Group #27157654 — subgroups

Subgroups — #27157654

Order 12416.

216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₂×#15690877	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715897	C₂	1	1	yes	maximal, characteristic
3	#20261601	C₂	1	4	yes	maximal
4	#20260580	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715897	C₂²	1	2	yes	—
6	C₂²×#11715897	C₂²	1	2	yes	—
7	#15690877	—	2	2	no	—
8	C₄⋊#11715897	—	2	2	no	—
9	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
11	#15690717	C₄	1	2	yes	—
12	#15690717	C₄	1	1	yes	characteristic
13	#95×C₉₇	C₄	1	1	yes	characteristic
14	C₂×#11715897	—	2	2	no	—
15	C₂×#11715897	—	2	2	no	—
16	C₂×#11715897	—	2	4	no	—
17	C₂×#11715897	—	2	4	no	—
18	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
21	C₄×Dic₉₇	—	2	2	no	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₄×Dic₉₇	D₄	1	2	yes	—
24	C₄×Dic₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
33	C₂×Dic₉₇	—	2	1	no	—
34	C₂×Dic₉₇	—	2	1	no	—
35	C₂×Dic₉₇	—	2	2	no	—
36	C₂×Dic₉₇	—	2	2	no	—
37	C₂×Dic₉₇	—	2	2	no	—
38	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
39	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₉₇	#41	1	2	yes	—
42	C₂×Dic₉₇	#41	1	2	yes	—
43	#11715897	—	4	2	no	—
44	#11715897	—	4	2	no	—
45	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₉₇	#117	1	2	yes	—
47	C₂²×C₉₇	#131	1	2	yes	—
48	C₂²×C₉₇	#131	1	1	yes	characteristic
49	C₂²×C₉₇	#131	1	1	yes	characteristic
50	C₄×C₉₇	—	2	2	no	—
51	C₄×C₉₇	—	4	2	no	—
52	Dic₉₇	—	2	2	no	—
53	Dic₉₇	—	4	2	no	—
54	Dic₉₇	#124	1	1	yes	characteristic
55	Dic₉₇	#124	1	1	yes	characteristic
56	Dic₉₇	#125	1	2	yes	—
57	C₂×C₉₇	#382	1	1	yes	characteristic
58	C₂×C₉₇	#424	1	1	yes	characteristic
59	C₂×C₉₇	#424	1	1	yes	characteristic
60	C₂×C₉₇	#433	1	2	yes	—
61	C₂×C₉₇	#433	1	2	yes	—
62	#1832	—	97	1	no	maximal, Sylow
63	C₉₇	#1832	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	97	1	no	—
65	C₂×C₄⋊C₈	—	97	1	no	—
66	#376	—	97	1	no	—
67	#336	—	97	4	no	—
68	C₂²×C₈	—	97	2	no	—
69	C₂²×C₈	—	97	2	no	—
70	C₂×C₄²	—	97	1	no	—
71	C₂×C₄²	—	97	2	no	—
72	C₄⋊C₈	—	194	2	no	—
73	C₄⋊C₈	—	194	2	no	—
74	#95	—	97	2	no	—
75	#95	—	97	1	no	—
76	#95	#87604	1	1	yes	characteristic
77	C₂²×C₄	—	97	1	no	—
78	C₂²×C₄	—	97	1	no	—
79	C₂²×C₄	—	97	2	no	—
80	C₂²×C₄	C₂×#87604	1	2	yes	—
81	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
82	C₂×C₈	—	194	2	no	—
83	C₂×C₈	—	194	2	no	—
84	C₂×C₈	—	194	4	no	—
85	C₂×C₈	—	194	4	no	—
86	C₄²	—	97	2	no	—
87	C₄²	—	97	2	no	—
88	C₄²	—	194	2	no	—
89	C₄²	—	194	2	no	—
90	C₂³	C₂²×#87604	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	97	1	no	—
96	C₂×C₄	—	97	1	no	—
97	C₂×C₄	—	97	2	no	—
98	C₂×C₄	—	97	2	no	—
99	C₂×C₄	—	97	2	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	—	194	2	no	—
102	C₂×C₄	—	194	2	no	—
103	C₂×C₄	—	194	1	no	—
104	C₂×C₄	—	194	1	no	—
105	C₂×C₄	C₄⋊#87604	1	2	yes	—
106	C₈	—	388	2	no	—
107	C₈	—	388	2	no	—
108	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
109	C₂²	#15690889	1	2	yes	—
110	C₂²	#15690897	1	2	yes	—
111	C₂²	#15690897	1	1	yes	characteristic
112	C₂²	#15690898	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	97	1	no	—
116	C₄	—	97	1	no	—
117	C₄	—	97	2	no	—
118	C₄	—	388	2	no	—
119	C₄	—	194	2	no	—
120	C₂	#20261679	1	1	yes	characteristic
121	C₂	#20261741	1	1	yes	characteristic
122	C₂	#20261742	1	1	yes	characteristic
123	C₂	#20261759	1	2	yes	—
124	C₂	#20261760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690877

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715897

C₂

yes

maximal, characteristic

#20261601

C₂

yes

maximal

#20260580

C₂

yes

maximal, characteristic

C₂²×#11715897

C₂²

yes

—

C₂²×#11715897

C₂²

yes

—

#15690877

—

C₄⋊#11715897

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690717

C₄

yes

—

#15690717

C₄

yes

characteristic

#95×C₉₇

C₄

yes

—

—

—

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂³

yes

characteristic

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

#41

yes

—

C₂×Dic₉₇

#41

yes

—

#11715897

—

#11715897

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

#131

yes

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

yes

Dic₉₇

yes

Dic₉₇

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#433

yes

—

#1832

—

maximal, Sylow

C₉₇

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#87604

yes

—

C₂²×C₄

C₂×#87604

yes

characteristic

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₂³

C₂²×#87604

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

C₄⋊#87604

yes

—

106

C₈

—

388

—

107

C₈

—

388

—

108

C₂²

C₂×C₄⋊#87604

yes

characteristic

109

C₂²

#15690889

yes

—

110

C₂²

#15690897

yes

—

111

C₂²

#15690897

yes

characteristic

112

C₂²

#15690898

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

388

—

119

C₄

—

194

—

120

C₂

#20261679

yes

characteristic

121

C₂

#20261741

yes

characteristic

122

C₂

#20261742

yes

characteristic

123

C₂

#20261759

yes

—

124

C₂

#20261760

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	#95×C₉₇	C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	#95	#87604
hypocenter	C₉₇	#1832
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#87604

commutator, socle, prime radical

C₂³×C₉₇

C₂²×C₄

Fitting

#95×C₉₇

C₄

cosocle

C₂²×Dic₉₇

C₂³

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
63. C₉₇	#1832	semidirect

H ≅ G/N

type

63. C₉₇

#1832

semidirect

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

subgroup	index
C₂×#15690877	2
C₂×C₄⋊#11715897	2
#20261601	2
#20260580	2
#1832	97

subgroup

index

Subgroups — #27157654

Conjugacy classes of subgroups

Complements

Maximal subgroups