Group #20261352 — subgroups

Subgroups — #20261352

Order 6208.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#15690809	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690803	C₂	1	2	yes	maximal, complemented
3	#15690797	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690797	C₂	1	2	yes	maximal, complemented
5	#15690791	C₂	1	2	yes	maximal, complemented
6	#15690789	C₂	1	1	yes	maximal, characteristic, complemented
7	#15690787	C₂	1	2	yes	maximal, complemented
8	#15690775	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#15690769	C₂	1	1	yes	maximal, characteristic, complemented
11	#121×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₄	C₂²	1	2	yes	—
14	C₂²×Dic₉₇	C₂²	1	2	yes	—
15	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	#13095490	C₂²	1	2	yes	—
19	#13095490	C₂²	1	2	yes	—
20	#13095490	C₂²	1	2	yes	—
21	C₂²⋊Dic₉₇	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₉₇	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
26	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
27	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₉₇	C₂²	1	2	yes	—
30	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
31	C₂×D₄×C₉₇	C₂²	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
35	C₂³×C₉₇	C₂³	1	2	yes	—
36	C₂³×C₉₇	C₂³	1	1	yes	characteristic
37	D₄×C₉₇	—	2	2	no	—
38	D₄×C₉₇	—	4	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	2	yes	—
42	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₉₇	—	4	2	no	—
44	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₉₇	C₂³	1	2	yes	—
47	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₉₇	C₂³	1	2	yes	—
50	Dic₁₉₄	—	2	2	no	—
51	Dic₁₉₄	—	4	2	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	—	2	2	no	—
54	C₂²×C₉₇	—	2	2	no	—
55	C₂²×C₉₇	—	2	1	no	—
56	C₂²×C₉₇	—	2	2	no	—
57	C₂²×C₉₇	—	2	1	no	—
58	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
59	C₄×C₉₇	—	2	1	no	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	#41	1	2	yes	—
62	Dic₉₇	—	2	2	no	—
63	Dic₉₇	—	2	1	no	—
64	Dic₉₇	—	2	2	no	—
65	Dic₉₇	—	2	1	no	—
66	Dic₉₇	—	2	1	no	—
67	Dic₉₇	—	2	1	no	—
68	C₂×C₉₇	—	4	1	no	—
69	C₂×C₉₇	—	4	2	no	—
70	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
73	C₉₇	#542	1	1	yes	characteristic, complemented, Sylow
74	#542	—	97	1	no	maximal, Sylow
75	C₄⋊Q₈	—	97	1	no	—
76	#126	—	97	2	no	—
77	#124	—	97	1	no	—
78	#124	—	97	2	no	—
79	#123	—	97	2	no	—
80	C₂²⋊Q₈	—	97	1	no	—
81	C₂²⋊Q₈	—	97	2	no	—
82	#121	D₉₇	1	1	yes	characteristic
83	C₄×Q₈	—	97	1	no	—
84	C₄×D₄	—	97	1	no	—
85	#117	—	97	1	no	—
86	C₂×Q₈	—	97	2	no	—
87	C₂×Q₈	—	97	1	no	—
88	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
89	C₂×D₄	C₂×D₉₇	1	2	yes	—
90	C₂²×C₄	—	97	2	no	—
91	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
92	C₄⋊C₄	—	97	2	no	—
93	C₄⋊C₄	—	97	2	no	—
94	C₄⋊C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	1	no	—
96	C₄⋊C₄	—	97	1	no	—
97	C₄⋊C₄	—	97	1	no	—
98	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
99	C₂²⋊C₄	—	97	1	no	—
100	C₂²⋊C₄	—	97	1	no	—
101	C₂²⋊C₄	—	97	2	no	—
102	C₂²⋊C₄	—	97	2	no	—
103	C₂²⋊C₄	—	97	2	no	—
104	C₂²⋊C₄	—	97	2	no	—
105	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
106	C₄²	—	97	1	no	—
107	C₄²	—	97	1	no	—
108	C₄²	—	97	2	no	—
109	C₂³	C₂²×D₉₇	1	1	yes	characteristic
110	C₂³	C₂²×D₉₇	1	2	yes	—
111	Q₈	—	388	2	no	—
112	Q₈	—	194	2	no	—
113	D₄	—	2	2	no	—
114	D₄	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	97	1	no	—
117	C₂×C₄	—	97	1	no	—
118	C₂×C₄	—	97	2	no	—
119	C₂×C₄	—	97	1	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	97	2	no	—
122	C₂×C₄	—	388	2	no	—
123	C₂×C₄	C₂²×D₉₇	1	2	yes	—
124	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂³×D₉₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	194	2	no	—
136	C₄	—	194	1	no	—
137	C₄	—	194	2	no	—
138	C₄	—	194	1	no	—
139	C₄	—	194	1	no	—
140	C₄	—	194	1	no	—
141	C₄	#13095510	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×#13095510	1	1	yes	characteristic
145	C₂	#15690855	1	1	yes	characteristic
146	C₂	#15690857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690809

C₂

yes

maximal, characteristic, complemented

#15690803

C₂

yes

maximal, complemented

#15690797

C₂

yes

maximal, characteristic, complemented

#15690797

C₂

yes

maximal, complemented

#15690791

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, characteristic, complemented

#15690787

C₂

yes

maximal, complemented

#15690775

C₂

yes

maximal, characteristic, complemented

D₄×Dic₉₇

C₂

yes

maximal, characteristic, complemented

#15690769

C₂

yes

maximal, characteristic, complemented

#121×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂²

yes

—

C₂²×Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

—

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

C₂²

yes

—

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂³×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#41

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

H(5,𝔽₂)

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

characteristic

C₉₇

#542

yes

characteristic, complemented, Sylow

#542

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

D₉₇

yes

characteristic

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂×D₄

C₂×D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

C₂×D₉₇

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

C₂²×D₉₇

yes

characteristic

110

C₂³

C₂²×D₉₇

yes

—

111

Q₈

—

388

—

112

Q₈

—

194

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

388

—

123

C₂×C₄

C₂²×D₉₇

yes

—

124

C₂×C₄

C₂²×D₉₇

yes

characteristic

125

C₂×C₄

C₂²×D₉₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₉₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

194

—

136

C₄

—

194

—

137

C₄

—

194

—

138

C₄

—

194

—

139

C₄

—

194

—

140

C₄

—

194

—

141

C₄

#13095510

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#13095510

yes

characteristic

145

C₂

#15690855

yes

characteristic

146

C₂

#15690857

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#121×C₉₇	C₂
hypercenter	#121	D₉₇
hypocenter	C₉₇	#542
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#121×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #15690809	C₂	semidirect
2. #15690803	C₂	semidirect
3. #15690797	C₂	semidirect
4. #15690797	C₂	semidirect
5. #15690791	C₂	semidirect
6. #15690789	C₂	semidirect
7. #15690787	C₂	semidirect
8. #15690775	C₂	semidirect
9. D₄×Dic₉₇	C₂	semidirect
10. #15690769	C₂	semidirect
73. C₉₇	#542	semidirect

H ≅ G/N

type

1. #15690809

C₂

semidirect

2. #15690803

C₂

semidirect

3. #15690797

C₂

semidirect

4. #15690797

C₂

semidirect

5. #15690791

C₂

semidirect

6. #15690789

C₂

semidirect

7. #15690787

C₂

semidirect

8. #15690775

C₂

semidirect

9. D₄×Dic₉₇

C₂

semidirect

10. #15690769

C₂

semidirect

73. C₉₇

#542

semidirect

Subgroups — #20261352

Conjugacy classes of subgroups

Complements