Group #22572888 — subgroups

Subgroups — #22572888

Order 8576.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16864125	C₂	1	1	yes	maximal, characteristic
3	#16863747	C₂	1	3	yes	maximal
4	#16863692	C₂	1	3	yes	maximal
5	#16863607	C₂	1	3	yes	maximal
6	C₄×Dic₂₆₈	C₂	1	3	yes	maximal
7	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	C₂²	1	3	yes	—
9	#14524043	C₂²	1	3	yes	—
10	#14524043	C₂²	1	3	yes	—
11	#14524042	—	2	3	no	—
12	#14524034	—	2	3	no	—
13	Q₈×Dic₆₇	—	2	1	no	—
14	Q₈×Dic₆₇	—	2	1	no	—
15	C₄×Dic₁₃₄	C₂²	1	3	yes	—
16	C₄×Dic₁₃₄	C₂²	1	3	yes	—
17	#14523977	C₂²	1	3	yes	—
18	#14523968	C₂²	1	3	yes	—
19	#14523968	C₂²	1	6	yes	—
20	#14523968	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
23	C₄×C₈×C₆₇	C₂²	1	3	yes	—
24	C₂×Dic₁₃₄	—	2	3	no	—
25	C₂×Dic₁₃₄	C₂³	1	3	yes	—
26	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
27	Dic₂₆₈	—	2	6	no	—
28	C₄⋊Dic₆₇	—	2	3	no	—
29	C₄⋊Dic₆₇	—	2	3	no	—
30	C₄⋊Dic₆₇	—	2	3	no	—
31	C₄⋊Dic₆₇	C₂³	1	3	yes	—
32	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
33	Dic₆₇⋊C₄	—	2	3	no	—
34	Dic₆₇⋊C₄	—	2	3	no	—
35	C₄×Dic₆₇	—	2	3	no	—
36	C₄×Dic₆₇	—	2	3	no	—
37	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂³	1	3	yes	—
39	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	D₄	1	3	yes	—
41	C₄²×C₆₇	C₂³	1	3	yes	—
42	Q₈×C₆₇	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
45	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₇	—	2	1	no	—
47	C₂×Dic₆₇	—	2	1	no	—
48	C₂×Dic₆₇	—	2	3	no	—
49	C₂×Dic₆₇	—	2	3	no	—
50	Dic₁₃₄	—	2	1	no	—
51	Dic₁₃₄	—	2	3	no	—
52	Dic₁₃₄	—	2	3	no	—
53	Dic₁₃₄	—	8	3	no	—
54	Dic₁₃₄	#41	1	2	yes	—
55	C₈×C₆₇	—	2	3	no	—
56	C₈×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₇	—	2	3	no	—
59	C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
60	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
61	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₆₇	—	2	2	no	—
63	Dic₆₇	—	4	3	no	—
64	Dic₆₇	—	4	3	no	—
65	Dic₆₇	—	4	1	no	—
66	C₂×C₆₇	#548	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₆₇	#577	1	1	yes	characteristic
69	#3363	—	67	1	no	maximal, Sylow
70	C₆₇	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	—	67	1	no	—
72	#557	—	67	1	no	—
73	#487	—	67	3	no	—
74	#477	—	67	3	no	—
75	#462	—	67	3	no	—
76	C₈×Q₈	D₆₇	1	1	yes	characteristic
77	C₄×Q₁₆	—	67	3	no	—
78	C₂×Q₁₆	—	67	3	no	—
79	C₄⋊Q₈	—	67	3	no	—
80	C₄⋊Q₈	—	67	3	no	—
81	C₄⋊Q₈	—	134	3	no	—
82	#125	—	134	3	no	—
83	C₄×Q₈	—	67	3	no	—
84	C₄×Q₈	—	67	3	no	—
85	C₄×Q₈	—	134	1	no	—
86	C₄×Q₈	—	134	1	no	—
87	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
88	#107	—	67	3	no	—
89	C₄⋊C₈	C₂×D₆₇	1	3	yes	—
90	#103	—	67	1	no	—
91	#103	—	67	3	no	—
92	#103	—	67	6	no	—
93	C₄×C₈	C₂×D₆₇	1	3	yes	—
94	C₂×Q₈	—	67	1	no	—
95	C₂×Q₈	—	67	3	no	—
96	C₂×Q₈	—	134	3	no	—
97	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
98	Q₁₆	—	134	6	no	—
99	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₆₇	1	3	yes	—
101	C₄⋊C₄	—	67	3	no	—
102	C₄⋊C₄	—	67	1	no	—
103	C₄⋊C₄	—	134	3	no	—
104	C₄⋊C₄	—	134	3	no	—
105	C₄⋊C₄	—	134	3	no	—
106	C₄⋊C₄	—	134	3	no	—
107	C₄⋊C₄	—	134	3	no	—
108	C₄⋊C₄	D₂₆₈	1	3	yes	—
109	C₄²	—	134	3	no	—
110	C₄²	—	134	3	no	—
111	C₄²	C₂²×D₆₇	1	3	yes	—
112	Q₈	—	536	3	no	—
113	Q₈	—	67	2	no	—
114	Q₈	—	134	3	no	—
115	Q₈	—	134	3	no	—
116	Q₈	—	134	1	no	—
117	Q₈	Dic₂₆₈	1	4	yes	complemented
118	C₂×C₄	—	134	1	no	—
119	C₂×C₄	—	134	3	no	—
120	C₂×C₄	—	134	3	no	—
121	C₂×C₄	—	134	1	no	—
122	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
123	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
124	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
125	C₈	—	2	3	no	—
126	C₈	—	2	1	no	—
127	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	268	3	no	—
130	C₄	—	268	3	no	—
131	C₄	—	268	1	no	—
132	C₄	—	134	2	no	—
133	C₄	C₂×Dic₂₆₈	1	6	yes	—
134	C₄	#41×D₆₇	1	1	yes	characteristic
135	C₄	#14524091	1	1	yes	characteristic
136	C₂	#16864058	1	1	yes	characteristic
137	C₂	C₂²×Dic₂₆₈	1	1	yes	characteristic
138	C₂	#16864208	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₃₄

C₂

yes

maximal, characteristic

#16864125

C₂

yes

maximal, characteristic

#16863747

C₂

yes

maximal

#16863692

C₂

yes

maximal

#16863607

C₂

yes

maximal

C₄×Dic₂₆₈

C₂

yes

maximal

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

#14524043

C₂²

yes

—

#14524043

C₂²

yes

—

#14524042

—

#14524034

—

Q₈×Dic₆₇

—

Q₈×Dic₆₇

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14523977

C₂²

yes

—

#14523968

C₂²

yes

—

#14523968

C₂²

yes

—

#14523968

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂×Dic₁₃₄

C₂³

yes

characteristic

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

characteristic

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

Q₈×C₆₇

Q₁₆

yes

complemented

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

#41

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#548

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

C₈×Q₈

D₆₇

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

#125

—

134

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×Q₈

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

#103

—

#103

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

D₂₆₈

yes

characteristic

Q₁₆

—

134

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

100

C₂×C₈

C₂²×D₆₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

D₂₆₈

yes

—

109

C₄²

—

134

—

110

C₄²

—

134

—

111

C₄²

C₂²×D₆₇

yes

—

112

Q₈

—

536

—

113

Q₈

—

114

Q₈

—

134

—

115

Q₈

—

134

—

116

Q₈

—

134

—

117

Q₈

Dic₂₆₈

yes

complemented

118

C₂×C₄

—

134

—

119

C₂×C₄

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

C₂×D₂₆₈

yes

—

123

C₂×C₄

C₂×D₂₆₈

yes

—

124

C₂×C₄

C₂³×D₆₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×D₂₆₈

yes

characteristic

128

C₄

—

129

C₄

—

268

—

130

C₄

—

268

—

131

C₄

—

268

—

132

C₄

—

134

—

133

C₄

C₂×Dic₂₆₈

yes

—

134

C₄

#41×D₆₇

yes

characteristic

135

C₄

#14524091

yes

characteristic

136

C₂

#16864058

yes

characteristic

137

C₂

C₂²×Dic₂₆₈

yes

characteristic

138

C₂

#16864208

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	#3363
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₈×Q₈

D₆₇

hypocenter

C₆₇

#3363

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
42. Q₈×C₆₇	Q₁₆	semidirect
70. C₆₇	#3363	semidirect
117. Q₈	Dic₂₆₈	semidirect

H ≅ G/N

type

42. Q₈×C₆₇

Q₁₆

semidirect

70. C₆₇

#3363

semidirect

117. Q₈

Dic₂₆₈

semidirect

Subgroups — #22572888

Conjugacy classes of subgroups

Complements