Group #27157922 — subgroups

Subgroups — #27157922

Order 12416.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#20261740	C₂	1	1	yes	maximal, characteristic
2	C₄²×#87604	C₂	1	1	yes	maximal, characteristic
3	#20261605	C₂	1	4	yes	maximal
4	#20261291	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095521	C₂²	1	2	yes	—
6	C₂×C₄×#87604	—	2	2	no	—
7	C₂×C₄×#87604	C₂²	1	2	yes	—
8	#15690877	—	2	2	no	—
9	#15690809	C₄	1	2	yes	complemented
10	#15690802	C₄	1	1	yes	characteristic, complemented
11	#15690788	—	2	2	no	—
12	#15690775	—	2	2	no	—
13	#15690768	C₂²	1	2	yes	—
14	C₄²×D₉₇	C₂²	1	1	yes	characteristic
15	#125×C₉₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#87604	—	2	2	no	—
17	#13095521	—	2	4	no	—
18	C₂×#11715897	—	2	2	no	—
19	C₄×#87604	—	2	8	no	—
20	C₄×#87604	—	2	2	no	—
21	C₄×#87604	Q₈	1	4	yes	complemented
22	C₂×Dic₁₉₄	—	2	2	no	—
23	C₂×C₄×D₉₇	D₄	1	2	yes	—
24	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
26	#13095490	—	2	2	no	—
27	#13095490	—	2	2	no	—
28	#13095488	—	2	2	no	—
29	C₄×Dic₉₇	—	2	2	no	—
30	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₉₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₉₇	—	2	2	no	—
33	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
34	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#87604	—	2	8	no	—
37	C₂×#87604	—	2	4	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₉₇	—	2	4	no	—
45	C₄×D₉₇	C₂×Q₈	1	2	yes	—
46	C₄×D₉₇	#41	1	2	yes	—
47	Dic₁₉₄	—	4	4	no	—
48	#11715897	—	4	2	no	—
49	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₉₇	—	2	2	no	—
51	C₂×D₉₇	C₂²⋊Q₈	1	2	yes	—
52	#87604	—	2	8	no	—
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	—	4	2	no	—
55	C₄×C₉₇	C₄×Q₈	1	2	yes	—
56	Dic₉₇	—	4	2	no	—
57	Dic₉₇	#104	1	4	yes	—
58	Dic₉₇	#124	1	1	yes	characteristic
59	Dic₉₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₉₇	#391	1	1	yes	characteristic
61	C₂×C₉₇	C₂×#104	1	2	yes	—
62	D₉₇	—	2	2	no	—
63	#1994	—	97	1	no	maximal, Sylow
64	C₉₇	#1994	1	1	yes	characteristic, complemented, Sylow
65	#533	—	97	1	no	—
66	#423	—	97	1	no	—
67	C₄³	—	97	1	no	—
68	#339	—	97	4	no	—
69	C₂×M₁₆	—	97	2	no	—
70	C₄⋊Q₈	—	97	2	no	—
71	#125	—	97	1	no	—
72	#125	#87604	1	1	yes	characteristic, complemented
73	C₂²⋊Q₈	—	194	2	no	—
74	C₄×Q₈	—	194	2	no	—
75	#117	—	97	2	no	—
76	C₂×C₄²	—	97	1	no	—
77	C₂×C₄²	—	97	2	no	—
78	C₂×C₄²	—	194	2	no	—
79	C₄⋊C₈	—	194	2	no	—
80	C₂×Q₈	—	194	2	no	—
81	C₂²×C₄	—	97	1	no	—
82	C₂²×C₄	—	97	2	no	—
83	C₂²×C₄	—	194	2	no	—
84	M₁₆	—	194	4	no	—
85	C₂×C₈	—	194	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	97	2	no	—
88	C₄⋊C₄	—	194	2	no	—
89	C₄⋊C₄	—	194	2	no	—
90	C₄⋊C₄	C₂×#87604	1	2	yes	—
91	C₂²⋊C₄	—	194	2	no	—
92	C₄²	—	97	2	no	—
93	C₄²	—	97	1	no	—
94	C₄²	—	97	4	no	—
95	C₄²	—	194	2	no	—
96	C₄²	—	194	2	no	—
97	C₄²	—	194	8	no	—
98	C₄²	C₂×#87604	1	1	yes	characteristic
99	C₂³	—	97	1	no	—
100	Q₈	—	388	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	97	1	no	—
103	C₂×C₄	—	97	2	no	—
104	C₂×C₄	—	97	2	no	—
105	C₂×C₄	—	97	2	no	—
106	C₂×C₄	—	194	4	no	—
107	C₂×C₄	—	194	2	no	—
108	C₂×C₄	—	194	4	no	—
109	C₂×C₄	—	194	8	no	—
110	C₂×C₄	C₂²⋊#87604	1	2	yes	—
111	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
112	C₈	—	388	2	no	—
113	C₂²	—	97	2	no	—
114	C₂²	—	194	2	no	—
115	C₂²	C₂×C₂²⋊#87604	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	97	1	no	—
119	C₄	—	97	1	no	—
120	C₄	—	97	4	no	—
121	C₄	—	388	2	no	—
122	C₄	—	194	8	no	—
123	C₄	Q₈×#87604	1	2	yes	—
124	C₂	—	194	2	no	—
125	C₂	(C₂×#87604)⋊Q₈	1	1	yes	characteristic
126	C₂	D₉₇⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261740

C₂

yes

maximal, characteristic

C₄²×#87604

C₂

yes

maximal, characteristic

#20261605

C₂

yes

maximal

#20261291

C₂

yes

maximal, characteristic

C₂×#13095521

C₂²

yes

—

C₂×C₄×#87604

—

C₂×C₄×#87604

C₂²

yes

—

#15690877

—

#15690809

C₄

yes

complemented

#15690802

C₄

yes

characteristic, complemented

#15690788

—

#15690775

—

#15690768

C₂²

yes

—

C₄²×D₉₇

C₂²

yes

characteristic

#125×C₉₇

C₄

yes

characteristic, complemented

C₂²×#87604

—

—

—

—

—

Q₈

yes

complemented

C₂×Dic₁₉₄

—

C₂×C₄×D₉₇

D₄

yes

—

C₂×C₄×D₉₇

C₂³

yes

characteristic

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

#13095490

—

#13095488

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂²×D₉₇

C₂×D₄

yes

characteristic

C₂×#87604

—

C₂×#87604

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂²⋊C₄

yes

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₄×D₉₇

—

C₄×D₉₇

C₂×Q₈

yes

—

C₄×D₉₇

#41

yes

—

Dic₁₉₄

—

#11715897

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

C₂²⋊Q₈

yes

—

#87604

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

Dic₉₇

—

Dic₉₇

#104

yes

—

Dic₉₇

#124

yes

characteristic

Dic₉₇

C₄⋊Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

D₉₇

—

#1994

—

maximal, Sylow

C₉₇

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₄³

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#87604

yes

characteristic, complemented

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

194

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

194

—

C₄⋊C₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂×#87604

yes

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

C₂×#87604

yes

characteristic

C₂³

—

100

Q₈

—

388

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

194

—

107

C₂×C₄

—

194

—

108

C₂×C₄

—

194

—

109

C₂×C₄

—

194

—

110

C₂×C₄

C₂²⋊#87604

yes

—

111

C₂×C₄

C₂²×#87604

yes

characteristic

112

C₈

—

388

—

113

C₂²

—

114

C₂²

—

194

—

115

C₂²

C₂×C₂²⋊#87604

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

388

—

122

C₄

—

194

—

123

C₄

Q₈×#87604

yes

—

124

C₂

—

194

—

125

C₂

(C₂×#87604)⋊Q₈

yes

characteristic

126

C₂

D₉₇⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#125×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₉₇	C₂×D₄
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#125	#87604
hypocenter	C₉₇	#1994
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87604

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87604

Fitting

#125×C₉₇

C₄

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₉₇

C₂×D₄

cosocle

C₂×C₄×D₉₇

C₂³

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #15690809	C₄	semidirect
10. #15690802	C₄	semidirect
15. #125×C₉₇	C₄	semidirect
21. C₄×#87604	Q₈	semidirect
64. C₉₇	#1994	semidirect
72. #125	#87604	semidirect

H ≅ G/N

type

9. #15690809

C₄

semidirect

10. #15690802

C₄

semidirect

15. #125×C₉₇

C₄

semidirect

21. C₄×#87604

Q₈

semidirect

64. C₉₇

#1994

semidirect

72. #125

#87604

semidirect

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

subgroup	index
#20261740	2
C₄²×#87604	2
#20261605	2
#20261291	2
#1994	97

subgroup

index

C₄²×#87604

Subgroups — #27157922

Conjugacy classes of subgroups

Complements

Maximal subgroups