Group #22560495 — subgroups

Subgroups — #22560495

Order 8576.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16863424	C₂	1	2	yes	maximal, complemented
2	#16863370	C₂	1	1	yes	maximal, characteristic
3	#16863363	C₂	1	1	yes	maximal, characteristic
4	#16863167	C₂	1	2	yes	maximal, complemented
5	#568×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	#14524002	C₂²	1	2	yes	—
9	Dic₆₇⋊C₈	—	2	4	no	—
10	#14523958	C₄	1	4	yes	—
11	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
12	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
13	#131×C₆₇	—	2	4	no	—
14	C₂×M₁₆×C₆₇	—	2	2	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
16	#11761612	—	2	2	no	—
17	C₂×C₆₇⋊C₈	—	2	1	no	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
22	C₄×Dic₆₇	—	2	2	no	—
23	#41×C₆₇	—	2	2	no	—
24	#41×C₆₇	—	2	2	no	—
25	C₂×Q₈×C₆₇	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₆₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₆₇	—	2	4	no	—
31	M₁₆×C₆₇	—	2	2	no	—
32	M₁₆×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	D₄	1	4	yes	—
35	C₂³×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₆₇	—	2	2	no	—
38	D₄×C₆₇	—	2	4	no	—
39	D₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#41	1	2	yes	—
47	C₂×Dic₆₇	—	2	4	no	—
48	C₈×C₆₇	—	2	4	no	—
49	C₆₇⋊C₈	—	4	1	no	—
50	C₆₇⋊C₈	—	4	1	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	—	4	1	no	—
53	C₂²×C₆₇	—	4	2	no	—
54	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₆₇	C₄×D₄	1	2	yes	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	#120	1	1	yes	characteristic
58	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₆₇	#123	1	1	yes	characteristic
60	Dic₆₇	—	4	4	no	—
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	—	2	1	no	—
63	C₂×C₆₇	—	4	2	no	—
64	C₂×C₆₇	#385	1	1	yes	characteristic
65	#1868	—	67	1	no	maximal, Sylow
66	C₆₇	#1868	1	1	yes	characteristic, complemented, Sylow
67	#568	D₆₇	1	1	yes	characteristic
68	#424	—	67	2	no	—
69	#410	—	67	1	no	—
70	#408	—	67	1	no	—
71	#344	—	67	2	no	—
72	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	67	1	no	—
76	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
77	C₂²×C₈	—	67	1	no	—
78	#117	—	67	2	no	—
79	C₄⋊C₈	—	134	4	no	—
80	#99	—	67	4	no	—
81	C₂²⋊C₈	—	134	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₁₃₄	1	1	yes	characteristic
85	C₂×D₄	Dic₁₃₄	1	1	yes	characteristic
86	C₂×D₄	C₄×D₆₇	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	134	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	134	1	no	—
95	C₂×C₈	—	134	1	no	—
96	C₂×C₈	—	134	2	no	—
97	C₂×C₈	C₂²⋊D₆₇	1	4	yes	—
98	C₄⋊C₄	—	134	2	no	—
99	C₂²⋊C₄	—	67	4	no	—
100	C₄²	—	134	2	no	—
101	C₂³	Dic₆₇⋊C₄	1	2	yes	—
102	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	134	4	no	—
109	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
113	C₂×C₄	#11761628	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	268	1	no	—
116	C₈	—	268	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
121	C₂²	C₄×C₂²⋊D₆₇	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	268	4	no	—
124	C₄	#14524012	1	1	yes	characteristic
125	C₄	#14524022	1	2	yes	—
126	C₄	#14524026	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#16863269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863424

C₂

yes

maximal, complemented

#16863370

C₂

yes

maximal, characteristic

#16863363

C₂

yes

maximal, characteristic

#16863167

C₂

yes

maximal, complemented

#568×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

Dic₆₇⋊C₈

—

#14523958

C₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂×#41×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂×D₄×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

C₄⋊C₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#123

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#568

D₆₇

yes

—

—

—

—

C₂×#41

C₂×D₆₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

#117

—

C₄⋊C₈

—

134

—

#99

—

C₂²⋊C₈

—

134

—

#41

—

#41

—

C₂×Q₈

Dic₁₃₄

yes

characteristic

C₂×D₄

Dic₁₃₄

yes

characteristic

C₂×D₄

C₄×D₆₇

yes

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

100

C₄²

—

134

—

101

C₂³

Dic₆₇⋊C₄

yes

—

102

C₂³

C₂×C₄×D₆₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

134

—

109

C₂×C₄

Dic₆₇⋊C₄

yes

—

110

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

112

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

113

C₂×C₄

#11761628

yes

—

114

C₈

—

115

C₈

—

268

—

116

C₈

—

268

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×Dic₆₇⋊C₄

yes

characteristic

121

C₂²

C₄×C₂²⋊D₆₇

yes

—

122

C₄

—

123

C₄

—

268

—

124

C₄

#14524012

yes

characteristic

125

C₄

#14524022

yes

—

126

C₄

#14524026

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16863269

yes

characteristic

	subgroup	G/H
center	C₄	#14524012
commutator	C₂³×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#568×C₆₇	C₂
socle	C₂×C₆₇	#385
hypercenter	#568	D₆₇
hypocenter	C₆₇	#1868
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#14524012

commutator

C₂³×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#568×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #16863424	C₂	semidirect
4. #16863167	C₂	semidirect
66. C₆₇	#1868	semidirect

H ≅ G/N

type

1. #16863424

C₂

semidirect

4. #16863167

C₂

semidirect

66. C₆₇

#1868

semidirect

Subgroups — #22560495

Conjugacy classes of subgroups

Complements