Group #22562013 — subgroups

Subgroups — #22562013

Order 8576.

138 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16863351	C₂	1	1	yes	maximal, characteristic
2	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	#16863171	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₄	1	1	yes	characteristic
9	#14524002	C₄	1	1	yes	characteristic
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14523953	C₄	1	1	yes	characteristic
13	#14523953	C₄	1	1	yes	characteristic
14	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	#11761612	—	2	2	no	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
23	C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
24	C₂²⋊Dic₆₇	C₈	1	1	yes	characteristic
25	C₂²⋊Dic₆₇	C₈	1	1	yes	characteristic
26	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
28	C₆₇⋊C₁₆	—	2	1	no	—
29	C₆₇⋊C₁₆	—	2	1	no	—
30	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	1	no	—
32	C₂×C₈×C₆₇	—	2	1	no	—
33	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₁₆×C₆₇	—	2	1	no	—
36	C₁₆×C₆₇	—	2	1	no	—
37	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
46	C₈×C₆₇	—	2	1	no	—
47	C₈×C₆₇	#41	1	1	yes	characteristic
48	C₈×C₆₇	#41	1	1	yes	characteristic
49	C₆₇⋊C₈	—	2	1	no	—
50	C₆₇⋊C₈	M₁₆	1	2	yes	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	—	2	1	no	—
54	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	#117	1	1	yes	characteristic
57	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
58	Dic₆₇	—	2	1	no	—
59	Dic₆₇	—	2	1	no	—
60	Dic₆₇	—	2	1	no	—
61	Dic₆₇	—	2	1	no	—
62	C₂×C₆₇	—	4	1	no	—
63	C₂×C₆₇	#431	1	1	yes	characteristic
64	C₂×C₆₇	#504	1	1	yes	characteristic
65	C₂×C₆₇	#504	1	1	yes	characteristic
66	#2164	—	67	1	no	maximal, Sylow
67	C₆₇	#2164	1	1	yes	characteristic, complemented, Sylow
68	#405	—	67	1	no	—
69	C₄⋊C₁₆	—	67	1	no	—
70	C₄⋊C₁₆	—	67	1	no	—
71	C₂²⋊C₁₆	—	67	1	no	—
72	C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
73	#346	—	67	1	no	—
74	C₄×C₁₆	—	67	1	no	—
75	C₂×M₁₆	—	67	1	no	—
76	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
77	#117	—	67	1	no	—
78	C₂×C₁₆	—	67	1	no	—
79	C₂×C₁₆	—	67	1	no	—
80	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
82	#97	—	67	1	no	—
83	#97	—	67	1	no	—
84	C₄×C₈	—	67	1	no	—
85	C₄×C₈	—	67	1	no	—
86	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
87	M₁₆	—	134	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	67	1	no	—
91	C₂×C₈	—	67	1	no	—
92	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
94	C₄⋊C₄	—	67	1	no	—
95	C₄⋊C₄	—	67	1	no	—
96	C₂²⋊C₄	—	67	1	no	—
97	C₂²⋊C₄	—	67	1	no	—
98	C₄²	—	67	1	no	—
99	C₄²	—	67	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	134	1	no	—
103	C₁₆	—	134	1	no	—
104	C₂³	C₈×D₆₇	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	67	1	no	—
108	C₂×C₄	—	67	1	no	—
109	C₂×C₄	—	67	1	no	—
110	C₂×C₄	—	67	1	no	—
111	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	67	2	no	—
115	C₈	—	134	1	no	—
116	C₈	#11761627	1	1	yes	characteristic
117	C₈	#11761627	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	134	1	no	—
124	C₄	—	134	1	no	—
125	C₄	—	134	1	no	—
126	C₄	—	134	1	no	—
127	C₄	#14523999	1	1	yes	characteristic
128	C₄	M₁₆×D₆₇	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	#16863446	1	1	yes	characteristic
131	C₂	#16863812	1	1	yes	characteristic
132	C₂	#16863814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863351

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic

#16863171

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₆₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₄

yes

characteristic

#14524002

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523953

C₄

yes

characteristic

#14523953

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₆₇

C₈

yes

characteristic

C₄⋊Dic₆₇

C₈

yes

characteristic

C₂²⋊Dic₆₇

C₈

yes

characteristic

C₂²⋊Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₆₇

yes

characteristic

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

134

—

103

C₁₆

—

134

—

104

C₂³

C₈×D₆₇

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₈×D₆₇

yes

characteristic

112

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

134

—

116

C₈

#11761627

yes

characteristic

117

C₈

#11761627

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₈×D₆₇

yes

characteristic

122

C₄

—

123

C₄

—

134

—

124

C₄

—

134

—

125

C₄

—

134

—

126

C₄

—

134

—

127

C₄

#14523999

yes

characteristic

128

C₄

M₁₆×D₆₇

yes

characteristic

129

C₂

—

130

C₂

#16863446

yes

characteristic

131

C₂

#16863812

yes

characteristic

132

C₂

#16863814

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₂²⋊C₁₆×C₆₇	C₂
hypercenter	C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	#2164
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

C₂²⋊C₁₆×C₆₇

C₂

hypercenter

C₂²⋊C₁₆

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. Dic₆₇⋊C₁₆	C₂	semidirect
3. Dic₆₇⋊C₁₆	C₂	semidirect
5. #16863171	C₂	semidirect
6. C₁₆×Dic₆₇	C₂	semidirect
67. C₆₇	#2164	semidirect

H ≅ G/N

type

2. Dic₆₇⋊C₁₆

C₂

semidirect

3. Dic₆₇⋊C₁₆

C₂

semidirect

5. #16863171

C₂

semidirect

6. C₁₆×Dic₆₇

C₂

semidirect

67. C₆₇

#2164

semidirect

Subgroups — #22562013

Conjugacy classes of subgroups

Complements