Group #22562005 — subgroups

Subgroups — #22562005

Order 8576.

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

normal subgroups only

This group has 160 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	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆₇	C₄	1	1	yes	characteristic
8	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₆₇	C₄	1	1	yes	characteristic
10	C₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×Dic₆₇	C₈	1	1	yes	characteristic
16	C₂²×Dic₆₇	C₈	1	1	yes	characteristic
17	C₂×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₆₇	C₈	1	1	yes	characteristic
21	C₄×Dic₆₇	C₈	1	1	yes	characteristic
22	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
24	C₆₇⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	—	2	2	no	—
27	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₁₆×C₆₇	—	2	2	no	—
30	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	1	no	—
32	C₂×C₄×C₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₆₇	—	2	2	no	—
36	C₂×Dic₆₇	C₁₆	1	4	yes	complemented
37	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
41	C₈×C₆₇	—	2	1	no	—
42	C₈×C₆₇	#41	1	2	yes	—
43	C₆₇⋊C₈	—	2	1	no	—
44	C₆₇⋊C₈	M₁₆	1	2	yes	—
45	C₂²×C₆₇	—	2	1	no	—
46	C₂²×C₆₇	—	2	1	no	—
47	C₂²×C₆₇	C₂×C₁₆	1	2	yes	—
48	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	#117	1	1	yes	characteristic
51	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
52	Dic₆₇	—	2	1	no	—
53	Dic₆₇	C₂×C₁₆	1	2	yes	—
54	Dic₆₇	C₂×C₁₆	1	2	yes	—
55	Dic₆₇	M₃₂	1	2	yes	—
56	C₂×C₆₇	—	2	2	no	—
57	C₂×C₆₇	#431	1	1	yes	characteristic
58	C₂×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
60	#2162	—	67	1	no	maximal, Sylow
61	C₆₇	#2162	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	67	1	no	—
63	C₄⋊C₁₆	—	67	2	no	—
64	C₂²⋊C₁₆	—	67	1	no	—
65	C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
66	C₄×C₁₆	—	67	2	no	—
67	C₂²×C₈	—	67	1	no	—
68	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
69	C₂×C₄²	—	67	1	no	—
70	C₂×C₁₆	—	67	2	no	—
71	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
72	C₄×C₈	—	67	1	no	—
73	C₄×C₈	—	67	1	no	—
74	C₄×C₈	—	67	1	no	—
75	C₄×C₈	—	67	1	no	—
76	C₂²×C₄	—	67	1	no	—
77	C₂²×C₄	—	67	1	no	—
78	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	67	1	no	—
81	C₂×C₈	—	67	1	no	—
82	C₂×C₈	—	134	2	no	—
83	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
85	C₄²	—	67	1	no	—
86	C₄²	—	67	1	no	—
87	C₄²	—	67	1	no	—
88	C₄²	—	67	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	134	2	no	—
91	C₂³	C₈×D₆₇	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	67	4	no	—
95	C₂×C₄	—	67	1	no	—
96	C₂×C₄	—	67	1	no	—
97	C₂×C₄	—	67	1	no	—
98	C₂×C₄	—	67	1	no	—
99	C₂×C₄	—	134	2	no	—
100	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	67	2	no	—
104	C₈	—	134	1	no	—
105	C₈	#11761627	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₁₆×D₆₇	1	2	yes	—
109	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	67	2	no	—
112	C₄	—	67	2	no	—
113	C₄	—	67	2	no	—
114	C₄	—	134	1	no	—
115	C₄	#14523999	1	1	yes	characteristic
116	C₄	M₁₆×D₆₇	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#16863446	1	1	yes	characteristic
119	C₂	C₂×C₁₆×D₆₇	1	1	yes	characteristic
120	C₂	M₃₂×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic

C₁₆×Dic₆₇

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₈

yes

characteristic

C₂²×Dic₆₇

C₈

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

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×Dic₆₇

C₁₆

yes

complemented

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

#41

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₁₆

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

C₂×C₁₆

yes

—

Dic₆₇

C₂×C₁₆

yes

—

Dic₆₇

M₃₂

yes

—

C₂×C₆₇

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

C₂²×C₁₆

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₆₇

yes

characteristic

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

C₂³

C₈×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

100

C₂×C₄

C₈×D₆₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

134

—

105

C₈

#11761627

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₁₆×D₆₇

yes

—

109

C₂²

C₂×C₈×D₆₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

134

—

115

C₄

#14523999

yes

characteristic

116

C₄

M₁₆×D₆₇

yes

characteristic

117

C₂

—

118

C₂

#16863446

yes

characteristic

119

C₂

C₂×C₁₆×D₆₇

yes

characteristic

120

C₂

M₃₂×D₆₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂²×C₁₆
Fitting	C₂²⋊C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₈
hypercenter	C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	#2162
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator

C₂×C₆₇

C₂²×C₁₆

Fitting

C₂²⋊C₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×C₈

hypercenter

C₂²⋊C₁₆

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. Dic₆₇⋊C₁₆	C₂	semidirect
4. C₁₆×Dic₆₇	C₂	semidirect
36. C₂×Dic₆₇	C₁₆	semidirect
61. C₆₇	#2162	semidirect

H ≅ G/N

type

2. Dic₆₇⋊C₁₆

C₂

semidirect

4. C₁₆×Dic₆₇

C₂

semidirect

36. C₂×Dic₆₇

C₁₆

semidirect

61. C₆₇

#2162

semidirect

Subgroups — #22562005

Conjugacy classes of subgroups

Complements