Group #22560074 — subgroups

Subgroups — #22560074

Order 8576.

296 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₇	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863150	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₈	C₄	1	4	yes	complemented
11	C₈×Dic₆₇	—	2	4	no	—
12	C₆₇⋊#95	C₄	1	2	yes	—
13	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	C₄	1	4	yes	complemented
16	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
17	C₂×C₆₇⋊C₈	—	2	1	no	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
22	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
23	C₄⋊Dic₆₇	C₈	1	8	yes	complemented
24	C₄×Dic₆₇	—	2	2	no	—
25	C₄×Dic₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	—	2	2	no	—
36	C₂×C₄×C₆₇	C₄²	1	2	yes	—
37	C₂×C₄×C₆₇	C₄²	1	2	yes	—
38	C₂×C₄×C₆₇	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₂×Dic₆₇	—	2	4	no	—
46	C₂×Dic₆₇	—	2	4	no	—
47	C₂×Dic₆₇	C₂×C₈	1	8	yes	—
48	C₈×C₆₇	—	2	4	no	—
49	C₆₇⋊C₈	—	2	2	no	—
50	C₆₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₆₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₆₇	C₂²×C₈	1	2	yes	—
55	C₂²×C₆₇	#131	1	2	yes	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	C₄×C₈	1	4	yes	complemented
58	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₆₇	#117	1	1	yes	characteristic
60	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
61	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
62	Dic₆₇	—	2	8	no	—
63	C₂×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×C₆₇	#405	1	1	yes	characteristic
66	C₂×C₆₇	C₈×D₄	1	2	yes	—
67	C₂×C₆₇	C₈×Q₈	1	2	yes	—
68	C₈×C₄⋊C₄	—	67	1	no	maximal, Sylow
69	C₆₇	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic
71	C₂×C₄×C₈	—	67	1	no	—
72	C₂×C₄×C₈	—	67	2	no	—
73	C₄×C₄⋊C₄	—	67	1	no	—
74	#336	—	67	2	no	—
75	C₂²×C₈	—	67	1	no	—
76	C₂²×C₈	—	67	1	no	—
77	C₂²×C₈	C₂×D₆₇	1	2	yes	—
78	C₂×C₄⋊C₄	—	67	2	no	—
79	C₂×C₄²	—	67	2	no	—
80	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
81	C₄⋊C₈	Dic₆₇	1	4	yes	complemented
82	C₄×C₈	—	67	4	no	—
83	C₄×C₈	—	134	4	no	—
84	#95	—	67	2	no	—
85	C₂²×C₄	—	67	4	no	—
86	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	67	2	no	—
92	C₂×C₈	—	67	2	no	—
93	C₂×C₈	—	67	2	no	—
94	C₂×C₈	—	67	2	no	—
95	C₂×C₈	—	134	1	no	—
96	C₂×C₈	—	134	1	no	—
97	C₂×C₈	C₂×Dic₆₇	1	4	yes	—
98	C₄⋊C₄	—	67	8	no	—
99	C₄²	—	134	2	no	—
100	C₄²	—	134	2	no	—
101	C₄²	C₄×D₆₇	1	2	yes	—
102	C₄²	C₂×Dic₆₇	1	2	yes	—
103	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	67	8	no	—
106	C₂×C₄	—	134	4	no	—
107	C₂×C₄	—	134	4	no	—
108	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
109	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
110	C₂×C₄	C₈×D₆₇	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
113	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
115	C₂×C₄	#11761627	1	1	yes	characteristic
116	C₂×C₄	#11761629	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	—	67	4	no	—
119	C₈	—	134	2	no	—
120	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
121	C₂²	#14524005	1	1	yes	characteristic
122	C₂²	#14524008	1	1	yes	characteristic
123	C₂²	C₂×C₈×D₆₇	1	2	yes	—
124	C₂²	#14524051	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	134	8	no	—
127	C₄	C₈×Dic₆₇	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
129	C₄	#14524001	1	1	yes	characteristic
130	C₄	D₄×Dic₆₇	1	1	yes	characteristic
131	C₄	Q₈×Dic₆₇	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
133	C₂	C₂×C₈×Dic₆₇	1	1	yes	characteristic
134	C₂	#16863354	1	1	yes	characteristic
135	C₂	#16863461	1	2	yes	—
136	C₂	#16863520	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

#16863150

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₄

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₄×C₆₇⋊C₈

C₄

yes

complemented

C₈×Dic₆₇

—

C₆₇⋊#95

C₄

yes

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

C₈

yes

complemented

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

—

C₂×C₄×C₆₇

C₄²

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

C₄×Q₈

yes

characteristic

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

#131

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₈

yes

complemented

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×C₆₇

#405

yes

characteristic

C₂×C₆₇

C₈×D₄

yes

—

C₂×C₆₇

C₈×Q₈

yes

—

C₈×C₄⋊C₄

—

maximal, Sylow

C₆₇

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

D₆₇

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

Dic₆₇

yes

complemented

C₄×C₈

—

C₄×C₈

—

134

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₄⋊C₄

—

C₄²

—

134

—

100

C₄²

—

134

—

101

C₄²

C₄×D₆₇

yes

—

102

C₄²

C₂×Dic₆₇

yes

—

103

C₂³

C₂×C₄×D₆₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

134

—

107

C₂×C₄

—

134

—

108

C₂×C₄

C₄×Dic₆₇

yes

—

109

C₂×C₄

C₄×Dic₆₇

yes

—

110

C₂×C₄

C₈×D₆₇

yes

—

111

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

112

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

113

C₂×C₄

D₄×D₆₇

yes

characteristic

114

C₂×C₄

Q₈×D₆₇

yes

characteristic

115

C₂×C₄

#11761627

yes

characteristic

116

C₂×C₄

#11761629

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

134

—

120

C₂²

C₂×C₄×Dic₆₇

yes

characteristic

121

C₂²

#14524005

yes

characteristic

122

C₂²

#14524008

yes

characteristic

123

C₂²

C₂×C₈×D₆₇

yes

—

124

C₂²

#14524051

yes

—

125

C₄

—

126

C₄

—

134

—

127

C₄

C₈×Dic₆₇

yes

complemented

128

C₄

C₄⋊C₄×D₆₇

yes

characteristic

129

C₄

#14524001

yes

characteristic

130

C₄

D₄×Dic₆₇

yes

characteristic

131

C₄

Q₈×Dic₆₇

yes

characteristic

132

C₂

C₄⋊C₄×Dic₆₇

yes

characteristic

133

C₂

C₂×C₈×Dic₆₇

yes

characteristic

134

C₂

#16863354

yes

characteristic

135

C₂

#16863461

yes

—

136

C₂

#16863520

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄×C₈
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₇

commutator

C₂×C₆₇

C₂×C₄×C₈

Fitting

C₂×C₄⋊C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₆₇

hypocenter

C₆₇

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
10. C₄×C₆₇⋊C₈	C₄	semidirect
15. C₄⋊C₈×C₆₇	C₄	semidirect
23. C₄⋊Dic₆₇	C₈	semidirect
50. C₆₇⋊C₈	C₄⋊C₄	semidirect
57. C₄×C₆₇	C₄×C₈	semidirect
69. C₆₇	C₈×C₄⋊C₄	semidirect
81. C₄⋊C₈	Dic₆₇	semidirect
127. C₄	C₈×Dic₆₇	semidirect

H ≅ G/N

type

10. C₄×C₆₇⋊C₈

C₄

semidirect

15. C₄⋊C₈×C₆₇

C₄

semidirect

23. C₄⋊Dic₆₇

C₈

semidirect

50. C₆₇⋊C₈

C₄⋊C₄

semidirect

57. C₄×C₆₇

C₄×C₈

semidirect

69. C₆₇

C₈×C₄⋊C₄

semidirect

81. C₄⋊C₈

Dic₆₇

semidirect

127. C₄

C₈×Dic₆₇

semidirect

Subgroups — #22560074

Conjugacy classes of subgroups

Complements