Group #27142538 — subgroups

Subgroups — #27142538

Order 12416.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690735	C₂	1	4	yes	maximal
3	#20260605	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
6	#15690810	—	2	2	no	—
7	#15690810	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₉₇	—	2	2	no	—
9	C₂×C₄⋊Dic₉₇	—	2	2	no	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	#15690735	—	2	8	no	—
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉₇	—	2	2	no	—
15	C₂×Dic₁₉₄	—	2	4	no	—
16	C₂×Dic₁₉₄	—	2	4	no	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₉₇	—	2	2	no	—
20	C₂²×Dic₉₇	—	2	2	no	—
21	C₄⋊Dic₉₇	—	2	8	no	—
22	C₄⋊Dic₉₇	—	2	4	no	—
23	C₄⋊Dic₉₇	—	4	4	no	—
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	D₄	1	4	yes	—
29	C₄²×C₉₇	C₂×C₄	1	2	yes	—
30	C₄²×C₉₇	D₄	1	2	yes	—
31	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
34	C₂×C₄×C₉₇	Q₁₆	1	4	yes	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₉₇	#41	1	2	yes	—
39	C₂×Dic₉₇	—	2	4	no	—
40	C₂×Dic₉₇	—	4	2	no	—
41	C₂×Dic₉₇	—	4	2	no	—
42	C₂×Dic₉₇	—	4	2	no	—
43	C₂×Dic₉₇	—	4	4	no	—
44	Dic₁₉₄	—	2	8	no	—
45	Dic₁₉₄	—	4	4	no	—
46	C₈×C₉₇	—	2	4	no	—
47	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₉₇	#124	1	1	yes	characteristic
49	C₂²×C₉₇	#127	1	1	yes	characteristic
50	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₉₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₉₇	#103	1	8	yes	complemented
53	C₄×C₉₇	C₄×D₄	1	2	yes	—
54	C₄×C₉₇	#121	1	2	yes	—
55	Dic₉₇	—	4	4	no	—
56	Dic₉₇	—	8	2	no	—
57	C₂×C₉₇	#390	1	1	yes	characteristic
58	C₂×C₉₇	C₂×#103	1	2	yes	—
59	C₂×C₉₇	#487	1	2	yes	—
60	C₂×C₉₇	#492	1	1	yes	characteristic
61	C₂×C₉₇	#494	1	1	yes	characteristic
62	#1956	—	97	1	no	maximal, Sylow
63	C₉₇	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	97	1	no	—
65	C₂×#103	—	97	4	no	—
66	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
67	#384	—	97	1	no	—
68	C₂²×Q₈	—	97	2	no	—
69	C₂²×C₈	C₂×D₉₇	1	2	yes	—
70	C₄⋊Q₈	—	97	4	no	—
71	C₄⋊Q₈	—	194	2	no	—
72	C₂×C₄⋊C₄	—	97	2	no	—
73	C₂×C₄⋊C₄	—	194	2	no	—
74	C₂×C₄⋊C₄	—	194	2	no	—
75	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
76	#103	—	194	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	97	4	no	—
79	C₂×Q₈	—	194	4	no	—
80	C₂×Q₈	—	194	4	no	—
81	C₂×Q₈	—	194	2	no	—
82	C₂²×C₄	—	194	2	no	—
83	C₂²×C₄	—	194	2	no	—
84	C₂²×C₄	D₃₈₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₃₈₈	1	4	yes	—
89	C₄⋊C₄	—	388	4	no	—
90	C₄⋊C₄	—	194	4	no	—
91	C₄⋊C₄	—	194	8	no	—
92	C₄²	C₄×D₉₇	1	2	yes	—
93	C₄²	C₂²⋊D₉₇	1	2	yes	—
94	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
95	Q₈	—	388	4	no	—
96	Q₈	—	194	8	no	—
97	C₂×C₄	—	388	2	no	—
98	C₂×C₄	—	388	4	no	—
99	C₂×C₄	—	388	2	no	—
100	C₂×C₄	—	388	2	no	—
101	C₂×C₄	—	194	4	no	—
102	C₂×C₄	#13095488	1	4	yes	—
103	C₂×C₄	#13095500	1	4	yes	—
104	C₂×C₄	Dic₃₈₈	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
108	C₂×C₄	#13095511	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×#13095488	1	1	yes	characteristic
111	C₂²	#15690798	1	1	yes	characteristic
112	C₂²	#15690807	1	1	yes	characteristic
113	C₂²	C₂×#13095500	1	2	yes	—
114	C₂²	C₂×Dic₃₈₈	1	2	yes	—
115	C₄	—	776	2	no	—
116	C₄	—	388	4	no	—
117	C₄	#15690735	1	8	yes	complemented
118	C₄	C₄×D₃₈₈	1	2	yes	—
119	C₄	#15690784	1	2	yes	—
120	C₂	#20260637	1	1	yes	characteristic
121	C₂	C₂×#15690735	1	2	yes	—
122	C₂	#20261086	1	2	yes	—
123	C₂	#20261108	1	1	yes	characteristic
124	C₂	#20261115	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690735

C₂

yes

maximal

#20260605

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690735

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

#103

yes

complemented

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

#121

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

C₂×#103

yes

—

C₂×C₉₇

#487

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₉₇

yes

characteristic

#384

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₃₈₈

yes

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₄×D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×D₃₈₈

yes

characteristic

Q₈

—

388

—

Q₈

—

194

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

100

C₂×C₄

—

388

—

101

C₂×C₄

—

194

—

102

C₂×C₄

#13095488

yes

—

103

C₂×C₄

#13095500

yes

—

104

C₂×C₄

Dic₃₈₈

yes

—

105

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

107

C₂×C₄

C₂×D₃₈₈

yes

—

108

C₂×C₄

#13095511

yes

—

109

C₈

—

110

C₂²

C₂×#13095488

yes

characteristic

111

C₂²

#15690798

yes

characteristic

112

C₂²

#15690807

yes

characteristic

113

C₂²

C₂×#13095500

yes

—

114

C₂²

C₂×Dic₃₈₈

yes

—

115

C₄

—

776

—

116

C₄

—

388

—

117

C₄

#15690735

yes

complemented

118

C₄

C₄×D₃₈₈

yes

—

119

C₄

#15690784

yes

—

120

C₂

#20260637

yes

characteristic

121

C₂

C₂×#15690735

yes

—

122

C₂

#20261086

yes

—

123

C₂

#20261108

yes

characteristic

124

C₂

#20261115

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1956
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1956

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #15690810	C₄	semidirect
52. C₄×C₉₇	#103	semidirect
63. C₉₇	#1956	semidirect
117. C₄	#15690735	semidirect

H ≅ G/N

type

7. #15690810

C₄

semidirect

52. C₄×C₉₇

#103

semidirect

63. C₉₇

#1956

semidirect

117. C₄

#15690735

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690735	2
#20260605	2
C₂×C₄×C₈×C₉₇	2
#1956	97

subgroup

index

C₂×C₄×C₈×C₉₇

Subgroups — #27142538

Conjugacy classes of subgroups

Complements

Maximal subgroups