Group #27157659 — subgroups

Subgroups — #27157659

Order 12416.

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

normal subgroups only

This group has 224 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	#20261681	C₂	1	1	yes	maximal, characteristic
2	#20261679	C₂	1	1	yes	maximal, characteristic
3	#20261609	C₂	1	4	yes	maximal
4	#97×D₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#87604	—	2	2	no	—
6	C₂×C₄⋊#87604	C₂²	1	2	yes	—
7	C₂×C₄⋊#87604	C₂²	1	2	yes	—
8	(C₂×#87604)⋊C₄	—	2	2	no	—
9	C₂×C₈×D₉₇	C₂²	1	2	yes	—
10	C₄²×D₉₇	C₂²	1	1	yes	characteristic
11	#15690721	C₄	1	1	yes	characteristic, complemented
12	#15690720	C₄	1	2	yes	complemented
13	#97×C₉₇	C₄	1	1	yes	characteristic, complemented
14	C₂²×#87604	—	2	2	no	—
15	C₂²×#87604	—	2	2	no	—
16	C₄⋊#87604	—	2	4	no	—
17	C₄⋊#87604	—	2	4	no	—
18	C₄⋊#87604	—	4	4	no	—
19	C₂×C₄×D₉₇	D₄	1	2	yes	—
20	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
21	C₂×#11715895	C₂×C₄	1	2	yes	—
22	C₈×D₉₇	—	2	2	no	—
23	C₈×D₉₇	—	2	2	no	—
24	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₉₇	Q₈	1	2	yes	—
26	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
27	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×#87604	—	4	2	no	—
30	C₂×#87604	—	4	2	no	—
31	C₂×#87604	—	4	4	no	—
32	C₂×#87604	—	4	2	no	—
33	C₂×#87604	—	4	2	no	—
34	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₄×D₉₇	—	2	2	no	—
39	C₄×D₉₇	—	2	2	no	—
40	C₄×D₉₇	#41	1	2	yes	—
41	C₄×D₉₇	#41	1	2	yes	—
42	C₈×C₉₇	—	2	2	no	—
43	#11715895	—	2	2	no	—
44	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×D₉₇	#124	1	1	yes	characteristic
46	C₂×D₉₇	#124	1	1	yes	characteristic
47	C₂×D₉₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₂×D₉₇	#137	1	2	yes	—
49	#87604	—	8	2	no	—
50	#87604	—	8	2	no	—
51	C₄×C₉₇	—	2	2	no	—
52	C₄×C₉₇	#117	1	2	yes	—
53	Dic₉₇	—	2	2	no	—
54	Dic₉₇	#125	1	2	yes	—
55	C₂×C₉₇	#382	1	1	yes	characteristic
56	C₂×C₉₇	#428	1	2	yes	—
57	D₉₇	#489	1	2	yes	—
58	D₉₇	#490	1	1	yes	characteristic
59	D₉₇	#491	1	1	yes	characteristic
60	#1836	—	97	1	no	maximal, Sylow
61	C₉₇	#1836	1	1	yes	characteristic, complemented, Sylow
62	C₂×#97	—	97	1	no	—
63	#384	—	97	1	no	—
64	#382	—	97	1	no	—
65	#340	—	97	4	no	—
66	C₂²×C₈	—	97	2	no	—
67	C₂×C₄⋊C₄	—	97	2	no	—
68	C₂×C₄⋊C₄	—	97	2	no	—
69	C₂×C₄⋊C₄	—	194	2	no	—
70	C₂×C₄²	—	97	1	no	—
71	#97	—	97	2	no	—
72	#97	—	97	1	no	—
73	#97	#87604	1	1	yes	characteristic, complemented
74	#95	—	194	2	no	—
75	C₂²×C₄	—	97	1	no	—
76	C₂²×C₄	—	97	2	no	—
77	C₂²×C₄	—	194	2	no	—
78	C₂²×C₄	—	194	2	no	—
79	C₂×C₈	—	97	2	no	—
80	C₂×C₈	—	194	2	no	—
81	C₂×C₈	—	194	2	no	—
82	C₂×C₈	C₂×#87604	1	2	yes	—
83	C₄⋊C₄	—	388	4	no	—
84	C₄⋊C₄	—	194	4	no	—
85	C₄⋊C₄	—	194	4	no	—
86	C₄²	—	97	2	no	—
87	C₄²	—	97	1	no	—
88	C₄²	C₂×#87604	1	1	yes	characteristic
89	C₂³	—	97	1	no	—
90	C₂×C₄	—	97	1	no	—
91	C₂×C₄	—	97	2	no	—
92	C₂×C₄	—	97	2	no	—
93	C₂×C₄	—	97	2	no	—
94	C₂×C₄	—	388	4	no	—
95	C₂×C₄	—	388	2	no	—
96	C₂×C₄	—	388	2	no	—
97	C₂×C₄	—	388	2	no	—
98	C₂×C₄	—	388	2	no	—
99	C₂×C₄	—	194	2	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	C₄⋊#87604	1	2	yes	—
102	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	194	2	no	—
105	C₂²	—	97	1	no	—
106	C₂²	—	97	1	no	—
107	C₂²	—	97	2	no	—
108	C₂²	—	97	2	no	—
109	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	776	2	no	—
112	C₄	—	776	2	no	—
113	C₄	—	97	2	no	—
114	C₄	—	194	2	no	—
115	C₄	#15690889	1	2	yes	—
116	C₂	—	97	1	no	—
117	C₂	—	97	2	no	—
118	C₂	—	97	1	no	—
119	C₂	#20261679	1	1	yes	characteristic
120	C₂	#20261746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261681

C₂

yes

maximal, characteristic

#20261679

C₂

yes

maximal, characteristic

#20261609

C₂

yes

maximal

#97×D₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#87604

—

C₂×C₄⋊#87604

C₂²

yes

—

C₂×C₄⋊#87604

C₂²

yes

—

(C₂×#87604)⋊C₄

—

C₂×C₈×D₉₇

C₂²

yes

—

C₄²×D₉₇

C₂²

yes

characteristic

#15690721

C₄

yes

characteristic, complemented

#15690720

C₄

yes

complemented

#97×C₉₇

C₄

yes

characteristic, complemented

C₂²×#87604

—

C₂²×#87604

—

C₄⋊#87604

—

C₄⋊#87604

—

C₄⋊#87604

—

C₂×C₄×D₉₇

D₄

yes

—

C₂×C₄×D₉₇

C₂³

yes

characteristic

C₂×#11715895

C₂×C₄

yes

—

C₈×D₉₇

—

C₈×D₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

Q₈

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂²×D₉₇

C₂×D₄

yes

—

—

—

—

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×Q₈

yes

characteristic

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

#41

yes

—

C₄×D₉₇

#41

yes

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂×D₉₇

yes

C₂×D₉₇

yes

C₂×D₉₇

yes

—

C₂×D₉₇

#137

yes

—

#87604

—

#87604

—

C₄×C₉₇

—

C₄×C₉₇

#117

yes

—

Dic₉₇

—

Dic₉₇

#125

yes

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

#428

yes

—

D₉₇

#489

yes

—

D₉₇

yes

D₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#97

—

#384

—

#382

—

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#95

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×#87604

yes

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

C₂×#87604

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

100

C₂×C₄

—

194

—

101

C₂×C₄

C₄⋊#87604

yes

—

102

C₂×C₄

C₂²×#87604

yes

characteristic

103

C₈

—

104

C₈

—

194

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊#87604

yes

characteristic

110

C₄

—

111

C₄

—

776

—

112

C₄

—

776

—

113

C₄

—

114

C₄

—

194

—

115

C₄

#15690889

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#20261679

yes

characteristic

120

C₂

#20261746

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#97×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂²×D₉₇	C₂×D₄
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#97	#87604
hypocenter	C₉₇	#1836
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#87604

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87604

Fitting

#97×C₉₇

C₄

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂²×D₉₇

C₂×D₄

cosocle

C₂×C₄×D₉₇

C₂³

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
11. #15690721	C₄	semidirect
12. #15690720	C₄	semidirect
13. #97×C₉₇	C₄	semidirect
61. C₉₇	#1836	semidirect
73. #97	#87604	semidirect

H ≅ G/N

type

11. #15690721

C₄

semidirect

12. #15690720

C₄

semidirect

13. #97×C₉₇

C₄

semidirect

61. C₉₇

#1836

semidirect

73. #97

#87604

semidirect

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

subgroup	index
#20261681	2
#20261679	2
#20261609	2
#97×D₉₇	2
#1836	97

subgroup

index

#97×D₉₇

Subgroups — #27157659

Conjugacy classes of subgroups

Complements

Maximal subgroups