Group #22569146 — subgroups

Subgroups — #22569146

Order 8576.

348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 181 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	#16863953	C₂	1	1	yes	maximal, characteristic
2	#16863525	C₂	1	8	yes	maximal, complemented
3	#16863424	C₂	1	2	yes	maximal
4	#16863354	C₂	1	2	yes	maximal
5	C₂×#14523954	C₂	1	1	yes	maximal, characteristic
6	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#14524043	C₄	1	2	yes	—
10	#14524035	C₄	1	2	yes	—
11	#14524022	C₄	1	4	yes	—
12	C₄×Dic₁₃₄	C₂²	1	4	yes	—
13	#14524002	C₂²	1	2	yes	—
14	Dic₆₇⋊C₈	C₂²	1	8	yes	—
15	#14523954	C₂²	1	4	yes	—
16	#14523953	C₂²	1	4	yes	—
17	C₈×Dic₆₇	C₂²	1	4	yes	—
18	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
21	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	—
22	C₂×C₆₇⋊C₈	—	2	2	no	—
23	C₂×C₆₇⋊C₈	Q₈	1	4	yes	complemented
24	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
25	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
26	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
28	Dic₆₇⋊C₄	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₆₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	C₂³	1	4	yes	—
35	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	4	2	no	—
40	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₇	#41	1	2	yes	—
45	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₇	C₂²×C₄	1	8	yes	—
47	Dic₁₃₄	—	4	4	no	—
48	C₈×C₆₇	—	2	4	no	—
49	C₆₇⋊C₈	—	2	2	no	—
50	C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₆₇	—	2	2	no	—
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	C₄×Q₈	1	2	yes	—
57	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
59	Dic₆₇	—	2	8	no	—
60	C₂×C₆₇	—	2	2	no	—
61	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₆₇	#568	1	2	yes	—
63	#2961	—	67	1	no	maximal, Sylow
64	C₆₇	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	—	67	1	no	—
66	#446	—	67	8	no	—
67	#424	—	67	2	no	—
68	#423	D₆₇	1	1	yes	characteristic
69	#405	—	67	2	no	—
70	C₂×#97	—	67	1	no	—
71	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
72	C₂²×C₈	—	67	1	no	—
73	C₂²×C₈	—	67	1	no	—
74	C₄⋊Q₈	—	67	2	no	—
75	#125	—	67	2	no	—
76	C₂²⋊Q₈	—	67	4	no	—
77	C₄×Q₈	—	67	4	no	—
78	#117	—	67	2	no	—
79	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
80	C₄⋊C₈	—	67	8	no	—
81	C₄⋊C₈	C₂×D₆₇	1	4	yes	—
82	#97	—	67	4	no	—
83	#97	—	67	4	no	—
84	C₄×C₈	—	67	4	no	—
85	C₂×Q₈	—	67	4	no	—
86	C₂²×C₄	C₄×D₆₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	67	4	no	—
90	C₂×C₈	—	67	2	no	—
91	C₂×C₈	—	67	2	no	—
92	C₂×C₈	—	134	2	no	—
93	C₂×C₈	C₂²×D₆₇	1	4	yes	—
94	C₄⋊C₄	—	67	4	no	—
95	C₄⋊C₄	—	67	8	no	—
96	C₄⋊C₄	—	67	4	no	—
97	C₂²⋊C₄	—	67	4	no	—
98	C₄²	—	67	4	no	—
99	C₄²	C₄×D₆₇	1	1	yes	characteristic
100	C₄²	C₄×D₆₇	1	1	yes	characteristic
101	C₄²	C₂²×D₆₇	1	2	yes	—
102	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
103	Q₈	—	268	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	67	8	no	—
106	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₆₇	1	2	yes	—
110	C₂×C₄	#11761627	1	2	yes	—
111	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	67	4	no	—
114	C₈	—	134	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#14524008	1	2	yes	—
117	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	134	8	no	—
121	C₄	#14524008	1	2	yes	—
122	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
123	C₄	C₂×#11761627	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×#14524008	1	1	yes	characteristic
126	C₂	#16864162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal, characteristic

#16863525

C₂

yes

maximal, complemented

#16863424

C₂

yes

maximal

#16863354

C₂

yes

maximal

C₂×#14523954

C₂

yes

maximal, characteristic

#423×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524043

C₄

yes

—

#14524035

C₄

yes

—

#14524022

C₄

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

#14523954

C₂²

yes

—

#14523953

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

C₂²×C₄

yes

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×Q₈

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₇

#568

yes

—

#2961

—

maximal, Sylow

C₆₇

#2961

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

#405

—

C₂×#97

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

100

C₄²

C₄×D₆₇

yes

characteristic

101

C₄²

C₂²×D₆₇

yes

—

102

C₂³

C₂×C₄×D₆₇

yes

characteristic

103

Q₈

—

268

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₆₇

yes

—

107

C₂×C₄

C₂×C₄×D₆₇

yes

—

108

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

109

C₂×C₄

Q₈×D₆₇

yes

—

110

C₂×C₄

#11761627

yes

—

111

C₂×C₄

C₂³×D₆₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

134

—

115

C₂²

—

116

C₂²

#14524008

yes

—

117

C₂²

C₂²×C₄×D₆₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

134

—

121

C₄

#14524008

yes

—

122

C₄

C₂×Q₈×D₆₇

yes

characteristic

123

C₄

C₂×#11761627

yes

characteristic

124

C₂

—

125

C₂

C₂×#14524008

yes

characteristic

126

C₂

#16864162

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#423×C₆₇	C₂
hypercenter	#423	D₆₇
hypocenter	C₆₇	#2961
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

#423×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #16863525	C₂	semidirect
23. C₂×C₆₇⋊C₈	Q₈	semidirect
64. C₆₇	#2961	semidirect

H ≅ G/N

type

2. #16863525

C₂

semidirect

23. C₂×C₆₇⋊C₈

Q₈

semidirect

64. C₆₇

#2961

semidirect

Subgroups — #22569146

Conjugacy classes of subgroups

Complements