Group #20260621 — subgroups

Subgroups — #20260621

Order 6208.

154 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 75 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×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690717	C₂	1	1	yes	maximal, characteristic
4	#15690717	C₂	1	1	yes	maximal, characteristic
5	#15690717	C₂	1	1	yes	maximal, characteristic
6	#15690717	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₉₇	—	2	2	no	—
13	C₄×Dic₉₇	—	2	2	no	—
14	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₉₇	C₄	1	4	yes	complemented
18	C₂³×C₉₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	1	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₉₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₉₇	—	2	1	no	—
29	C₂×Dic₉₇	—	2	1	no	—
30	C₂×Dic₉₇	—	2	1	no	—
31	C₂×Dic₉₇	—	2	1	no	—
32	C₂×Dic₉₇	—	2	1	no	—
33	C₂×Dic₉₇	—	2	1	no	—
34	C₂×Dic₉₇	—	2	1	no	—
35	C₂×Dic₉₇	—	2	1	no	—
36	C₂×Dic₉₇	—	2	1	no	—
37	C₂×Dic₉₇	—	2	1	no	—
38	C₂×Dic₉₇	D₄	1	2	yes	—
39	C₂×Dic₉₇	Q₈	1	2	yes	—
40	C₂²×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₉₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₉₇	#41	1	1	yes	characteristic
44	C₂²×C₉₇	#41	1	1	yes	characteristic
45	C₂²×C₉₇	#41	1	1	yes	characteristic
46	C₂²×C₉₇	#41	1	1	yes	characteristic
47	C₄×C₉₇	—	2	2	no	—
48	C₄×C₉₇	—	2	2	no	—
49	C₄×C₉₇	—	2	2	no	—
50	Dic₉₇	—	2	2	no	—
51	Dic₉₇	—	2	2	no	—
52	Dic₉₇	—	4	1	no	—
53	Dic₉₇	—	4	1	no	—
54	C₂×C₉₇	#117	1	1	yes	characteristic
55	C₂×C₉₇	C₄×D₄	1	1	yes	characteristic
56	C₂×C₉₇	C₄×Q₈	1	1	yes	characteristic
57	C₂×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₉₇	#123	1	1	yes	characteristic
59	C₂×C₉₇	#125	1	1	yes	characteristic
60	C₂×C₉₇	#126	1	1	yes	characteristic
61	C₉₇	#387	1	1	yes	characteristic, complemented, Sylow
62	#387	—	97	1	no	maximal, Sylow
63	C₂×C₄⋊C₄	—	97	1	no	—
64	C₂×C₄⋊C₄	D₉₇	1	1	yes	characteristic
65	C₂×C₄²	—	97	1	no	—
66	#95	—	97	1	no	—
67	#95	—	97	1	no	—
68	#95	—	97	1	no	—
69	#95	—	97	1	no	—
70	C₂²×C₄	—	97	1	no	—
71	C₂²×C₄	—	97	1	no	—
72	C₂²×C₄	—	97	1	no	—
73	C₂²×C₄	—	97	1	no	—
74	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
77	C₄⋊C₄	—	194	2	no	—
78	C₄⋊C₄	Dic₉₇	1	4	yes	complemented
79	C₄²	—	194	2	no	—
80	C₂³	C₂²×D₉₇	1	1	yes	characteristic
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	97	2	no	—
88	C₂×C₄	—	97	2	no	—
89	C₂×C₄	—	194	1	no	—
90	C₂×C₄	—	194	1	no	—
91	C₂×C₄	—	194	1	no	—
92	C₂×C₄	—	194	1	no	—
93	C₂×C₄	—	194	1	no	—
94	C₂×C₄	—	194	1	no	—
95	C₂×C₄	—	194	1	no	—
96	C₂×C₄	—	194	1	no	—
97	C₂×C₄	—	194	1	no	—
98	C₂×C₄	—	194	1	no	—
99	C₂×C₄	C₂×Dic₉₇	1	2	yes	—
100	C₂×C₄	C₂×Dic₉₇	1	2	yes	—
101	C₂×C₄	C₂×Dic₉₇	1	2	yes	—
102	C₂²	C₂²×Dic₉₇	1	1	yes	characteristic
103	C₂²	D₄×D₉₇	1	1	yes	characteristic
104	C₂²	Q₈×D₉₇	1	1	yes	characteristic
105	C₂²	#13095510	1	1	yes	characteristic
106	C₂²	#13095511	1	1	yes	characteristic
107	C₂²	#13095511	1	1	yes	characteristic
108	C₂²	#13095512	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	388	1	no	—
113	C₄	—	388	1	no	—
114	C₄	—	194	2	no	—
115	C₄	—	194	2	no	—
116	C₂	#15690769	1	1	yes	characteristic
117	C₂	D₄×Dic₉₇	1	1	yes	characteristic
118	C₂	Q₈×Dic₉₇	1	1	yes	characteristic
119	C₂	#15690788	1	1	yes	characteristic
120	C₂	#15690795	1	1	yes	characteristic
121	C₂	#15690800	1	1	yes	characteristic
122	C₂	#15690805	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₉₇

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₇

C₂

yes

maximal, characteristic

#15690717

C₂

yes

maximal, characteristic

#15690717

C₂

yes

maximal, characteristic

#15690717

C₂

yes

maximal, characteristic

#15690717

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₄

yes

complemented

C₂³×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×C₄

yes

—

C₂×C₄×C₉₇

C₂×C₄

yes

—

C₂×C₄×C₉₇

C₂×C₄

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

D₄

yes

—

C₂×Dic₉₇

Q₈

yes

—

C₂²×C₉₇

C₂²×C₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

C₂×Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#117

yes

characteristic

C₂×C₉₇

C₄×D₄

yes

characteristic

C₂×C₉₇

C₄×Q₈

yes

characteristic

C₂×C₉₇

C₂²⋊Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

characteristic, complemented, Sylow

#387

—

maximal, Sylow

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₉₇

yes

characteristic

C₂×C₄²

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

194

—

C₄⋊C₄

Dic₉₇

yes

complemented

C₄²

—

194

—

C₂³

C₂²×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂×Dic₉₇

yes

—

100

C₂×C₄

C₂×Dic₉₇

yes

—

101

C₂×C₄

C₂×Dic₉₇

yes

—

102

C₂²

C₂²×Dic₉₇

yes

characteristic

103

C₂²

D₄×D₉₇

yes

characteristic

104

C₂²

Q₈×D₉₇

yes

characteristic

105

C₂²

#13095510

yes

characteristic

106

C₂²

#13095511

yes

characteristic

107

C₂²

#13095511

yes

characteristic

108

C₂²

#13095512

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

388

—

113

C₄

—

388

—

114

C₄

—

194

—

115

C₄

—

194

—

116

C₂

#15690769

yes

characteristic

117

C₂

D₄×Dic₉₇

yes

characteristic

118

C₂

Q₈×Dic₉₇

yes

characteristic

119

C₂

#15690788

yes

characteristic

120

C₂

#15690795

yes

characteristic

121

C₂

#15690800

yes

characteristic

122

C₂

#15690805

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₉₇	C₂
socle	C₂³×C₉₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#387
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₉₇

C₂

socle

C₂³×C₉₇

C₂³

hypercenter

C₂×C₄⋊C₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
17. C₄⋊C₄×C₉₇	C₄	semidirect
61. C₉₇	#387	semidirect
78. C₄⋊C₄	Dic₉₇	semidirect

H ≅ G/N

type

17. C₄⋊C₄×C₉₇

C₄

semidirect

61. C₉₇

#387

semidirect

78. C₄⋊C₄

Dic₉₇

semidirect

Subgroups — #20260621

Conjugacy classes of subgroups

Complements