Group #22562010 — subgroups

Subgroups — #22562010

Order 8576.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₆₇)⋊C₁₆	C₂	1	2	yes	maximal
4	#16863172	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₆₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	C₄²×D₆₇	C₄	1	1	yes	characteristic
9	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₆₇	C₈	1	1	yes	characteristic
16	C₂×C₄×D₆₇	C₈	1	1	yes	characteristic
17	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₆₇	—	2	2	no	—
21	C₈×D₆₇	—	2	2	no	—
22	C₄×Dic₆₇	C₈	1	1	yes	characteristic
23	C₄×Dic₆₇	C₈	1	1	yes	characteristic
24	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
25	C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₆₇	—	2	2	no	—
30	C₂²×D₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
37	C₄×D₆₇	—	2	2	no	—
38	C₄×D₆₇	—	2	1	no	—
39	C₄×D₆₇	—	2	1	no	—
40	C₄×D₆₇	—	2	2	no	—
41	C₈×C₆₇	M₁₆	1	2	yes	—
42	C₈×C₆₇	#41	1	2	yes	—
43	C₆₇⋊C₈	—	2	1	no	—
44	C₆₇⋊C₈	—	2	1	no	—
45	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₆₇	—	2	1	no	—
47	C₂×D₆₇	—	2	1	no	—
48	C₂×D₆₇	M₃₂	1	2	yes	—
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	—	2	1	no	—
51	C₄×C₆₇	#117	1	1	yes	characteristic
52	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₆₇	—	2	1	no	—
54	Dic₆₇	—	2	1	no	—
55	Dic₆₇	M₃₂	1	2	yes	—
56	C₂×C₆₇	#431	1	1	yes	characteristic
57	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
59	D₆₇	—	2	2	no	—
60	#2163	—	67	1	no	maximal, Sylow
61	C₆₇	#2163	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	67	1	no	—
63	C₄⋊C₁₆	—	67	2	no	—
64	C₂²⋊C₁₆	—	67	2	no	—
65	#346	—	67	1	no	—
66	#346	D₆₇	1	1	yes	characteristic, complemented
67	C₂²×C₈	—	67	1	no	—
68	C₂²×C₈	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
76	C₂²×C₄	—	67	1	no	—
77	C₂²×C₄	—	67	1	no	—
78	C₂²×C₄	—	67	1	no	—
79	C₂×C₈	—	67	1	no	—
80	C₂×C₈	—	67	1	no	—
81	C₂×C₈	—	134	2	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	C₄×D₆₇	1	1	yes	characteristic
89	C₁₆	—	2	2	no	—
90	C₁₆	—	134	2	no	—
91	C₂³	—	67	1	no	—
92	C₂×C₄	—	67	1	no	—
93	C₂×C₄	—	67	1	no	—
94	C₂×C₄	—	67	1	no	—
95	C₂×C₄	—	134	2	no	—
96	C₂×C₄	—	134	2	no	—
97	C₂×C₄	—	134	1	no	—
98	C₂×C₄	—	134	1	no	—
99	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
100	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
102	C₈	—	134	1	no	—
103	C₈	—	134	1	no	—
104	C₈	#11761611	1	2	yes	—
105	C₈	#11761628	1	2	yes	—
106	C₂²	—	67	2	no	—
107	C₂²	—	134	1	no	—
108	C₂²	—	134	1	no	—
109	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	67	2	no	—
113	C₄	—	134	1	no	—
114	C₄	—	134	1	no	—
115	C₄	#14524000	1	1	yes	characteristic
116	C₄	C₂×#11761611	1	1	yes	characteristic
117	C₂	—	134	2	no	—
118	C₂	#16863447	1	1	yes	characteristic
119	C₂	M₃₂×D₆₇	1	1	yes	characteristic
120	C₂	M₃₂×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₆₇

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₆₇)⋊C₁₆

C₂

yes

maximal

#16863172

C₂

yes

maximal, characteristic, complemented

#346×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₇

C₄

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

characteristic

C₄²×D₆₇

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₄×C₆₇⋊C₈

C₄

yes

characteristic

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₈

yes

characteristic

C₂×C₄×D₆₇

C₈

yes

characteristic

C₂×C₄×D₆₇

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

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

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

M₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

M₃₂

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

M₃₂

yes

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

characteristic

D₆₇

—

#2163

—

maximal, Sylow

C₆₇

#2163

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

D₆₇

yes

characteristic, complemented

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₈×D₆₇

yes

characteristic

100

C₂×C₄

C₈×D₆₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

102

C₈

—

134

—

103

C₈

—

134

—

104

C₈

#11761611

yes

—

105

C₈

#11761628

yes

—

106

C₂²

—

107

C₂²

—

134

—

108

C₂²

—

134

—

109

C₂²

C₂×C₈×D₆₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

134

—

114

C₄

—

134

—

115

C₄

#14524000

yes

characteristic

116

C₄

C₂×#11761611

yes

characteristic

117

C₂

—

134

—

118

C₂

#16863447

yes

characteristic

119

C₂

M₃₂×D₆₇

yes

characteristic

120

C₂

M₃₂×D₆₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	#346×C₆₇	C₂
hypercenter	#346	D₆₇
hypocenter	C₆₇	#2163
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

#346×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₆₇⋊C₁₆	C₂	semidirect
4. #16863172	C₂	semidirect
5. #346×C₆₇	C₂	semidirect
61. C₆₇	#2163	semidirect
66. #346	D₆₇	semidirect

H ≅ G/N

type

2. Dic₆₇⋊C₁₆

C₂

semidirect

4. #16863172

C₂

semidirect

5. #346×C₆₇

C₂

semidirect

61. C₆₇

#2163

semidirect

66. #346

D₆₇

semidirect

Subgroups — #22562010

Conjugacy classes of subgroups

Complements