Group #22560994 — subgroups

Subgroups — #22560994

Order 8576.

196 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#14523979	C₂	1	1	yes	maximal, characteristic
2	#16863439	C₂	1	2	yes	maximal
3	C₂×#14523977	C₂	1	1	yes	maximal, characteristic
4	#16863164	C₂	1	2	yes	maximal
5	#405×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	#14523979	C₄	1	4	yes	complemented
9	#14523977	—	2	1	no	—
10	#14523977	—	2	1	no	—
11	#14523976	—	2	2	no	—
12	#14523974	—	2	2	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#117×C₆₇	C₂²	1	1	yes	characteristic
16	#97×C₆₇	—	2	1	no	—
17	C₄×C₈×C₆₇	—	2	1	no	—
18	C₂²×Dic₆₇	—	2	2	no	—
19	#11761612	—	2	2	no	—
20	#11761612	C₂×C₄	1	4	yes	—
21	C₂×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₆₇	—	2	1	no	—
26	M₁₆×C₆₇	Q₈	1	2	yes	—
27	C₂×C₈×C₆₇	—	2	1	no	—
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	Q₈	1	2	yes	—
32	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₆₇	—	2	1	no	—
35	C₄²×C₆₇	—	2	1	no	—
36	C₂³×C₆₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	1	no	—
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₂×Dic₆₇	—	4	2	no	—
46	C₂×Dic₆₇	—	8	2	no	—
47	C₈×C₆₇	—	2	2	no	—
48	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₆₇⋊C₈	—	2	4	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	—	4	1	no	—
57	C₄×C₆₇	—	4	1	no	—
58	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
60	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
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	—	67	1	no	—
71	#428	—	67	2	no	—
72	C₂×#107	—	67	1	no	—
73	#405	D₆₇	1	1	yes	characteristic
74	#343	—	67	2	no	—
75	C₂×M₁₆	—	67	2	no	—
76	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
78	#117	C₂×D₆₇	1	1	yes	characteristic
79	C₂×C₄⋊C₄	—	67	2	no	—
80	#108	—	67	4	no	—
81	#107	—	134	1	no	—
82	#107	—	134	1	no	—
83	#107	—	134	2	no	—
84	#106	—	134	2	no	—
85	#97	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₂²×C₄	—	134	2	no	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	M₁₆	—	2	1	no	—
90	M₁₆	—	67	4	no	—
91	M₁₆	—	134	2	no	—
92	M₁₆	Dic₁₃₄	1	2	yes	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	134	2	no	—
95	C₂×C₈	Dic₁₃₄	1	2	yes	—
96	C₂×C₈	C₄×D₆₇	1	2	yes	—
97	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
99	C₄⋊C₄	—	134	2	no	—
100	C₄⋊C₄	—	134	2	no	—
101	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
102	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₂³	#11761628	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	268	2	no	—
109	C₂×C₄	—	536	2	no	—
110	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
115	C₂×C₄	#11761628	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	134	4	no	—
118	C₈	Dic₆₇⋊C₄	1	2	yes	—
119	C₈	Dic₆₇⋊C₄	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	C₄×Dic₁₃₄	1	1	yes	characteristic
123	C₂²	#14524017	1	1	yes	characteristic
124	C₂²	#14524035	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	536	2	no	—
128	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
129	C₄	C₄×D₂₆₈	1	1	yes	characteristic
130	C₄	#14524022	1	1	yes	characteristic
131	C₄	#14524043	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16863294	1	1	yes	characteristic
135	C₂	#16863666	1	1	yes	characteristic
136	C₂	#16863671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523979

C₂

yes

maximal, characteristic

#16863439

C₂

yes

maximal

C₂×#14523977

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal

#405×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523979

C₄

yes

—

—

—

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

Q₈

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

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

—

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

—

C₄×C₆₇

—

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

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

#343

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

#108

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

#106

—

134

—

#97

—

C₄×C₈

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

Dic₁₃₄

yes

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

100

C₄⋊C₄

—

134

—

101

C₄⋊C₄

D₂₆₈

yes

characteristic

102

C₄⋊C₄

D₂₆₈

yes

characteristic

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

#11761628

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

268

—

109

C₂×C₄

—

536

—

110

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

112

C₂×C₄

C₂×D₂₆₈

yes

characteristic

113

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

114

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

115

C₂×C₄

#11761628

yes

characteristic

116

C₈

—

117

C₈

—

134

—

118

C₈

Dic₆₇⋊C₄

yes

—

119

C₈

Dic₆₇⋊C₄

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄×Dic₁₃₄

yes

characteristic

123

C₂²

#14524017

yes

characteristic

124

C₂²

#14524035

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

536

—

128

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

129

C₄

C₄×D₂₆₈

yes

characteristic

130

C₄

#14524022

yes

characteristic

131

C₄

#14524043

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#16863294

yes

characteristic

135

C₂

#16863666

yes

characteristic

136

C₂

#16863671

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#405×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#405	D₆₇
hypocenter	C₆₇	#1952
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#14524017

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#405×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #14523979	C₄	semidirect
69. C₆₇	#1952	semidirect

H ≅ G/N

type

8. #14523979

C₄

semidirect

69. C₆₇

#1952

semidirect

Subgroups — #22560994

Conjugacy classes of subgroups

Complements