Group #27150670 — subgroups

Subgroups — #27150670

Order 12416.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20260865	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₁₉₄	C₂	1	4	yes	maximal, complemented
4	#20260761	C₂	1	2	yes	maximal
5	#20260759	C₂	1	1	yes	maximal, characteristic
6	#20260690	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095495	C₂²	1	2	yes	—
9	#15690810	C₄	1	2	yes	—
10	#15690802	C₄	1	2	yes	—
11	#15690789	C₄	1	4	yes	—
12	C₄×Dic₁₉₄	C₂²	1	4	yes	—
13	#15690769	C₂²	1	2	yes	—
14	C₄⋊#11715895	C₂²	1	4	yes	—
15	#15690739	C₂²	1	8	yes	—
16	#15690720	C₂²	1	4	yes	—
17	C₈×Dic₉₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₉₇	C₂²	1	2	yes	—
22	C₄×C₈×C₉₇	C₂²	1	2	yes	—
23	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
24	#13095495	—	2	4	no	—
25	C₂×#11715895	C₂³	1	4	yes	—
26	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
28	#13095490	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₉₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₉₇	—	2	2	no	—
34	C₂×C₈×C₉₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₉₇	C₂³	1	2	yes	—
36	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
38	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₉₇	C₂³	1	2	yes	—
41	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	—	2	4	no	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₉₇	C₂²×C₄	1	8	yes	—
50	Dic₁₉₄	—	2	8	no	—
51	C₈×C₉₇	—	2	2	no	—
52	C₈×C₉₇	C₂×Q₈	1	2	yes	—
53	C₈×C₉₇	C₂×Q₈	1	2	yes	—
54	#11715895	—	2	4	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	C₄×Q₈	1	2	yes	—
60	C₄×C₉₇	#131	1	4	yes	—
61	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
63	Dic₉₇	—	2	8	no	—
64	C₂×C₉₇	—	2	2	no	—
65	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₉₇	C₂×#131	1	2	yes	—
67	#2960	—	97	1	no	maximal, Sylow
68	C₉₇	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	97	1	no	—
70	#446	—	97	4	no	—
71	C₈×Q₈	—	97	4	no	—
72	#424	—	97	2	no	—
73	#423	—	97	1	no	—
74	#405	—	97	2	no	—
75	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
76	C₂×M₁₆	—	97	2	no	—
77	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	97	2	no	—
80	#125	—	97	2	no	—
81	C₂²⋊Q₈	—	97	4	no	—
82	C₄×Q₈	—	97	4	no	—
83	#117	—	97	2	no	—
84	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
85	C₄⋊C₈	—	97	8	no	—
86	C₄⋊C₈	—	97	4	no	—
87	#97	—	97	4	no	—
88	C₄×C₈	—	97	4	no	—
89	C₄×C₈	C₂×D₉₇	1	2	yes	—
90	C₄×C₈	C₂×D₉₇	1	2	yes	—
91	C₂×Q₈	—	97	4	no	—
92	C₂²×C₄	C₄×D₉₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
94	M₁₆	—	194	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	97	4	no	—
97	C₂×C₈	Dic₁₉₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₉₇	1	2	yes	—
99	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
101	C₄⋊C₄	—	97	8	no	—
102	C₄⋊C₄	—	97	4	no	—
103	C₄⋊C₄	—	97	4	no	—
104	C₂²⋊C₄	—	97	4	no	—
105	C₄²	—	97	4	no	—
106	C₄²	C₄×D₉₇	1	1	yes	characteristic
107	C₄²	C₄×D₉₇	1	1	yes	characteristic
108	C₄²	C₂²×D₉₇	1	2	yes	—
109	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
110	Q₈	—	194	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	97	8	no	—
113	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
117	C₂×C₄	#13095511	1	2	yes	—
118	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	194	4	no	—
121	C₈	C₂×Dic₁₉₄	1	2	yes	—
122	C₈	C₂×Dic₁₉₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₁₉₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	194	8	no	—
128	C₄	C₄×Dic₁₉₄	1	2	yes	—
129	C₄	#15690816	1	4	yes	—
130	C₄	C₂²×Dic₁₉₄	1	1	yes	characteristic
131	C₄	C₂×#13095511	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₁₉₄	1	1	yes	characteristic
134	C₂	C₂×#15690816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

#20260865

C₂

yes

maximal, complemented

C₈×Dic₁₉₄

C₂

yes

maximal, complemented

#20260761

C₂

yes

maximal

#20260759

C₂

yes

maximal, characteristic

#20260690

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690810

C₄

yes

—

#15690802

C₄

yes

—

#15690789

C₄

yes

—

C₄×Dic₁₉₄

C₂²

yes

—

#15690769

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

#15690739

C₂²

yes

—

#15690720

C₂²

yes

—

C₈×Dic₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

#13095495

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

C₂×C₄

yes

—

C₂²⋊Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

Q₈

yes

complemented

C₂×C₈×C₉₇

C₂³

yes

—

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

C₂²×C₄

yes

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

C₂×Q₈

yes

—

C₈×C₉₇

C₂×Q₈

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×Q₈

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#131

yes

—

C₄×C₉₇

C₂²×Q₈

yes

characteristic

C₄×C₉₇

C₂×#41

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₉₇

C₂×#131

yes

—

#2960

—

maximal, Sylow

C₉₇

#2960

yes

characteristic, complemented, Sylow

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₉₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

—

C₄×C₈

C₂×D₉₇

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₉₄

yes

complemented

C₂×C₈

C₂²×D₉₇

yes

—

C₂×C₈

C₂²×D₉₇

yes

characteristic

100

C₂×C₈

C₂²×D₉₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₉₇

yes

characteristic

107

C₄²

C₄×D₉₇

yes

characteristic

108

C₄²

C₂²×D₉₇

yes

—

109

C₂³

C₂×C₄×D₉₇

yes

characteristic

110

Q₈

—

194

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₉₇

yes

—

114

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

115

C₂×C₄

C₂×C₄×D₉₇

yes

—

116

C₂×C₄

C₂×Dic₁₉₄

yes

—

117

C₂×C₄

#13095511

yes

—

118

C₂×C₄

C₂³×D₉₇

yes

characteristic

119

C₈

—

120

C₈

—

194

—

121

C₈

C₂×Dic₁₉₄

yes

—

122

C₈

C₂×Dic₁₉₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₁₉₄

yes

—

125

C₂²

C₂²×C₄×D₉₇

yes

characteristic

126

C₄

—

127

C₄

—

194

—

128

C₄

C₄×Dic₁₉₄

yes

—

129

C₄

#15690816

yes

—

130

C₄

C₂²×Dic₁₉₄

yes

characteristic

131

C₄

C₂×#13095511

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₁₉₄

yes

characteristic

134

C₂

C₂×#15690816

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#2960
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂×C₄×C₉₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20260865	C₂	semidirect
3. C₈×Dic₁₉₄	C₂	semidirect
34. C₂×C₈×C₉₇	Q₈	semidirect
68. C₉₇	#2960	semidirect
97. C₂×C₈	Dic₁₉₄	semidirect

H ≅ G/N

type

2. #20260865

C₂

semidirect

3. C₈×Dic₁₉₄

C₂

semidirect

34. C₂×C₈×C₉₇

Q₈

semidirect

68. C₉₇

#2960

semidirect

97. C₂×C₈

Dic₁₉₄

semidirect

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

subgroup	index
#20261292	2
#20260865	2
C₈×Dic₁₉₄	2
#20260761	2
#20260759	2
#20260690	2
C₂×C₄×C₈×C₉₇	2
#2960	97

subgroup

index

C₈×Dic₁₉₄

C₂×C₄×C₈×C₉₇

Subgroups — #27150670

Conjugacy classes of subgroups

Complements

Maximal subgroups