Group #20260615 — subgroups

Subgroups — #20260615

Order 6208.

286 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂³×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₉₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690717	C₂	1	2	yes	maximal, complemented
5	C₂²×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₇	—	2	2	no	—
7	C₂²×Dic₉₇	—	2	4	no	—
8	C₂²×Dic₉₇	C₂²	1	2	yes	—
9	C₂²×Dic₉₇	C₂²	1	2	yes	—
10	C₂²⋊Dic₉₇	—	2	2	no	—
11	C₂²⋊Dic₉₇	—	2	4	no	—
12	C₂⁴×C₉₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₉₇	—	2	2	no	—
15	C₂×D₄×C₉₇	—	2	2	no	—
16	C₂×D₄×C₉₇	C₄	1	4	yes	complemented
17	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
18	C₂³×C₉₇	—	2	2	no	—
19	C₂³×C₉₇	—	2	2	no	—
20	C₂³×C₉₇	—	2	1	no	—
21	C₂³×C₉₇	—	2	1	no	—
22	C₂³×C₉₇	—	2	2	no	—
23	C₂³×C₉₇	—	2	2	no	—
24	C₂³×C₉₇	C₂×C₄	1	2	yes	—
25	C₂³×C₉₇	C₂×C₄	1	2	yes	complemented
26	C₂³×C₉₇	D₄	1	2	yes	—
27	C₂³×C₉₇	D₄	1	2	yes	—
28	C₂³×C₉₇	C₂³	1	1	yes	characteristic
29	D₄×C₉₇	—	2	8	no	—
30	C₂×C₄×C₉₇	—	2	1	no	—
31	C₂×C₄×C₉₇	—	2	1	no	—
32	C₂×C₄×C₉₇	C₂×C₄	1	2	yes	complemented
33	C₂×Dic₉₇	—	2	8	no	—
34	C₂×Dic₉₇	—	2	2	no	—
35	C₂×Dic₉₇	—	2	2	no	—
36	C₂×Dic₉₇	—	2	2	no	—
37	C₂×Dic₉₇	—	2	2	no	—
38	C₂×Dic₉₇	—	2	2	no	—
39	C₂×Dic₉₇	D₄	1	4	yes	complemented
40	C₂²×C₉₇	—	2	4	no	—
41	C₂²×C₉₇	—	2	2	no	—
42	C₂²×C₉₇	—	2	4	no	—
43	C₂²×C₉₇	—	2	2	no	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	—	2	4	no	—
46	C₂²×C₉₇	—	4	1	no	—
47	C₂²×C₉₇	—	4	2	no	—
48	C₂²×C₉₇	—	4	1	no	—
49	C₂²×C₉₇	C₂²⋊C₄	1	4	yes	complemented
50	C₂²×C₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₉₇	C₂×D₄	1	2	yes	—
52	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₉₇	#41	1	2	yes	—
55	C₄×C₉₇	—	2	2	no	—
56	Dic₉₇	—	2	4	no	—
57	Dic₉₇	—	4	2	no	—
58	C₂×C₉₇	—	2	4	no	—
59	C₂×C₉₇	—	4	2	no	—
60	C₂×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₉₇	C₄×D₄	1	2	yes	—
62	C₂×C₉₇	#120	1	1	yes	characteristic
63	C₂×C₉₇	#121	1	2	yes	—
64	C₂×C₉₇	#123	1	1	yes	characteristic
65	C₉₇	#386	1	1	yes	characteristic, complemented, Sylow
66	#386	—	97	1	no	maximal, Sylow
67	C₂²×D₄	D₉₇	1	1	yes	characteristic
68	C₂³×C₄	—	97	1	no	—
69	C₂×C₂²⋊C₄	—	97	2	no	—
70	C₂×C₂²⋊C₄	—	97	1	no	—
71	#95	—	97	2	no	—
72	C₂⁴	C₂×D₉₇	1	1	yes	characteristic
73	C₂⁴	C₂×D₉₇	1	1	yes	characteristic
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	Dic₉₇	1	4	yes	complemented
77	C₂²×C₄	—	97	2	no	—
78	C₂²×C₄	—	97	2	no	—
79	C₂²×C₄	—	194	4	no	—
80	C₂²×C₄	—	194	2	no	—
81	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
82	C₂²⋊C₄	—	194	4	no	—
83	C₂²⋊C₄	—	194	2	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	1	no	—
88	C₂³	—	2	1	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₂×Dic₉₇	1	2	yes	—
91	C₂³	C₂×Dic₉₇	1	2	yes	complemented
92	C₂³	C₂²⋊D₉₇	1	2	yes	—
93	C₂³	C₂²⋊D₉₇	1	2	yes	—
94	C₂³	C₂²×D₉₇	1	1	yes	characteristic
95	D₄	—	2	8	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	97	4	no	—
99	C₂×C₄	—	194	2	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	—	194	2	no	—
102	C₂×C₄	—	194	8	no	—
103	C₂×C₄	—	194	2	no	—
104	C₂×C₄	—	194	2	no	—
105	C₂×C₄	C₂×Dic₉₇	1	2	yes	complemented
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	4	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂²⋊Dic₉₇	1	4	yes	complemented
116	C₂²	C₂²×Dic₉₇	1	1	yes	characteristic
117	C₂²	D₄×D₉₇	1	2	yes	—
118	C₂²	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
119	C₂²	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
120	C₂²	#13095510	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	388	2	no	—
123	C₄	—	194	4	no	—
124	C₂	—	2	4	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
127	C₂	D₄×Dic₉₇	1	2	yes	—
128	C₂	#15690778	1	1	yes	characteristic
129	C₂	#15690782	1	2	yes	—
130	C₂	#15690791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₉₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₉₇

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₉₇

C₂

yes

maximal, characteristic, complemented

#15690717

C₂

yes

maximal, complemented

C₂²×D₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

C₂²

yes

—

C₂²×Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₂⁴×C₉₇

C₂²

yes

characteristic

C₂⁴×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₄

yes

complemented

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂×C₄

yes

complemented

C₂³×C₉₇

D₄

yes

—

C₂³×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×C₄

yes

complemented

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

D₄

yes

complemented

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊C₄

yes

complemented

C₂²×C₉₇

C₂²×C₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

#41

yes

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₉₇

C₄×D₄

yes

—

C₂×C₉₇

#120

yes

characteristic

C₂×C₉₇

#121

yes

—

C₂×C₉₇

#123

yes

characteristic

C₉₇

#386

yes

characteristic, complemented, Sylow

#386

—

maximal, Sylow

C₂²×D₄

D₉₇

yes

characteristic

C₂³×C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

C₂×D₉₇

yes

characteristic

C₂⁴

C₂×D₉₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₉₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₂²⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₉₇

yes

—

C₂³

C₂×Dic₉₇

yes

complemented

C₂³

C₂²⋊D₉₇

yes

—

C₂³

C₂²⋊D₉₇

yes

—

C₂³

C₂²×D₉₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

C₂×Dic₉₇

yes

complemented

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²⋊Dic₉₇

yes

complemented

116

C₂²

C₂²×Dic₉₇

yes

characteristic

117

C₂²

D₄×D₉₇

yes

—

118

C₂²

C₂×C₂²⋊D₉₇

yes

characteristic

119

C₂²

C₂×C₂²⋊D₉₇

yes

characteristic

120

C₂²

#13095510

yes

—

121

C₄

—

122

C₄

—

388

—

123

C₄

—

194

—

124

C₂

—

125

C₂

—

126

C₂

C₂×C₂²⋊Dic₉₇

yes

characteristic

127

C₂

D₄×Dic₉₇

yes

—

128

C₂

#15690778

yes

characteristic

129

C₂

#15690782

yes

—

130

C₂

#15690791

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₄
Fitting	C₂²×D₄×C₉₇	C₂
socle	C₂³×C₉₇	C₂³
hypercenter	C₂²×D₄	D₉₇
hypocenter	C₉₇	#386
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂²×C₄

Fitting

C₂²×D₄×C₉₇

C₂

socle

C₂³×C₉₇

C₂³

hypercenter

C₂²×D₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂³×Dic₉₇	C₂	semidirect
2. C₂×C₂²⋊Dic₉₇	C₂	semidirect
3. C₂×C₂²⋊Dic₉₇	C₂	semidirect
4. #15690717	C₂	semidirect
16. C₂×D₄×C₉₇	C₄	semidirect
25. C₂³×C₉₇	C₂×C₄	semidirect
32. C₂×C₄×C₉₇	C₂×C₄	semidirect
39. C₂×Dic₉₇	D₄	semidirect
49. C₂²×C₉₇	C₂²⋊C₄	semidirect
65. C₉₇	#386	semidirect
76. C₂×D₄	Dic₉₇	semidirect
91. C₂³	C₂×Dic₉₇	semidirect
105. C₂×C₄	C₂×Dic₉₇	semidirect
115. C₂²	C₂²⋊Dic₉₇	semidirect

H ≅ G/N

type

1. C₂³×Dic₉₇

C₂

semidirect

2. C₂×C₂²⋊Dic₉₇

C₂

semidirect

3. C₂×C₂²⋊Dic₉₇

C₂

semidirect

4. #15690717

C₂

semidirect

16. C₂×D₄×C₉₇

C₄

semidirect

25. C₂³×C₉₇

C₂×C₄

semidirect

32. C₂×C₄×C₉₇

C₂×C₄

semidirect

39. C₂×Dic₉₇

D₄

semidirect

49. C₂²×C₉₇

C₂²⋊C₄

semidirect

65. C₉₇

#386

semidirect

76. C₂×D₄

Dic₉₇

semidirect

91. C₂³

C₂×Dic₉₇

semidirect

105. C₂×C₄

C₂×Dic₉₇

semidirect

115. C₂²

C₂²⋊Dic₉₇

semidirect

Subgroups — #20260615

Conjugacy classes of subgroups

Complements