Group #22578095 — subgroups

Subgroups — #22578095

Order 8768.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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	#17984083	C₂	1	2	yes	maximal, complemented
2	#17984079	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984079	C₂	1	2	yes	maximal, complemented
4	#17984072	C₂	1	2	yes	maximal
5	#17984064	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#17984046	C₂	1	1	yes	maximal, characteristic, complemented
9	#17984044	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊D₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
18	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
19	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
20	#14555489	C₂²	1	2	yes	—
21	#14555489	C₂²	1	2	yes	—
22	#14555489	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
24	#14555487	C₂²	1	2	yes	—
25	#14555487	C₂²	1	1	yes	characteristic
26	#14555487	C₂²	1	2	yes	—
27	C₄×Dic₁₃₇	C₂²	1	2	yes	—
28	C₄×Dic₁₃₇	C₂²	1	2	yes	—
29	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
35	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²⋊D₁₃₇	—	2	2	no	—
37	C₂×Dic₁₃₇	—	2	4	no	—
38	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₃₇	C₂³	1	2	yes	—
40	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₃₇	C₂³	1	2	yes	—
43	C₄×D₁₃₇	—	2	2	no	—
44	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
45	Q₈×C₁₃₇	—	2	2	no	—
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₁₃₇	—	2	1	no	—
52	C₂²×C₁₃₇	—	2	1	no	—
53	C₂²×C₁₃₇	#41	1	2	yes	—
54	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₃₇	—	2	1	no	—
56	C₂×D₁₃₇	—	2	1	no	—
57	C₂×D₁₃₇	—	2	1	no	—
58	C₄×C₁₃₇	—	2	1	no	—
59	C₄×C₁₃₇	—	2	2	no	—
60	C₄×C₁₃₇	—	2	2	no	—
61	C₄×C₁₃₇	#41	1	2	yes	—
62	Dic₁₃₇	—	2	1	no	—
63	Dic₁₃₇	—	2	2	no	—
64	Dic₁₃₇	—	2	2	no	—
65	Dic₁₃₇	—	2	1	no	—
66	Dic₁₃₇	—	2	1	no	—
67	C₂×C₁₃₇	—	2	2	no	—
68	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₃₇	—	4	1	no	—
72	C₁₃₇	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	137	1	no	maximal, Sylow
74	#126	—	137	2	no	—
75	#125	—	137	2	no	—
76	#125	—	137	1	no	—
77	#123	—	137	2	no	—
78	C₂²⋊Q₈	—	137	1	no	—
79	C₂²⋊Q₈	D₁₃₇	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	137	1	no	—
81	C₄×D₄	—	137	1	no	—
82	#117	—	137	1	no	—
83	#117	—	137	2	no	—
84	C₂×C₄⋊C₄	—	137	1	no	—
85	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
86	C₂×D₄	—	137	1	no	—
87	C₂²×C₄	—	137	2	no	—
88	C₂²×C₄	—	137	1	no	—
89	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
90	C₄⋊C₄	—	137	2	no	—
91	C₄⋊C₄	—	137	2	no	—
92	C₄⋊C₄	—	137	1	no	—
93	C₄⋊C₄	—	137	1	no	—
94	C₄⋊C₄	—	137	1	no	—
95	C₄⋊C₄	—	137	2	no	—
96	C₄⋊C₄	—	137	2	no	—
97	C₄⋊C₄	—	137	2	no	—
98	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
100	C₂²⋊C₄	—	137	1	no	—
101	C₂²⋊C₄	—	137	2	no	—
102	C₂²⋊C₄	—	137	1	no	—
103	C₂²⋊C₄	—	137	2	no	—
104	C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
105	C₄²	—	137	2	no	—
106	C₄²	—	137	2	no	—
107	C₄²	—	137	1	no	—
108	C₂³	—	137	1	no	—
109	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	274	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	274	4	no	—
114	C₂×C₄	—	274	2	no	—
115	C₂×C₄	—	137	1	no	—
116	C₂×C₄	—	137	1	no	—
117	C₂×C₄	—	137	2	no	—
118	C₂×C₄	—	137	1	no	—
119	C₂×C₄	—	137	2	no	—
120	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
121	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
122	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	274	1	no	—
127	C₂²	—	274	1	no	—
128	C₂²	—	274	1	no	—
129	C₂²	#14555511	1	2	yes	—
130	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	274	1	no	—
135	C₄	—	274	2	no	—
136	C₄	—	274	2	no	—
137	C₄	—	274	1	no	—
138	C₄	—	274	1	no	—
139	C₄	#14555509	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	548	1	no	—
142	C₂	C₂×#14555509	1	1	yes	characteristic
143	C₂	C₂×#14555511	1	1	yes	characteristic
144	C₂	#17984135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984083

C₂

yes

maximal, complemented

#17984079

C₂

yes

maximal, characteristic, complemented

#17984079

C₂

yes

maximal, complemented

#17984072

C₂

yes

maximal

#17984064

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₃₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₇

C₂

yes

maximal, characteristic

#17984046

C₂

yes

maximal, characteristic, complemented

#17984044

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

—

#14555489

C₂²

yes

—

#14555489

C₂²

yes

characteristic

C₂²⋊Dic₁₃₇

C₂²

yes

characteristic

#14555487

C₂²

yes

—

#14555487

C₂²

yes

characteristic

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂²⋊C₄×C₁₃₇

C₂²

yes

—

C₂²×D₁₃₇

C₂³

yes

characteristic

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#41

yes

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

2₋¹⁺⁴

yes

characteristic

D₁₃₇

—

C₁₃₇

#552

yes

characteristic, complemented, Sylow

#552

—

137

maximal, Sylow

#126

—

137

—

#125

—

137

—

#125

—

137

—

#123

—

137

—

C₂²⋊Q₈

—

137

—

C₂²⋊Q₈

D₁₃₇

yes

characteristic, complemented

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

#117

—

137

—

#117

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

C₂×D₁₃₇

yes

—

100

C₂²⋊C₄

—

137

—

101

C₂²⋊C₄

—

137

—

102

C₂²⋊C₄

—

137

—

103

C₂²⋊C₄

—

137

—

104

C₂²⋊C₄

C₂×D₁₃₇

yes

—

105

C₄²

—

137

—

106

C₄²

—

137

—

107

C₄²

—

137

—

108

C₂³

—

137

—

109

C₂³

C₂²×D₁₃₇

yes

characteristic

110

Q₈

—

111

D₄

—

274

—

112

C₂×C₄

—

113

C₂×C₄

—

274

—

114

C₂×C₄

—

274

—

115

C₂×C₄

—

137

—

116

C₂×C₄

—

137

—

117

C₂×C₄

—

137

—

118

C₂×C₄

—

137

—

119

C₂×C₄

—

137

—

120

C₂×C₄

C₂²×D₁₃₇

yes

—

121

C₂×C₄

C₂²×D₁₃₇

yes

—

122

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

123

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

274

—

127

C₂²

—

274

—

128

C₂²

—

274

—

129

C₂²

#14555511

yes

—

130

C₂²

C₂³×D₁₃₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

274

—

135

C₄

—

274

—

136

C₄

—

274

—

137

C₄

—

274

—

138

C₄

—

274

—

139

C₄

#14555509

yes

—

140

C₂

—

141

C₂

—

548

—

142

C₂

C₂×#14555509

yes

characteristic

143

C₂

C₂×#14555511

yes

characteristic

144

C₂

#17984135

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₂²⋊Q₈×C₁₃₇	C₂
hypercenter	C₂²⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	#552
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₂²⋊Q₈×C₁₃₇

C₂

hypercenter

C₂²⋊Q₈

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #17984083	C₂	semidirect
2. #17984079	C₂	semidirect
3. #17984079	C₂	semidirect
5. #17984064	C₂	semidirect
6. Q₈×Dic₁₃₇	C₂	semidirect
8. #17984046	C₂	semidirect
9. #17984044	C₂	semidirect
10. C₂×C₄⋊Dic₁₃₇	C₂	semidirect
11. C₂²⋊Q₈×C₁₃₇	C₂	semidirect
72. C₁₃₇	#552	semidirect
79. C₂²⋊Q₈	D₁₃₇	semidirect

H ≅ G/N

type

1. #17984083

C₂

semidirect

2. #17984079

C₂

semidirect

3. #17984079

C₂

semidirect

5. #17984064

C₂

semidirect

6. Q₈×Dic₁₃₇

C₂

semidirect

8. #17984046

C₂

semidirect

9. #17984044

C₂

semidirect

10. C₂×C₄⋊Dic₁₃₇

C₂

semidirect

11. C₂²⋊Q₈×C₁₃₇

C₂

semidirect

72. C₁₃₇

#552

semidirect

79. C₂²⋊Q₈

D₁₃₇

semidirect

Subgroups — #22578095

Conjugacy classes of subgroups

Complements