Group #27152238 — subgroups

Subgroups — C₂×#20261135 #27152238

Order 12416.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690801	C₂	1	2	yes	maximal
2	#20261135	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690743	C₂	1	2	yes	maximal
4	C₂×#15690742	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	#15690801	—	2	4	no	—
9	#15690801	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₉₇	—	2	2	no	—
11	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
12	C₂×#13095490	—	2	2	no	—
13	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
14	#15690743	C₂²	1	8	yes	—
15	#15690742	C₂²	1	4	yes	—
16	C₈×Dic₉₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
19	#106×C₉₇	C₂²	1	4	yes	—
20	C₂²×Dic₉₇	—	2	2	no	—
21	C₂²×Dic₉₇	D₄	1	2	yes	—
22	C₂×#11715895	Q₈	1	4	yes	—
23	C₂×#11715895	C₂³	1	2	yes	—
24	C₄⋊Dic₉₇	—	2	2	no	—
25	C₄⋊Dic₉₇	—	2	8	no	—
26	C₄⋊Dic₉₇	C₂³	1	4	yes	—
27	#13095490	—	2	8	no	—
28	C₄×Dic₉₇	D₄	1	2	yes	—
29	C₄×Dic₉₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₉₇	—	2	2	no	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₉₇	Q₈	1	4	yes	—
33	C₂×C₈×C₉₇	C₂³	1	2	yes	—
34	C₄⋊C₄×C₉₇	—	2	2	no	—
35	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
36	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	—	2	4	no	—
38	C₂×C₄×C₉₇	—	4	4	no	—
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	2	4	no	—
45	C₂×Dic₉₇	—	4	4	no	—
46	C₂×Dic₉₇	C₂×D₄	1	4	yes	—
47	C₈×C₉₇	C₂×Q₈	1	4	yes	—
48	#11715895	C₂×Q₈	1	4	yes	—
49	C₂²×C₉₇	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₉₇	#135	1	4	yes	—
51	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₉₇	—	4	4	no	—
53	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
54	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
56	Dic₉₇	—	2	4	no	—
57	Dic₉₇	—	4	4	no	—
58	C₂×C₉₇	#499	1	4	yes	complemented
59	C₂×C₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₉₇	C₂×#135	1	2	yes	—
61	C₂×#499	—	97	1	no	maximal, Sylow
62	C₉₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
63	C₂×#125	—	97	2	no	—
64	#499	—	97	8	no	—
65	C₂×#107	—	97	2	no	—
66	C₂×#106	—	97	1	no	—
67	C₂×#106	D₉₇	1	1	yes	characteristic
68	C₂×C₄×C₈	—	97	1	no	—
69	C₂²×C₈	—	97	1	no	—
70	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
71	#125	—	97	8	no	—
72	#125	—	194	4	no	—
73	C₂×C₄⋊C₄	—	97	2	no	—
74	C₂×C₄⋊C₄	—	194	2	no	—
75	C₂×C₄⋊C₄	—	194	2	no	—
76	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
77	C₂×C₄²	—	97	1	no	—
78	#107	—	97	8	no	—
79	#106	—	97	4	no	—
80	#106	C₂×D₉₇	1	4	yes	—
81	C₄×C₈	—	97	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	97	2	no	—
84	C₂²×C₄	—	194	2	no	—
85	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
86	C₂×C₈	—	97	4	no	—
87	C₂×C₈	—	97	2	no	—
88	C₂×C₈	Dic₁₉₄	1	4	yes	—
89	C₂×C₈	C₂²×D₉₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	97	4	no	—
92	C₄⋊C₄	—	194	8	no	—
93	C₄⋊C₄	—	194	8	no	—
94	C₄⋊C₄	—	194	2	no	—
95	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
96	C₄²	—	97	2	no	—
97	C₄²	—	97	2	no	—
98	C₂³	D₄×D₉₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	97	4	no	—
102	C₂×C₄	—	388	4	no	—
103	C₂×C₄	—	194	4	no	—
104	C₂×C₄	—	194	4	no	—
105	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₉₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
108	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
109	C₈	—	97	4	no	—
110	C₈	C₂×Dic₁₉₄	1	4	yes	—
111	C₂²	#15690809	1	2	yes	—
112	C₂²	#15690831	1	4	yes	—
113	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	388	4	no	—
116	C₄	—	194	4	no	—
117	C₄	#15690809	1	2	yes	—
118	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
119	C₄	C₂²×Dic₁₉₄	1	1	yes	characteristic
120	C₂	#20261135	1	4	yes	direct product factor
121	C₂	C₂×#15690809	1	1	yes	characteristic
122	C₂	C₂×#15690831	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690801

C₂

yes

maximal

#20261135

C₂

yes

maximal, direct product factor

C₂×#15690743

C₂

yes

maximal

C₂×#15690742

C₂

yes

maximal, characteristic

C₂×C₈×Dic₉₇

C₂

yes

maximal, characteristic

C₂×#106×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

characteristic

#15690801

—

#15690801

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690743

C₂²

yes

—

#15690742

C₂²

yes

—

C₈×Dic₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

#106×C₉₇

C₂²

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

D₄

yes

—

C₂×#11715895

Q₈

yes

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

Q₈

yes

—

C₂×C₈×C₉₇

C₂³

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×D₄

yes

—

C₈×C₉₇

C₂×Q₈

yes

—

#11715895

C₂×Q₈

yes

—

C₂²×C₉₇

C₄⋊Q₈

yes

—

C₂²×C₉₇

#135

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₄⋊Q₈

yes

—

C₄×C₉₇

C₂²×Q₈

yes

characteristic

C₄×C₉₇

C₂²×Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#499

yes

complemented

C₂×C₉₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₉₇

C₂×#135

yes

—

C₂×#499

—

maximal, Sylow

C₉₇

C₂×#499

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic

C₂×C₄×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#125

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₉₇

yes

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

C₂²×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂²×D₉₇

yes

—

C₄²

—

C₄²

—

C₂³

D₄×D₉₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

388

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

D₄×D₉₇

yes

characteristic

106

C₂×C₄

Q₈×D₉₇

yes

—

107

C₂×C₄

C₂×Dic₁₉₄

yes

—

108

C₂×C₄

C₂³×D₉₇

yes

characteristic

109

C₈

—

110

C₈

C₂×Dic₁₉₄

yes

—

111

C₂²

#15690809

yes

—

112

C₂²

#15690831

yes

—

113

C₂²

C₂×D₄×D₉₇

yes

characteristic

114

C₄

—

115

C₄

—

388

—

116

C₄

—

194

—

117

C₄

#15690809

yes

—

118

C₄

C₂×Q₈×D₉₇

yes

characteristic

119

C₄

C₂²×Dic₁₉₄

yes

characteristic

120

C₂

#20261135

yes

direct product factor

121

C₂

C₂×#15690809

yes

characteristic

122

C₂

C₂×#15690831

yes

—

	subgroup	G/H
center	C₂³	D₄×D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#106×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×#106	D₉₇
hypocenter	C₉₇	C₂×#499
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×#106×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #20261135	C₂	direct (N × H)
58. C₂×C₉₇	#499	semidirect
62. C₉₇	C₂×#499	semidirect
120. C₂	#20261135	direct (N × H)

H ≅ G/N

type

2. #20261135

C₂

direct (N × H)

58. C₂×C₉₇

#499

semidirect

62. C₉₇

C₂×#499

semidirect

120. C₂

#20261135

direct (N × H)

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

subgroup	index
C₂×#15690801	2
#20261135	2
C₂×#15690743	2
C₂×#15690742	2
C₂×C₈×Dic₉₇	2
C₂×#106×C₉₇	2
C₂×#499	97

subgroup

index

C₂×C₈×Dic₉₇

C₂×#106×C₉₇

C₂×#499

Subgroups — C₂×#20261135 #27152238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20261135 #27152238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261135 #27152238