Group #20261294 — subgroups

Subgroups — #20261294

Order 6208.

346 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×C₂²⋊D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690793	C₂	1	2	yes	maximal, complemented
3	#15690791	C₂	1	2	yes	maximal, complemented
4	#15690782	C₂	1	4	yes	maximal, complemented
5	#15690779	C₂	1	2	yes	maximal, complemented
6	#15690778	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₉₇	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₉₇	—	2	4	no	—
11	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	complemented
13	C₂×C₂²⋊D₉₇	C₂²	1	4	yes	complemented
14	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×Dic₉₇	C₂²	1	2	yes	complemented
16	#13095490	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₉₇	C₂²	1	2	yes	complemented
18	C₂²⋊Dic₉₇	C₂²	1	2	yes	complemented
19	C₂²⋊Dic₉₇	C₂²	1	4	yes	complemented
20	#13095488	C₂²	1	4	yes	complemented
21	C₂⁴×C₉₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₉₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₄×C₉₇	—	2	4	no	—
24	C₂×D₄×C₉₇	C₂²	1	2	yes	—
25	C₂×D₄×C₉₇	C₂²	1	2	yes	complemented
26	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
27	C₂³×C₉₇	—	2	2	no	—
28	C₂³×C₉₇	—	4	2	no	—
29	C₂³×C₉₇	—	4	2	no	—
30	C₂³×C₉₇	D₄	1	4	yes	complemented
31	C₂³×C₉₇	C₂³	1	2	yes	—
32	C₂³×C₉₇	C₂³	1	1	yes	characteristic
33	C₂³×C₉₇	C₂³	1	1	yes	characteristic
34	C₂³×C₉₇	C₂³	1	1	yes	characteristic
35	C₂²×D₉₇	—	2	2	no	—
36	C₂²×D₉₇	—	4	2	no	—
37	C₂²×D₉₇	C₂³	1	2	yes	—
38	D₄×C₉₇	—	2	4	no	—
39	D₄×C₉₇	—	2	4	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	C₂³	1	2	yes	—
42	C₂²⋊D₉₇	—	2	4	no	—
43	C₂²⋊D₉₇	—	2	4	no	—
44	C₂²⋊D₉₇	—	4	4	no	—
45	C₂×Dic₉₇	—	2	2	no	—
46	C₂×Dic₉₇	—	4	2	no	—
47	C₂×Dic₉₇	C₂³	1	4	yes	—
48	C₂×Dic₉₇	C₂³	1	2	yes	—
49	C₂²×C₉₇	—	2	2	no	—
50	C₂²×C₉₇	—	2	2	no	—
51	C₂²×C₉₇	—	2	2	no	—
52	C₂²×C₉₇	—	2	2	no	—
53	C₂²×C₉₇	—	2	4	no	—
54	C₂²×C₉₇	—	4	4	no	—
55	C₂²×C₉₇	—	4	4	no	—
56	C₂²×C₉₇	C₂×D₄	1	2	yes	—
57	C₂²×C₉₇	C₂×D₄	1	2	yes	—
58	C₂²×C₉₇	C₂×D₄	1	2	yes	—
59	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
60	C₂×D₉₇	—	2	2	no	—
61	C₂×D₉₇	—	2	4	no	—
62	C₂×D₉₇	—	4	4	no	—
63	C₄×C₉₇	—	2	2	no	—
64	Dic₉₇	—	2	2	no	—
65	Dic₉₇	—	2	4	no	—
66	C₂×C₉₇	—	2	2	no	—
67	C₂×C₉₇	—	2	2	no	—
68	C₂×C₉₇	—	2	2	no	—
69	C₂×C₉₇	—	4	2	no	—
70	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₉₇	H(5,𝔽₂)	1	2	yes	—
72	D₉₇	—	4	2	no	—
73	C₉₇	#534	1	1	yes	characteristic, complemented, Sylow
74	#534	—	97	1	no	maximal, Sylow
75	C₂²×D₄	—	97	1	no	—
76	C₂²×D₄	D₉₇	1	1	yes	characteristic, complemented
77	#123	—	97	2	no	—
78	#123	—	97	2	no	—
79	#121	—	97	4	no	—
80	#120	—	97	2	no	—
81	#120	—	97	2	no	—
82	C₂×C₂²⋊C₄	—	97	1	no	—
83	C₂⁴	—	97	1	no	—
84	C₂⁴	C₂×D₉₇	1	1	yes	characteristic
85	C₂⁴	C₂×D₉₇	1	1	yes	characteristic, complemented
86	C₂×D₄	—	2	4	no	—
87	C₂×D₄	—	97	4	no	—
88	C₂×D₄	—	97	2	no	—
89	C₂×D₄	—	97	2	no	—
90	C₂×D₄	—	194	4	no	—
91	C₂×D₄	C₂×D₉₇	1	2	yes	—
92	C₂×D₄	C₂×D₉₇	1	2	yes	complemented
93	C₂²×C₄	—	97	2	no	—
94	C₂²×C₄	—	97	1	no	—
95	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
96	C₄⋊C₄	—	97	4	no	—
97	C₂²⋊C₄	—	97	2	no	—
98	C₂²⋊C₄	—	97	2	no	—
99	C₂²⋊C₄	—	97	4	no	—
100	C₂²⋊C₄	—	97	4	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	4	2	no	—
104	C₂³	—	97	2	no	—
105	C₂³	—	388	2	no	—
106	C₂³	—	194	2	no	—
107	C₂³	C₂²⋊D₉₇	1	4	yes	complemented
108	C₂³	C₂²×D₉₇	1	2	yes	—
109	C₂³	C₂²×D₉₇	1	1	yes	characteristic
110	C₂³	C₂²×D₉₇	1	1	yes	characteristic
111	C₂³	C₂²×D₉₇	1	1	yes	characteristic
112	D₄	—	2	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	388	4	no	—
115	D₄	—	194	4	no	—
116	D₄	—	194	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	97	4	no	—
119	C₂×C₄	—	97	2	no	—
120	C₂×C₄	—	388	2	no	—
121	C₂×C₄	—	194	2	no	—
122	C₂×C₄	C₂²×D₉₇	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	388	4	no	—
131	C₂²	—	194	2	no	—
132	C₂²	—	194	4	no	—
133	C₂²	C₂×C₂²⋊D₉₇	1	2	yes	—
134	C₂²	C₂×C₂²⋊D₉₇	1	2	yes	—
135	C₂²	C₂×C₂²⋊D₉₇	1	2	yes	—
136	C₂²	C₂³×D₉₇	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	194	2	no	—
139	C₄	—	194	4	no	—
140	C₂	—	2	2	no	—
141	C₂	—	2	2	no	—
142	C₂	—	2	2	no	—
143	C₂	—	4	2	no	—
144	C₂	—	388	2	no	—
145	C₂	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
146	C₂	#15690855	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₉₇

C₂

yes

maximal, characteristic, complemented

#15690793

C₂

yes

maximal, complemented

#15690791

C₂

yes

maximal, complemented

#15690782

C₂

yes

maximal, complemented

#15690779

C₂

yes

maximal, complemented

#15690778

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₉₇

—

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

complemented

C₂×C₂²⋊D₉₇

C₂²

yes

complemented

C₂²×Dic₉₇

C₂²

yes

characteristic, complemented

C₂²×Dic₉₇

C₂²

yes

complemented

#13095490

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

complemented

C₂⁴×C₉₇

C₂²

yes

characteristic

C₂⁴×C₉₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂²

yes

—

C₂×D₄×C₉₇

C₂²

yes

complemented

C₂²×C₄×C₉₇

C₂²

yes

characteristic, complemented

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

D₄

yes

complemented

C₂³×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

H(5,𝔽₂)

yes

—

D₉₇

—

C₉₇

#534

yes

characteristic, complemented, Sylow

#534

—

maximal, Sylow

C₂²×D₄

—

C₂²×D₄

D₉₇

yes

characteristic, complemented

—

—

—

—

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂⁴

C₂×D₉₇

yes

characteristic

C₂⁴

C₂×D₉₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

C₂×D₉₇

yes

—

C₂×D₄

C₂×D₉₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic, complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

388

—

106

C₂³

—

194

—

107

C₂³

C₂²⋊D₉₇

yes

complemented

108

C₂³

C₂²×D₉₇

yes

—

109

C₂³

C₂²×D₉₇

yes

characteristic

110

C₂³

C₂²×D₉₇

yes

characteristic

111

C₂³

C₂²×D₉₇

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

388

—

115

D₄

—

194

—

116

D₄

—

194

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

388

—

121

C₂×C₄

—

194

—

122

C₂×C₄

C₂²×D₉₇

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

388

—

131

C₂²

—

194

—

132

C₂²

—

194

—

133

C₂²

C₂×C₂²⋊D₉₇

yes

—

134

C₂²

C₂×C₂²⋊D₉₇

yes

—

135

C₂²

C₂×C₂²⋊D₉₇

yes

—

136

C₂²

C₂³×D₉₇

yes

characteristic

137

C₄

—

138

C₄

—

194

—

139

C₄

—

194

—

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

—

388

—

145

C₂

C₂²×C₂²⋊D₉₇

yes

characteristic

146

C₂

#15690855

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₂²×D₄×C₉₇	C₂
hypercenter	C₂²×D₄	D₉₇
hypocenter	C₉₇	#534
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₂²×D₄×C₉₇

C₂

hypercenter

C₂²×D₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₉₇	C₂	semidirect
2. #15690793	C₂	semidirect
3. #15690791	C₂	semidirect
4. #15690782	C₂	semidirect
5. #15690779	C₂	semidirect
6. #15690778	C₂	semidirect
7. C₂×C₂²⋊Dic₉₇	C₂	semidirect
8. C₂²×D₄×C₉₇	C₂	semidirect
9. C₂³×D₉₇	C₂²	semidirect
12. C₂×C₂²⋊D₉₇	C₂²	semidirect
13. C₂×C₂²⋊D₉₇	C₂²	semidirect
14. C₂²×Dic₉₇	C₂²	semidirect
15. C₂²×Dic₉₇	C₂²	semidirect
16. #13095490	C₂²	semidirect
17. C₂²⋊Dic₉₇	C₂²	semidirect
18. C₂²⋊Dic₉₇	C₂²	semidirect
19. C₂²⋊Dic₉₇	C₂²	semidirect
20. #13095488	C₂²	semidirect
22. C₂⁴×C₉₇	C₂²	semidirect
25. C₂×D₄×C₉₇	C₂²	semidirect
26. C₂²×C₄×C₉₇	C₂²	semidirect
30. C₂³×C₉₇	D₄	semidirect
73. C₉₇	#534	semidirect
76. C₂²×D₄	D₉₇	semidirect
85. C₂⁴	C₂×D₉₇	semidirect
92. C₂×D₄	C₂×D₉₇	semidirect
95. C₂²×C₄	C₂×D₉₇	semidirect
107. C₂³	C₂²⋊D₉₇	semidirect

H ≅ G/N

type

1. C₂²×C₂²⋊D₉₇

C₂

semidirect

2. #15690793

C₂

semidirect

3. #15690791

C₂

semidirect

4. #15690782

C₂

semidirect

5. #15690779

C₂

semidirect

6. #15690778

C₂

semidirect

7. C₂×C₂²⋊Dic₉₇

C₂

semidirect

8. C₂²×D₄×C₉₇

C₂

semidirect

9. C₂³×D₉₇

C₂²

semidirect

12. C₂×C₂²⋊D₉₇

C₂²

semidirect

13. C₂×C₂²⋊D₉₇

C₂²

semidirect

14. C₂²×Dic₉₇

C₂²

semidirect

15. C₂²×Dic₉₇

C₂²

semidirect

16. #13095490

C₂²

semidirect

17. C₂²⋊Dic₉₇

C₂²

semidirect

18. C₂²⋊Dic₉₇

C₂²

semidirect

19. C₂²⋊Dic₉₇

C₂²

semidirect

20. #13095488

C₂²

semidirect

22. C₂⁴×C₉₇

C₂²

semidirect

25. C₂×D₄×C₉₇

C₂²

semidirect

26. C₂²×C₄×C₉₇

C₂²

semidirect

30. C₂³×C₉₇

D₄

semidirect

73. C₉₇

#534

semidirect

76. C₂²×D₄

D₉₇

semidirect

85. C₂⁴

C₂×D₉₇

semidirect

92. C₂×D₄

C₂×D₉₇

semidirect

95. C₂²×C₄

C₂×D₉₇

semidirect

107. C₂³

C₂²⋊D₉₇

semidirect

Subgroups — #20261294

Conjugacy classes of subgroups

Complements