Group #22560995 — subgroups

Subgroups — #22560995

Order 8576.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16863438	C₂	1	2	yes	maximal
2	C₂×#14523977	C₂	1	1	yes	maximal, characteristic
3	#16863351	C₂	1	1	yes	maximal, characteristic
4	#16863164	C₂	1	2	yes	maximal
5	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	1	yes	characteristic
7	#14524002	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	#14523977	—	2	2	no	—
10	#14523977	—	2	1	no	—
11	#14523977	—	2	1	no	—
12	#14523975	—	2	2	no	—
13	#14523953	—	2	1	no	—
14	C₈×Dic₆₇	—	2	1	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#108×C₆₇	C₄	1	4	yes	complemented
18	C₂²×Dic₆₇	—	2	2	no	—
19	#11761612	—	2	1	no	—
20	#11761612	Q₈	1	2	yes	—
21	C₂×C₆₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
25	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
26	C₂²⋊Dic₆₇	—	2	1	no	—
27	C₄×Dic₆₇	—	2	1	no	—
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₆₇	—	2	2	no	—
30	M₁₆×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	Q₈	1	2	yes	—
36	C₂³×C₆₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
43	C₂×Dic₆₇	—	2	1	no	—
44	C₂×Dic₆₇	—	2	1	no	—
45	C₂×Dic₆₇	—	4	2	no	—
46	C₂×Dic₆₇	—	8	2	no	—
47	C₈×C₆₇	—	2	4	no	—
48	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₆₇	#121	1	1	yes	characteristic
55	C₂²×C₆₇	#125	1	1	yes	characteristic
56	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
58	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₆₇	—	4	1	no	—
61	Dic₆₇	—	4	1	no	—
62	Dic₆₇	—	8	2	no	—
63	C₂×C₆₇	—	2	1	no	—
64	C₂×C₆₇	—	2	1	no	—
65	C₂×C₆₇	#389	1	1	yes	characteristic
66	C₂×C₆₇	#472	1	1	yes	characteristic
67	C₂×C₆₇	#473	1	1	yes	characteristic
68	#1952	—	67	1	no	maximal, Sylow
69	C₆₇	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	D₆₇	1	1	yes	characteristic
71	#428	—	67	2	no	—
72	C₂×#107	—	67	1	no	—
73	#405	—	67	1	no	—
74	#343	—	67	2	no	—
75	C₂×M₁₆	—	67	1	no	—
76	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
77	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
78	#117	—	67	1	no	—
79	C₂×C₄⋊C₄	—	67	2	no	—
80	#108	Dic₆₇	1	4	yes	complemented
81	#107	—	134	2	no	—
82	#107	—	134	1	no	—
83	#107	—	134	1	no	—
84	#106	—	134	2	no	—
85	#97	—	134	1	no	—
86	C₄×C₈	—	134	1	no	—
87	C₂²×C₄	—	134	2	no	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	67	2	no	—
91	M₁₆	—	134	1	no	—
92	M₁₆	C₂×Dic₆₇	1	4	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	134	1	no	—
95	C₂×C₈	Dic₁₃₄	1	2	yes	—
96	C₂×C₈	D₂₆₈	1	1	yes	characteristic
97	C₂×C₈	D₂₆₈	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
99	C₄⋊C₄	—	67	1	no	—
100	C₄⋊C₄	—	67	1	no	—
101	C₄⋊C₄	—	134	2	no	—
102	C₄⋊C₄	—	134	2	no	—
103	C₂²⋊C₄	—	134	1	no	—
104	C₄²	—	134	1	no	—
105	C₂³	#11761629	1	1	yes	characteristic
106	C₂×C₄	—	268	2	no	—
107	C₂×C₄	—	536	2	no	—
108	C₂×C₄	—	134	1	no	—
109	C₂×C₄	—	134	1	no	—
110	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
113	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
115	C₂×C₄	#11761627	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	—	134	2	no	—
118	C₈	C₄⋊Dic₆₇	1	2	yes	—
119	C₈	C₄⋊Dic₆₇	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	Q₈×Dic₆₇	1	1	yes	characteristic
123	C₂²	#14524018	1	1	yes	characteristic
124	C₂²	#14524034	1	1	yes	characteristic
125	C₄	—	268	1	no	—
126	C₄	—	268	1	no	—
127	C₄	—	536	2	no	—
128	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
129	C₄	D₄×Dic₆₇	1	1	yes	characteristic
130	C₄	#14524023	1	1	yes	characteristic
131	C₄	#14524042	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16863295	1	1	yes	characteristic
135	C₂	#16863667	1	1	yes	characteristic
136	C₂	#16863672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863438

C₂

yes

maximal

C₂×#14523977

C₂

yes

maximal, characteristic

#16863351

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

#11761612

—

#11761612

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

D₄

yes

characteristic

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#108

D₆₇

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₆₇

yes

complemented

#107

—

134

—

#107

—

134

—

#107

—

134

—

#106

—

134

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

C₂×Dic₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

134

—

102

C₄⋊C₄

—

134

—

103

C₂²⋊C₄

—

134

—

104

C₄²

—

134

—

105

C₂³

#11761629

yes

characteristic

106

C₂×C₄

—

268

—

107

C₂×C₄

—

536

—

108

C₂×C₄

—

134

—

109

C₂×C₄

—

134

—

110

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

111

C₂×C₄

D₄×D₆₇

yes

characteristic

112

C₂×C₄

C₂×D₂₆₈

yes

characteristic

113

C₂×C₄

Q₈×D₆₇

yes

characteristic

114

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

115

C₂×C₄

#11761627

yes

characteristic

116

C₈

—

117

C₈

—

134

—

118

C₈

C₄⋊Dic₆₇

yes

—

119

C₈

C₄⋊Dic₆₇

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

Q₈×Dic₆₇

yes

characteristic

123

C₂²

#14524018

yes

characteristic

124

C₂²

#14524034

yes

characteristic

125

C₄

—

268

—

126

C₄

—

268

—

127

C₄

—

536

—

128

C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

129

C₄

D₄×Dic₆₇

yes

characteristic

130

C₄

#14524023

yes

characteristic

131

C₄

#14524042

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#16863295

yes

characteristic

135

C₂

#16863667

yes

characteristic

136

C₂

#16863672

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#1952
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14524018

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₆₇	C₄	semidirect
69. C₆₇	#1952	semidirect
80. #108	Dic₆₇	semidirect

H ≅ G/N

type

17. #108×C₆₇

C₄

semidirect

69. C₆₇

#1952

semidirect

80. #108

Dic₆₇

semidirect

Subgroups — #22560995

Conjugacy classes of subgroups

Complements