Group #27152070 — subgroups

Subgroups — C₂×#20261099 #27152070

Order 12416.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690802	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261099	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690731	C₂	1	4	yes	maximal, complemented
4	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690802	—	2	2	no	—
8	#15690802	C₂²	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×#13095490	—	2	2	no	—
11	#15690731	C₂²	1	16	yes	complemented
12	#15690721	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₉₇	—	2	2	no	—
14	C₂²×D₄×C₉₇	C₂²	1	2	yes	—
15	#127×C₉₇	—	2	2	no	—
16	#127×C₉₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₇	—	2	2	no	—
19	C₂×#11715895	—	2	4	no	—
20	C₂×#11715895	C₂³	1	4	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	C₂³	1	4	yes	—
23	#13095490	—	2	8	no	—
24	C₂⁴×C₉₇	—	2	2	no	—
25	C₂×D₄×C₉₇	—	2	2	no	—
26	C₂×D₄×C₉₇	—	2	8	no	—
27	C₂×D₄×C₉₇	—	2	8	no	—
28	C₂×D₄×C₉₇	—	4	8	no	—
29	C₂×D₄×C₉₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₉₇	D₄	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₄²×C₉₇	D₄	1	2	yes	—
33	C₄²×C₉₇	C₂³	1	2	yes	—
34	C₂³×C₉₇	—	2	4	no	—
35	C₂³×C₉₇	—	4	4	no	—
36	C₂³×C₉₇	—	4	8	no	—
37	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
38	D₄×C₉₇	—	2	8	no	—
39	D₄×C₉₇	—	4	16	no	—
40	D₄×C₉₇	—	4	4	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	#41	1	4	yes	—
45	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₉₇	—	2	4	no	—
47	C₂×Dic₉₇	—	4	4	no	—
48	#11715895	—	2	4	no	—
49	C₂²×C₉₇	—	4	4	no	—
50	C₂²×C₉₇	—	4	8	no	—
51	C₂²×C₉₇	—	8	8	no	—
52	C₂²×C₉₇	#124	1	2	yes	—
53	C₂²×C₉₇	AGL(1,ℤ/8)	1	4	yes	—
54	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₉₇	—	2	4	no	—
56	C₄×C₉₇	#124	1	2	yes	—
57	C₄×C₉₇	C₂×#41	1	2	yes	—
58	Dic₉₇	—	4	4	no	—
59	C₂×C₉₇	—	8	4	no	—
60	C₂×C₉₇	#490	1	4	yes	complemented
61	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
62	C₂×C₉₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
63	C₂×#490	—	97	1	no	maximal, Sylow
64	C₉₇	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	D₉₇	1	1	yes	characteristic
66	C₂×#125	—	97	1	no	—
67	#490	—	97	8	no	—
68	C₂×#102	—	97	4	no	—
69	C₂×#97	—	97	1	no	—
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₉₇	1	2	yes	—
72	C₂²×C₈	—	97	2	no	—
73	#127	—	2	2	no	—
74	#127	C₂×D₉₇	1	4	yes	—
75	#125	—	97	4	no	—
76	#125	—	194	2	no	—
77	C₂×C₄⋊C₄	—	97	2	no	—
78	C₂×C₄⋊C₄	—	194	2	no	—
79	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
80	#102	—	97	16	no	—
81	#97	—	97	4	no	—
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₉₇	1	4	yes	—
88	C₂²×C₄	—	194	2	no	—
89	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₈	—	97	4	no	—
92	C₂×C₈	—	194	4	no	—
93	C₄⋊C₄	—	97	4	no	—
94	C₄⋊C₄	—	194	8	no	—
95	C₄⋊C₄	—	194	2	no	—
96	C₄²	C₂²⋊D₉₇	1	2	yes	—
97	C₄²	C₂²×D₉₇	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	4	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	388	4	no	—
107	C₂×C₄	—	194	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₉₇	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
110	C₂×C₄	#13095510	1	4	yes	—
111	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
112	C₈	—	194	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#15690797	1	2	yes	—
117	C₂²	#15690836	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	388	4	no	—
121	C₄	#15690797	1	2	yes	—
122	C₄	C₂×#13095510	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#20261099	1	4	yes	direct product factor
125	C₂	C₂×#15690797	1	1	yes	characteristic
126	C₂	C₂×#15690836	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic, complemented

#20261099

C₂

yes

maximal, direct product factor

C₂×#15690731

C₂

yes

maximal, complemented

C₂×#15690721

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690802

—

#15690802

C₂²

yes

complemented

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690731

C₂²

yes

complemented

#15690721

C₂²

yes

complemented

C₂²×D₄×C₉₇

—

C₂²×D₄×C₉₇

C₂²

yes

—

#127×C₉₇

—

#127×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₂⁴×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×#490

—

maximal, Sylow

C₉₇

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

D₉₇

yes

—

—

—

C₂×#97

—

C₂²×D₄

—

C₂²×D₄

C₂×D₉₇

yes

—

C₂²×C₈

—

#127

—

#127

C₂×D₉₇

yes

—

#125

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#102

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₉₇

yes

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²×D₉₇

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

388

—

107

C₂×C₄

—

194

—

108

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

110

C₂×C₄

#13095510

yes

—

111

C₂×C₄

C₂³×D₉₇

yes

characteristic

112

C₈

—

194

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#15690797

yes

—

117

C₂²

#15690836

yes

—

118

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

119

C₄

—

120

C₄

—

388

—

121

C₄

#15690797

yes

—

122

C₄

C₂×#13095510

yes

—

123

C₂

—

124

C₂

#20261099

yes

direct product factor

125

C₂

C₂×#15690797

yes

characteristic

126

C₂

C₂×#15690836

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting, prime radical	C₂×#127×C₉₇	C₂
socle	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×#127	D₉₇
hypocenter	C₉₇	C₂×#490
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting, prime radical

C₂×#127×C₉₇

C₂

socle

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15690802	C₂	semidirect
2. #20261099	C₂	direct (N × H)
3. C₂×#15690731	C₂	semidirect
4. C₂×#15690721	C₂	semidirect
8. #15690802	C₂²	semidirect
11. #15690731	C₂²	semidirect
12. #15690721	C₂²	semidirect
60. C₂×C₉₇	#490	semidirect
64. C₉₇	C₂×#490	semidirect
124. C₂	#20261099	direct (N × H)

H ≅ G/N

type

1. C₂×#15690802

C₂

semidirect

2. #20261099

C₂

direct (N × H)

3. C₂×#15690731

C₂

semidirect

4. C₂×#15690721

C₂

semidirect

8. #15690802

C₂²

semidirect

11. #15690731

C₂²

semidirect

12. #15690721

C₂²

semidirect

60. C₂×C₉₇

#490

semidirect

64. C₉₇

C₂×#490

semidirect

124. C₂

#20261099

direct (N × H)

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

subgroup	index
C₂×#15690802	2
#20261099	2
C₂×#15690731	2
C₂×#15690721	2
C₂×#127×C₉₇	2
C₂×#490	97

subgroup

index

C₂×#127×C₉₇

Subgroups — C₂×#20261099 #27152070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20261099 #27152070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261099 #27152070