Group #22560334 — subgroups

Subgroups — #22560334

Order 8576.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16863434	C₂	1	1	yes	maximal, characteristic
2	#16863351	C₂	1	1	yes	maximal, characteristic
3	#16863168	C₂	1	2	yes	maximal
4	#16863153	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	2	yes	—
8	#14524002	C₂²	1	1	yes	characteristic
9	#14523977	—	2	1	no	—
10	#14523975	—	2	1	no	—
11	#14523953	C₄	1	2	yes	complemented
12	C₈×Dic₆₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#108×C₆₇	—	2	1	no	—
16	#108×C₆₇	—	2	1	no	—
17	#11761612	—	2	1	no	—
18	#11761612	—	2	1	no	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₆₇	—	2	1	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₂²⋊Dic₆₇	—	2	1	no	—
23	C₂²⋊Dic₆₇	—	2	2	no	—
24	C₄×Dic₆₇	—	2	2	no	—
25	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
43	C₂×Dic₆₇	—	2	2	no	—
44	C₂×Dic₆₇	—	4	2	no	—
45	C₈×C₆₇	—	4	2	no	—
46	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇	—	4	1	no	—
52	C₂²×C₆₇	#117	1	1	yes	characteristic
53	C₂²×C₆₇	#124	1	1	yes	characteristic
54	C₂²×C₆₇	#125	1	1	yes	characteristic
55	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₆₇	#117	1	1	yes	characteristic
57	C₄×C₆₇	#124	1	1	yes	characteristic
58	C₄×C₆₇	#125	1	1	yes	characteristic
59	Dic₆₇	—	4	2	no	—
60	Dic₆₇	—	8	2	no	—
61	C₂×C₆₇	—	2	1	no	—
62	C₂×C₆₇	—	2	1	no	—
63	C₂×C₆₇	—	2	1	no	—
64	C₂×C₆₇	#382	1	1	yes	characteristic
65	#1833	—	67	1	no	maximal, Sylow
66	C₆₇	#1833	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	D₆₇	1	1	yes	characteristic
68	#427	—	67	1	no	—
69	#405	—	67	1	no	—
70	#344	—	67	2	no	—
71	H(3,ℤ/4)	—	67	2	no	—
72	C₂×M₁₆	—	67	1	no	—
73	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
74	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
75	#117	—	67	1	no	—
76	#117	—	67	2	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	134	1	no	—
80	#106	—	134	1	no	—
81	#97	—	67	2	no	—
82	C₄×C₈	—	67	2	no	—
83	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	134	1	no	—
87	M₁₆	—	134	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	67	2	no	—
90	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
91	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
92	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
93	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
96	C₄⋊C₄	—	134	2	no	—
97	C₄⋊C₄	—	134	1	no	—
98	C₂²⋊C₄	—	134	2	no	—
99	C₂²⋊C₄	—	134	1	no	—
100	C₄²	—	67	2	no	—
101	C₄²	—	134	2	no	—
102	C₂³	#11761629	1	1	yes	characteristic
103	C₂×C₄	—	268	2	no	—
104	C₂×C₄	—	134	2	no	—
105	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
108	C₂×C₄	#11761627	1	1	yes	characteristic
109	C₂×C₄	#11761627	1	1	yes	characteristic
110	C₂×C₄	#11761629	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	134	2	no	—
113	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
114	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
115	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
116	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#14524001	1	1	yes	characteristic
119	C₂²	#14524032	1	1	yes	characteristic
120	C₂²	#14524034	1	1	yes	characteristic
121	C₄	—	268	2	no	—
122	C₄	—	536	2	no	—
123	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₄	#14524001	1	1	yes	characteristic
125	C₄	#14524030	1	1	yes	characteristic
126	C₄	#14524034	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16863261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863434

C₂

yes

maximal, characteristic

#16863351

C₂

yes

maximal, characteristic

#16863168

C₂

yes

maximal

#16863153

C₂

yes

maximal

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

—

#14524002

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₆₇

C₄

yes

complemented

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

C₂×C₄⋊C₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#108

D₆₇

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

—

—

—

—

—

134

—

#106

—

134

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

100

C₄²

—

101

C₄²

—

134

—

102

C₂³

#11761629

yes

characteristic

103

C₂×C₄

—

268

—

104

C₂×C₄

—

134

—

105

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

107

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

108

C₂×C₄

#11761627

yes

characteristic

109

C₂×C₄

#11761627

yes

characteristic

110

C₂×C₄

#11761629

yes

characteristic

111

C₈

—

112

C₈

—

134

—

113

C₈

Dic₆₇⋊C₄

yes

characteristic

114

C₈

Dic₆₇⋊C₄

yes

characteristic

115

C₈

Dic₆₇⋊C₄

yes

characteristic

116

C₈

Dic₆₇⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#14524001

yes

characteristic

119

C₂²

#14524032

yes

characteristic

120

C₂²

#14524034

yes

characteristic

121

C₄

—

268

—

122

C₄

—

536

—

123

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

124

C₄

#14524001

yes

characteristic

125

C₄

#14524030

yes

characteristic

126

C₄

#14524034

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16863261

yes

characteristic

	subgroup	G/H
center	C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂×C₆₇	#382
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#1833
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#14524001

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #14523953	C₄	semidirect
12. C₈×Dic₆₇	C₄	semidirect
66. C₆₇	#1833	semidirect

H ≅ G/N

type

11. #14523953

C₄

semidirect

12. C₈×Dic₆₇

C₄

semidirect

66. C₆₇

#1833

semidirect

Subgroups — #22560334

Conjugacy classes of subgroups

Complements