Group #27140471 — subgroups

Subgroups — #27140471

Order 12416.

254 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20260759	C₂	1	2	yes	maximal, complemented
2	#20260484	C₂	1	2	yes	maximal, complemented
3	#20260479	C₂	1	2	yes	maximal, complemented
4	#532×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	2	no	—
6	C₄⋊#11715895	—	2	2	no	—
7	C₄⋊#11715895	C₂²	1	4	yes	—
8	C₂×#41×C₉₇	C₄	1	2	yes	—
9	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	#127×C₉₇	C₂²	1	1	yes	characteristic
11	#124×C₉₇	C₄	1	2	yes	—
12	#121×C₉₇	—	2	2	no	—
13	C₄×D₄×C₉₇	—	2	2	no	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#13095495	—	4	4	no	—
16	C₂×#11715895	—	2	4	no	—
17	#41×C₉₇	—	4	4	no	—
18	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
19	C₂×D₄×C₉₇	—	2	2	no	—
20	C₂×D₄×C₉₇	—	2	2	no	—
21	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉₇	—	2	2	no	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₉₇	—	2	2	no	—
26	C₂²⋊C₄×C₉₇	—	2	4	no	—
27	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₉₇	D₄	1	2	yes	—
29	C₄²×C₉₇	C₂³	1	1	yes	characteristic
30	C₂³×C₉₇	—	2	2	no	—
31	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₉₇	—	4	2	no	—
33	D₄×C₉₇	—	4	2	no	—
34	D₄×C₉₇	—	4	4	no	—
35	D₄×C₉₇	—	4	2	no	—
36	D₄×C₉₇	—	4	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	4	4	no	—
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
46	#11715895	—	4	4	no	—
47	C₂²×C₉₇	—	2	1	no	—
48	C₂²×C₉₇	—	2	2	no	—
49	C₂²×C₉₇	—	4	2	no	—
50	C₂²×C₉₇	—	4	4	no	—
51	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₉₇	—	2	1	no	—
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	—	4	2	no	—
55	C₄×C₉₇	#100	1	2	yes	—
56	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₉₇	#137	1	2	yes	—
58	C₂×C₉₇	—	4	1	no	—
59	C₂×C₉₇	—	8	2	no	—
60	C₂×C₉₇	C₂×#100	1	1	yes	characteristic
61	C₂×C₉₇	#417	1	2	yes	—
62	#1560	—	97	1	no	maximal, Sylow
63	C₉₇	#1560	1	1	yes	characteristic, complemented, Sylow
64	#532	D₉₇	1	1	yes	characteristic
65	#423	—	97	2	no	—
66	#333	—	97	2	no	—
67	#331	—	97	2	no	—
68	C₂×#41	Dic₉₇	1	2	yes	—
69	C₂×M₁₆	—	194	2	no	—
70	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
71	#127	C₂×D₉₇	1	1	yes	characteristic
72	#124	Dic₉₇	1	2	yes	—
73	#121	—	2	2	no	—
74	C₄×D₄	—	2	2	no	—
75	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
76	C₄⋊C₈	—	97	4	no	—
77	C₄⋊C₈	—	194	2	no	—
78	#41	—	4	4	no	—
79	C₂×Q₈	C₂×Dic₉₇	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
86	M₁₆	—	388	4	no	—
87	C₂×C₈	—	194	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
91	C₄²	C₂²⋊D₉₇	1	2	yes	—
92	C₄²	C₂²×D₉₇	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
109	C₈	—	388	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#15690727	1	2	yes	—
119	C₄	#15690835	1	2	yes	—
120	C₄	#15690840	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#15690727	1	1	yes	characteristic
124	C₂	#20260736	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal, complemented

#20260484

C₂

yes

maximal, complemented

#20260479

C₂

yes

maximal, complemented

#532×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

—

C₄⋊#11715895

—

C₄⋊#11715895

C₂²

yes

—

C₂×#41×C₉₇

C₄

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

#124×C₉₇

C₄

yes

—

#121×C₉₇

—

C₄×D₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#13095495

—

C₂×#11715895

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

C₂²⋊C₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#100

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

#137

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#100

yes

characteristic

C₂×C₉₇

#417

yes

—

#1560

—

maximal, Sylow

C₉₇

#1560

yes

characteristic, complemented, Sylow

#532

D₉₇

yes

—

—

—

C₂×#41

Dic₉₇

yes

—

C₂×M₁₆

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

#127

C₂×D₉₇

yes

characteristic

#124

Dic₉₇

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

#41

—

C₂×Q₈

C₂×Dic₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

—

M₁₆

—

388

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₉₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₉₇

yes

—

106

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

107

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

109

C₈

—

388

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#15690727

yes

—

119

C₄

#15690835

yes

—

120

C₄

#15690840

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#15690727

yes

characteristic

124

C₂

#20260736

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting, prime radical	#532×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#532	D₉₇
hypocenter	C₉₇	#1560
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting, prime radical

#532×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₄²×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #20260759	C₂	semidirect
2. #20260484	C₂	semidirect
3. #20260479	C₂	semidirect
63. C₉₇	#1560	semidirect

H ≅ G/N

type

1. #20260759

C₂

semidirect

2. #20260484

C₂

semidirect

3. #20260479

C₂

semidirect

63. C₉₇

#1560

semidirect

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

subgroup	index
#20260759	2
#20260484	2
#20260479	2
#532×C₉₇	2
#1560	97

subgroup

index

#532×C₉₇

Subgroups — #27140471

Conjugacy classes of subgroups

Complements

Maximal subgroups