Group #22561651 — subgroups

Subgroups — #22561651

Order 8576.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#14523977	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863398	C₂	1	2	yes	maximal
3	#16863362	C₂	1	1	yes	maximal, characteristic
4	#16863164	C₂	1	2	yes	maximal, complemented
5	#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	1	yes	characteristic
7	C₂×#11761612	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	#14523977	—	2	2	no	—
10	#14523968	—	2	2	no	—
11	#14523965	—	2	2	no	—
12	(C₂×D₆₇)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#11761628	—	2	2	no	—
16	#11761628	—	2	2	no	—
17	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
18	C₂×D₂₆₈	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₆₇	—	2	1	no	—
20	C₂²×Dic₆₇	—	2	2	no	—
21	C₂×C₄×D₆₇	—	2	1	no	—
22	#11761612	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	2	1	no	—
24	C₄⋊Dic₆₇	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₆₇	—	2	2	no	—
28	M₁₆×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	D₄	1	2	yes	—
32	C₂³×C₆₇	#41	1	1	yes	characteristic
33	C₂²×D₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	#41	1	2	yes	—
37	C₂×C₄×C₆₇	#41	1	2	yes	—
38	C₂²⋊D₆₇	—	4	2	no	—
39	C₂×Dic₆₇	—	2	1	no	—
40	C₂×Dic₆₇	—	4	2	no	—
41	C₂×Dic₆₇	—	8	2	no	—
42	D₂₆₈	—	2	2	no	—
43	C₄×D₆₇	—	4	1	no	—
44	C₄×D₆₇	—	4	1	no	—
45	Dic₁₃₄	—	2	2	no	—
46	C₈×C₆₇	—	2	2	no	—
47	C₈×C₆₇	—	4	2	no	—
48	C₆₇⋊C₈	—	4	1	no	—
49	C₂²×C₆₇	—	2	1	no	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	#121	1	1	yes	characteristic
52	C₂²×C₆₇	#126	1	2	yes	—
53	C₂×D₆₇	—	4	1	no	—
54	C₂×D₆₇	—	4	1	no	—
55	C₂×D₆₇	—	4	1	no	—
56	C₄×C₆₇	#123	1	2	yes	—
57	C₄×C₆₇	#123	1	1	yes	characteristic
58	C₄×C₆₇	#124	1	1	yes	characteristic
59	Dic₆₇	—	4	1	no	—
60	Dic₆₇	—	8	2	no	—
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	#397	1	1	yes	characteristic
63	C₂×C₆₇	#472	1	1	yes	characteristic
64	C₂×C₆₇	#473	1	1	yes	characteristic
65	D₆₇	—	8	1	no	—
66	#2078	—	67	1	no	maximal, Sylow
67	C₆₇	#2078	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	67	1	no	—
69	#417	—	67	2	no	—
70	#408	—	67	1	no	—
71	#343	—	67	2	no	—
72	#341	D₆₇	1	1	yes	characteristic, complemented
73	C₂×#41	—	67	1	no	—
74	C₂×M₁₆	—	67	1	no	—
75	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
76	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	67	2	no	—
78	#107	—	134	2	no	—
79	#103	—	134	2	no	—
80	#102	—	134	2	no	—
81	C₂²⋊C₈	—	134	2	no	—
82	#41	—	134	2	no	—
83	#41	—	134	2	no	—
84	C₂×Q₈	—	67	1	no	—
85	C₂×D₄	—	67	1	no	—
86	C₂×D₄	—	134	1	no	—
87	C₂²×C₄	—	134	1	no	—
88	C₂²×C₄	—	134	2	no	—
89	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	134	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	134	1	no	—
96	C₂×C₈	D₂₆₈	1	2	yes	—
97	C₄⋊C₄	—	134	2	no	—
98	C₄⋊C₄	—	134	2	no	—
99	C₂³	—	134	1	no	—
100	C₂³	#11761629	1	1	yes	characteristic
101	Q₈	—	134	2	no	—
102	D₄	—	268	2	no	—
103	D₄	—	134	2	no	—
104	C₂×C₄	—	268	1	no	—
105	C₂×C₄	—	268	1	no	—
106	C₂×C₄	—	268	2	no	—
107	C₂×C₄	—	536	2	no	—
108	C₂×C₄	—	134	1	no	—
109	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
111	C₂×C₄	#11761627	1	2	yes	—
112	C₂×C₄	#11761628	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	268	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	268	1	no	—
119	C₂²	—	268	1	no	—
120	C₂²	—	268	1	no	—
121	C₂²	#14524018	1	1	yes	characteristic
122	C₂²	#14524038	1	2	yes	—
123	C₄	—	268	1	no	—
124	C₄	—	536	2	no	—
125	C₄	#14524025	1	2	yes	—
126	C₄	#14524027	1	1	yes	characteristic
127	C₄	#14524031	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	536	1	no	—
130	C₂	#16863323	1	1	yes	characteristic
131	C₂	#16863667	1	1	yes	characteristic
132	C₂	#16863672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal, characteristic, complemented

#16863398

C₂

yes

maximal

#16863362

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal, complemented

#341×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523977

—

#14523968

—

#14523965

—

(C₂×D₆₇)⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#11761628

—

#11761628

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#126

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#123

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#2078

—

maximal, Sylow

C₆₇

#2078

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

134

—

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

134

—

#41

—

134

—

#41

—

134

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂³

—

134

—

100

C₂³

#11761629

yes

characteristic

101

Q₈

—

134

—

102

D₄

—

268

—

103

D₄

—

134

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

268

—

106

C₂×C₄

—

268

—

107

C₂×C₄

—

536

—

108

C₂×C₄

—

134

—

109

C₂×C₄

D₄×D₆₇

yes

characteristic

110

C₂×C₄

C₂×D₂₆₈

yes

characteristic

111

C₂×C₄

#11761627

yes

—

112

C₂×C₄

#11761628

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

268

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

268

—

119

C₂²

—

268

—

120

C₂²

—

268

—

121

C₂²

#14524018

yes

characteristic

122

C₂²

#14524038

yes

—

123

C₄

—

268

—

124

C₄

—

536

—

125

C₄

#14524025

yes

—

126

C₄

#14524027

yes

characteristic

127

C₄

#14524031

yes

characteristic

128

C₂

—

129

C₂

—

536

—

130

C₂

#16863323

yes

characteristic

131

C₂

#16863667

yes

characteristic

132

C₂

#16863672

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#341×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#341	D₆₇
hypocenter	C₆₇	#2078
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14524018

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#341×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#14523977	C₂	semidirect
4. #16863164	C₂	semidirect
5. #341×C₆₇	C₂	semidirect
67. C₆₇	#2078	semidirect
72. #341	D₆₇	semidirect

H ≅ G/N

type

1. C₂×#14523977

C₂

semidirect

4. #16863164

C₂

semidirect

5. #341×C₆₇

C₂

semidirect

67. C₆₇

#2078

semidirect

72. #341

D₆₇

semidirect

Subgroups — #22561651

Conjugacy classes of subgroups

Complements