Group #27142556 — subgroups

Subgroups — #27142556

Order 12416.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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	#20260601	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×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	C₂²	1	2	yes	—
10	#15690735	—	2	8	no	—
11	#15690717	—	2	2	no	—
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#97×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	C₂²×C₄×C₉₇	D₄	1	2	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	D₄	1	4	yes	—
28	C₄²×C₉₇	C₂×C₄	1	2	yes	—
29	C₄²×C₉₇	D₄	1	2	yes	—
30	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	—	2	4	no	—
32	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₉₇	#41	1	2	yes	—
37	C₂×Dic₉₇	—	2	4	no	—
38	C₂×Dic₉₇	—	4	2	no	—
39	C₂×Dic₉₇	—	4	2	no	—
40	C₂×Dic₉₇	—	4	2	no	—
41	C₂×Dic₉₇	—	4	4	no	—
42	Dic₁₉₄	—	2	8	no	—
43	Dic₁₉₄	—	4	4	no	—
44	C₈×C₉₇	—	2	4	no	—
45	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₉₇	#124	1	1	yes	characteristic
47	C₂²×C₉₇	#127	1	1	yes	characteristic
48	C₂²×C₉₇	#137	1	4	yes	—
49	C₄×C₉₇	—	2	4	no	—
50	C₄×C₉₇	C₄×D₄	1	2	yes	—
51	C₄×C₉₇	#121	1	2	yes	—
52	Dic₉₇	—	4	4	no	—
53	Dic₉₇	—	8	2	no	—
54	C₂×C₉₇	#390	1	1	yes	characteristic
55	C₂×C₉₇	#419	1	2	yes	—
56	C₂×C₉₇	#491	1	2	yes	—
57	C₂×C₉₇	#497	1	2	yes	—
58	#1960	—	97	1	no	maximal, Sylow
59	C₉₇	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	—	97	1	no	—
61	C₂×#103	—	97	4	no	—
62	C₂×#97	D₉₇	1	1	yes	characteristic
63	#382	—	97	1	no	—
64	C₂²×Q₈	—	97	2	no	—
65	C₂²×C₈	C₂×D₉₇	1	2	yes	—
66	C₄⋊Q₈	—	97	4	no	—
67	C₄⋊Q₈	—	194	2	no	—
68	C₂×C₄⋊C₄	—	97	2	no	—
69	C₂×C₄⋊C₄	—	194	2	no	—
70	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
71	#103	—	194	8	no	—
72	#97	—	2	2	no	—
73	#95	—	194	2	no	—
74	C₂×Q₈	—	97	4	no	—
75	C₂×Q₈	—	194	4	no	—
76	C₂×Q₈	—	194	4	no	—
77	C₂×Q₈	—	194	2	no	—
78	C₂²×C₄	—	194	2	no	—
79	C₂²×C₄	—	194	2	no	—
80	C₂²×C₄	D₃₈₈	1	2	yes	—
81	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₃₈₈	1	4	yes	—
85	C₄⋊C₄	—	194	4	no	—
86	C₄⋊C₄	—	194	8	no	—
87	C₄²	C₄×D₉₇	1	2	yes	—
88	C₄²	C₂²⋊D₉₇	1	2	yes	—
89	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
90	Q₈	—	388	4	no	—
91	Q₈	—	194	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	388	2	no	—
94	C₂×C₄	—	388	4	no	—
95	C₂×C₄	—	388	2	no	—
96	C₂×C₄	—	388	2	no	—
97	C₂×C₄	—	194	4	no	—
98	C₂×C₄	#13095488	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
102	C₂×C₄	#13095511	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×#13095488	1	1	yes	characteristic
105	C₂²	#15690798	1	1	yes	characteristic
106	C₂²	#15690807	1	1	yes	characteristic
107	C₂²	#15690843	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	776	2	no	—
110	C₄	—	388	4	no	—
111	C₄	C₄×D₃₈₈	1	2	yes	—
112	C₄	#15690784	1	2	yes	—
113	C₂	#20260637	1	1	yes	characteristic
114	C₂	#20260745	1	2	yes	—
115	C₂	#20261104	1	2	yes	—
116	C₂	#20261129	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690735

C₂

yes

maximal

#20260601

C₂

yes

maximal, characteristic

C₂×#97×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690735

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#97×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

—

C₂²×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

—

C₂×C₄×C₉₇

C₂²⋊C₄

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

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

#121

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#419

yes

—

C₂×C₉₇

#491

yes

—

C₂×C₉₇

#497

yes

—

#1960

—

maximal, Sylow

C₉₇

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

D₉₇

yes

characteristic

#382

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

#97

—

#95

—

194

—

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

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₄×D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×D₃₈₈

yes

characteristic

Q₈

—

388

—

Q₈

—

194

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

101

C₂×C₄

C₂×D₃₈₈

yes

—

102

C₂×C₄

#13095511

yes

—

103

C₈

—

104

C₂²

C₂×#13095488

yes

characteristic

105

C₂²

#15690798

yes

characteristic

106

C₂²

#15690807

yes

characteristic

107

C₂²

#15690843

yes

—

108

C₄

—

109

C₄

—

776

—

110

C₄

—

388

—

111

C₄

C₄×D₃₈₈

yes

—

112

C₄

#15690784

yes

—

113

C₂

#20260637

yes

characteristic

114

C₂

#20260745

yes

—

115

C₂

#20261104

yes

—

116

C₂

#20261129

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#97×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	#1960
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#97×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×#97

D₉₇

hypocenter

C₉₇

#1960

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #15690810	C₄	semidirect
59. C₉₇	#1960	semidirect

H ≅ G/N

type

7. #15690810

C₄

semidirect

59. C₉₇

#1960

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
#20260601	2
C₂×#97×C₉₇	2
#1960	97

subgroup

index

C₂×#97×C₉₇

Subgroups — #27142556

Conjugacy classes of subgroups

Complements

Maximal subgroups