

maximal, characteristic

#20260691

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

#533×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

characteristic

C₄⋊#11715895

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

#15690721

C₂²

yes

—

#15690721

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₄

yes

—

#125×C₉₇

C₄

yes

—

C₂²⋊Q₈×C₉₇

C₄

yes

—

C₄×Q₈×C₉₇

C₂²

yes

—

#117×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂³

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

—

—

C₂×Q₈

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×Q₈

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²×Q₈

yes

characteristic

C₄×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₉₇

#568

yes

—

#2961

—

maximal, Sylow

C₉₇

#2961

yes

characteristic, complemented, Sylow

#533

D₉₇

yes

—

—

—

—

C₂×#97

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₉₇

yes

—

#125

Dic₉₇

yes

—

C₂²⋊Q₈

Dic₉₇

yes

—

C₄×Q₈

C₂×D₉₇

yes

—

#117

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₂²⋊C₄

C₂×Dic₉₇

yes

—

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂×Dic₉₇

yes

characteristic

100

C₄²

C₂²×D₉₇

yes

—

101

C₄²

C₂²×D₉₇

yes

—

102

C₂³

C₂²×Dic₉₇

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₉₇

yes

—

106

C₂×C₄

C₂²×Dic₉₇

yes

—

107

C₂×C₄

C₂²×Dic₉₇

yes

—

108

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

109

C₂×C₄

Q₈×D₉₇

yes

—

110

C₂×C₄

#13095512

yes

—

111

C₂×C₄

C₂³×D₉₇

yes

characteristic

112

C₈

—

113

C₈

—

194

—

114

C₈

—

194

—

115

C₂²

—

116

C₂²

Q₈×Dic₉₇

yes

—

117

C₂²

C₂³×Dic₉₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₉₇

yes

—

122

C₄

C₂×Q₈×D₉₇

yes

characteristic

123

C₄

C₂×#13095512

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₉₇

yes

characteristic

126

C₂

#20261502

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#533×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#533	D₉₇
hypocenter	C₉₇	#2961
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Fitting

#533×C₉₇

C₂

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂×C₄×C₉₇

C₂⁴

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #20260866	C₂	semidirect
23. C₂×#11715895	Q₈	semidirect
64. C₉₇	#2961	semidirect

H ≅ G/N

type

1. #20260866

C₂

semidirect

23. C₂×#11715895

Q₈

semidirect

64. C₉₇

#2961

semidirect

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

subgroup	index
#20260866	2
#20260762	2
#20260759	2
#20260691	2
C₂×#15690721	2
#533×C₉₇	2
#2961	97

subgroup

index

#533×C₉₇

Subgroups — #27150677

Conjugacy classes of subgroups

Complements

Maximal subgroups