Group #20261345 — subgroups

Subgroups — #20261345

Order 6208.

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

normal subgroups only

This group has 192 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	#15690808	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690803	C₂	1	2	yes	maximal
3	#15690803	C₂	1	2	yes	maximal
4	#15690801	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690801	C₂	1	2	yes	maximal, complemented
6	#15690800	C₂	1	2	yes	maximal, complemented
7	#15690789	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#15690775	C₂	1	1	yes	maximal, characteristic, complemented
10	#15690769	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₉₇	C₂²	1	2	yes	—
17	C₄⋊Dic₉₇	C₂²	1	2	yes	—
18	#13095490	C₂²	1	2	yes	—
19	#13095490	C₂²	1	2	yes	—
20	#13095490	C₂²	1	2	yes	—
21	#13095490	C₂²	1	2	yes	—
22	#13095490	C₂²	1	2	yes	—
23	C₂²⋊Dic₉₇	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₉₇	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
26	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
27	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₉₇	C₂²	1	2	yes	—
29	C₄×Dic₉₇	C₂²	1	2	yes	—
30	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
35	C₂³×C₉₇	C₂³	1	1	yes	characteristic
36	Q₈×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₉₇	C₂³	1	2	yes	—
46	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₉₇	C₂³	1	2	yes	—
48	Dic₁₉₄	—	2	2	no	—
49	C₂²×C₉₇	—	2	1	no	—
50	C₂²×C₉₇	—	2	1	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
53	C₄×C₉₇	—	2	1	no	—
54	C₄×C₉₇	—	2	2	no	—
55	C₄×C₉₇	—	2	2	no	—
56	C₄×C₉₇	#41	1	2	yes	—
57	Dic₉₇	—	2	2	no	—
58	Dic₉₇	—	2	1	no	—
59	Dic₉₇	—	2	1	no	—
60	Dic₉₇	—	2	1	no	—
61	Dic₉₇	—	2	1	no	—
62	Dic₉₇	—	2	2	no	—
63	C₂×C₉₇	—	4	1	no	—
64	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
65	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
67	C₉₇	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	97	1	no	maximal, Sylow
69	C₄⋊Q₈	—	97	1	no	—
70	#126	—	97	2	no	—
71	#126	—	97	2	no	—
72	#125	—	97	2	no	—
73	#125	—	97	1	no	—
74	#125	—	97	2	no	—
75	C₂²⋊Q₈	—	97	1	no	—
76	C₂²⋊Q₈	D₉₇	1	1	yes	characteristic
77	C₄×Q₈	—	97	1	no	—
78	C₄×Q₈	—	97	1	no	—
79	#117	—	97	1	no	—
80	C₂×Q₈	—	97	1	no	—
81	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
83	C₄⋊C₄	—	97	2	no	—
84	C₄⋊C₄	—	97	2	no	—
85	C₄⋊C₄	—	97	2	no	—
86	C₄⋊C₄	—	97	2	no	—
87	C₄⋊C₄	—	97	2	no	—
88	C₄⋊C₄	—	97	1	no	—
89	C₄⋊C₄	—	97	1	no	—
90	C₄⋊C₄	—	97	1	no	—
91	C₄⋊C₄	—	97	2	no	—
92	C₄⋊C₄	—	97	2	no	—
93	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
95	C₂²⋊C₄	—	97	1	no	—
96	C₂²⋊C₄	—	97	1	no	—
97	C₂²⋊C₄	—	97	2	no	—
98	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
99	C₄²	—	97	1	no	—
100	C₄²	—	97	1	no	—
101	C₄²	—	97	2	no	—
102	C₄²	—	97	2	no	—
103	C₂³	C₂²×D₉₇	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	194	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	97	1	no	—
108	C₂×C₄	—	97	1	no	—
109	C₂×C₄	—	97	1	no	—
110	C₂×C₄	—	97	1	no	—
111	C₂×C₄	—	97	2	no	—
112	C₂×C₄	—	97	2	no	—
113	C₂×C₄	C₂²×D₉₇	1	2	yes	—
114	C₂×C₄	C₂²×D₉₇	1	2	yes	—
115	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₉₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	194	2	no	—
125	C₄	—	194	1	no	—
126	C₄	—	194	1	no	—
127	C₄	—	194	1	no	—
128	C₄	—	194	1	no	—
129	C₄	—	194	2	no	—
130	C₄	#13095510	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#13095510	1	1	yes	characteristic
133	C₂	#15690857	1	1	yes	characteristic
134	C₂	#15690858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690808

C₂

yes

maximal, characteristic, complemented

#15690803

C₂

yes

maximal

#15690803

C₂

yes

maximal

#15690801

C₂

yes

maximal, characteristic, complemented

#15690801

C₂

yes

maximal, complemented

#15690800

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, characteristic

Q₈×Dic₉₇

C₂

yes

maximal, characteristic, complemented

#15690775

C₂

yes

maximal, characteristic, complemented

#15690769

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

characteristic

C₄×Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

—

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

—

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#41

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

characteristic

C₉₇

#541

yes

characteristic, complemented, Sylow

#541

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

C₂×D₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂³

C₂²×D₉₇

yes

characteristic

104

Q₈

—

105

Q₈

—

194

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₉₇

yes

—

114

C₂×C₄

C₂²×D₉₇

yes

—

115

C₂×C₄

C₂²×D₉₇

yes

characteristic

116

C₂×C₄

C₂²×D₉₇

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₉₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

194

—

125

C₄

—

194

—

126

C₄

—

194

—

127

C₄

—

194

—

128

C₄

—

194

—

129

C₄

—

194

—

130

C₄

#13095510

yes

—

131

C₂

—

132

C₂

C₂×#13095510

yes

characteristic

133

C₂

#15690857

yes

characteristic

134

C₂

#15690858

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₂²⋊Q₈×C₉₇	C₂
hypercenter	C₂²⋊Q₈	D₉₇
hypocenter	C₉₇	#541
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₂²⋊Q₈×C₉₇

C₂

hypercenter

C₂²⋊Q₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15690808	C₂	semidirect
4. #15690801	C₂	semidirect
5. #15690801	C₂	semidirect
6. #15690800	C₂	semidirect
8. Q₈×Dic₉₇	C₂	semidirect
9. #15690775	C₂	semidirect
67. C₉₇	#541	semidirect

H ≅ G/N

type

1. #15690808

C₂

semidirect

4. #15690801

C₂

semidirect

5. #15690801

C₂

semidirect

6. #15690800

C₂

semidirect

8. Q₈×Dic₉₇

C₂

semidirect

9. #15690775

C₂

semidirect

67. C₉₇

#541

semidirect

Subgroups — #20261345

Conjugacy classes of subgroups

Complements