Group #27150777 — subgroups

Subgroups — #27150777

Order 12416.

452 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 217 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	#20260813	C₂	1	8	yes	maximal, complemented
2	#20260788	C₂	1	2	yes	maximal, complemented
3	#20260699	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	8	yes	complemented
6	C₄⋊#11715895	C₂²	1	4	yes	complemented
7	C₂²⋊#11715895	C₂²	1	4	yes	—
8	C₂²⋊#11715895	C₂²	1	8	yes	complemented
9	C₄×#11715895	C₂²	1	4	yes	complemented
10	C₂²×D₄×C₉₇	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₉₇	C₂²	1	2	yes	—
12	C₄×D₄×C₉₇	C₄	1	4	yes	—
13	C₄×D₄×C₉₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₉₇	C₄	1	2	yes	—
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	#13095495	—	4	8	no	—
18	C₂×#11715895	C₂³	1	8	yes	—
19	C₂⁴×C₉₇	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₉₇	—	2	4	no	—
21	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₉₇	—	4	4	no	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₉₇	C₂×C₄	1	8	yes	—
31	C₄²×C₉₇	C₂×C₄	1	2	yes	—
32	C₄²×C₉₇	C₂³	1	2	yes	—
33	C₂³×C₉₇	—	2	4	no	—
34	C₂³×C₉₇	—	4	2	no	—
35	C₂³×C₉₇	—	4	2	no	—
36	C₂³×C₉₇	C₂²×C₄	1	4	yes	—
37	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₉₇	—	2	8	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	8	no	—
41	C₂×C₄×C₉₇	—	4	4	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	—	4	1	no	—
44	C₂×C₄×C₉₇	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
50	#11715895	—	2	8	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	—	2	4	no	—
54	C₂²×C₉₇	—	2	4	no	—
55	C₂²×C₉₇	—	2	4	no	—
56	C₂²×C₉₇	—	4	8	no	—
57	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
58	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₉₇	—	2	1	no	—
60	C₄×C₉₇	—	2	4	no	—
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	C₂×M₁₆	1	4	yes	—
63	C₄×C₉₇	H(5,𝔽₂)	1	2	yes	—
64	C₂×C₉₇	—	2	2	no	—
65	C₂×C₉₇	—	4	4	no	—
66	C₂×C₉₇	#518	1	1	yes	characteristic
67	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
68	C₂×C₉₇	#568	1	1	yes	characteristic
69	#2975	—	97	1	no	maximal, Sylow
70	C₉₇	#2975	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×D₄	D₉₇	1	1	yes	characteristic
72	#436	—	97	8	no	—
73	#431	—	97	2	no	—
74	#407	—	97	4	no	—
75	C₂²×D₄	Dic₉₇	1	1	yes	characteristic
76	C₂³×C₄	C₂×D₉₇	1	2	yes	—
77	C₂×M₁₆	—	97	8	no	—
78	C₄×D₄	Dic₉₇	1	4	yes	—
79	C₄×D₄	C₂×D₉₇	1	4	yes	—
80	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	Dic₉₇	1	2	yes	—
82	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
83	C₄⋊C₈	—	97	4	no	—
84	C₂²⋊C₈	—	97	4	no	—
85	C₂²⋊C₈	—	97	8	no	—
86	C₄×C₈	—	97	4	no	—
87	C₂⁴	C₂×Dic₉₇	1	2	yes	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	C₂×Dic₉₇	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
94	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
97	M₁₆	—	388	8	no	—
98	C₂×C₈	—	97	8	no	—
99	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
100	C₂²⋊C₄	C₂×Dic₉₇	1	8	yes	—
101	C₄²	C₂×Dic₉₇	1	2	yes	—
102	C₄²	C₂²×D₉₇	1	2	yes	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂²×Dic₉₇	1	4	yes	—
107	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	#13095495	1	4	yes	complemented
115	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
117	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
118	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
119	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
120	C₈	—	194	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	4	8	no	—
127	C₂²	C₂×#13095495	1	2	yes	—
128	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂×#13095495	1	4	yes	—
133	C₄	#15690855	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	4	4	no	—
136	C₂	#20261210	1	1	yes	characteristic
137	C₂	C₂²×#13095495	1	1	yes	characteristic
138	C₂	#20261502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260813

C₂

yes

maximal, complemented

#20260788

C₂

yes

maximal, complemented

#20260699

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

complemented

C₄⋊#11715895

C₂²

yes

complemented

C₂²⋊#11715895

C₂²

yes

—

C₂²⋊#11715895

C₂²

yes

complemented

C₄×#11715895

C₂²

yes

complemented

C₂²×D₄×C₉₇

C₄

yes

characteristic

C₂³×C₄×C₉₇

C₂²

yes

—

C₄×D₄×C₉₇

C₄

yes

—

C₄×D₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#13095495

—

C₂×#11715895

C₂³

yes

—

C₂⁴×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

M₁₆

yes

complemented

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

H(5,𝔽₂)

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#518

yes

characteristic

C₂×C₉₇

C₂²×M₁₆

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

D₉₇

yes

—

—

—

C₂²×D₄

Dic₉₇

yes

characteristic

C₂³×C₄

C₂×D₉₇

yes

—

C₂×M₁₆

—

C₄×D₄

Dic₉₇

yes

—

C₄×D₄

C₂×D₉₇

yes

—

C₂×C₄⋊C₄

Dic₉₇

yes

characteristic

C₂×C₂²⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

C₂×Dic₉₇

yes

—

C₂×D₄

—

C₂×D₄

C₂×Dic₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

M₁₆

—

388

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

100

C₂²⋊C₄

C₂×Dic₉₇

yes

—

101

C₄²

C₂×Dic₉₇

yes

—

102

C₄²

C₂²×D₉₇

yes

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂²×Dic₉₇

yes

—

107

C₂³

C₂²×Dic₉₇

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#13095495

yes

complemented

115

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

116

C₂×C₄

C₂²×Dic₉₇

yes

—

117

C₂×C₄

C₂²×Dic₉₇

yes

—

118

C₂×C₄

C₂²×Dic₉₇

yes

—

119

C₂×C₄

C₂³×D₉₇

yes

characteristic

120

C₈

—

194

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#13095495

yes

—

128

C₂²

C₂³×Dic₉₇

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×#13095495

yes

—

133

C₄

#15690855

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20261210

yes

characteristic

137

C₂

C₂²×#13095495

yes

characteristic

138

C₂

#20261502

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₉₇	C₂
prime radical	C₂²×D₄×C₉₇	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₉₇
hypocenter	C₉₇	#2975
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Fitting

C₂×C₄×D₄×C₉₇

C₂

prime radical

C₂²×D₄×C₉₇

C₄

cosocle

C₂×C₄×C₉₇

C₂⁴

hypercenter

C₂×C₄×D₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #20260813	C₂	semidirect
2. #20260788	C₂	semidirect
3. #20260699	C₂	semidirect
5. C₂×#13095495	C₂²	semidirect
6. C₄⋊#11715895	C₂²	semidirect
8. C₂²⋊#11715895	C₂²	semidirect
9. C₄×#11715895	C₂²	semidirect
44. C₂×C₄×C₉₇	M₁₆	semidirect
70. C₉₇	#2975	semidirect
114. C₂×C₄	#13095495	semidirect

H ≅ G/N

type

1. #20260813

C₂

semidirect

2. #20260788

C₂

semidirect

3. #20260699

C₂

semidirect

5. C₂×#13095495

C₂²

semidirect

6. C₄⋊#11715895

C₂²

semidirect

8. C₂²⋊#11715895

C₂²

semidirect

9. C₄×#11715895

C₂²

semidirect

44. C₂×C₄×C₉₇

M₁₆

semidirect

70. C₉₇

#2975

semidirect

114. C₂×C₄

#13095495

semidirect

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

subgroup	index
#20260813	2
#20260788	2
#20260699	2
C₂×C₄×D₄×C₉₇	2
#2975	97

subgroup

index

C₂×C₄×D₄×C₉₇

Subgroups — #27150777

Conjugacy classes of subgroups

Complements

Maximal subgroups