Group #22559983 — subgroups

Subgroups — #22559983

Order 8576.

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

normal subgroups only

This group has 384 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	C₂³×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863150	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	—	2	4	no	—
7	C₂²×C₆₇⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₆₇	C₄	1	2	yes	—
11	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
12	C₈×Dic₆₇	—	2	4	no	—
13	C₆₇⋊#95	C₄	1	2	yes	—
14	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₆₇	C₄	1	4	yes	complemented
17	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₂×C₆₇⋊C₈	—	2	8	no	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₈	D₄	1	4	yes	—
23	C₂²⋊Dic₆₇	C₈	1	8	yes	complemented
24	C₄×Dic₆₇	—	2	2	no	—
25	C₄×Dic₆₇	—	2	2	no	—
26	C₂⁴×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	—	2	2	no	—
28	C₂²×C₄×C₆₇	—	2	2	no	—
29	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂³×C₆₇	—	2	2	no	—
37	C₂³×C₆₇	—	2	2	no	—
38	C₂³×C₆₇	C₄²	1	2	yes	—
39	C₂³×C₆₇	C₂×C₈	1	4	yes	—
40	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	C₄²	1	2	yes	—
46	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	#41	1	2	yes	—
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	2	4	no	—
52	C₂×Dic₆₇	C₂×C₈	1	8	yes	complemented
53	C₈×C₆₇	—	2	4	no	—
54	C₆₇⋊C₈	—	2	2	no	—
55	C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	complemented
56	C₂²×C₆₇	—	2	2	no	—
57	C₂²×C₆₇	—	2	2	no	—
58	C₂²×C₆₇	—	2	8	no	—
59	C₂²×C₆₇	C₄×C₈	1	4	yes	complemented
60	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₆₇	C₄×D₄	1	2	yes	—
62	C₂²×C₆₇	C₂²×C₈	1	2	yes	—
63	C₂²×C₆₇	#131	1	2	yes	—
64	C₄×C₆₇	—	2	2	no	—
65	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₆₇	#117	1	1	yes	characteristic
67	C₄×C₆₇	C₄×D₄	1	2	yes	—
68	Dic₆₇	—	2	8	no	—
69	C₂×C₆₇	—	2	4	no	—
70	C₂×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂×C₆₇	#405	1	1	yes	characteristic
73	C₂×C₆₇	C₈×D₄	1	4	yes	—
74	C₈×C₂²⋊C₄	—	67	1	no	maximal, Sylow
75	C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₈	—	67	1	no	—
77	C₂×C₂²⋊C₈	D₆₇	1	1	yes	characteristic
78	C₂×C₄×C₈	—	67	2	no	—
79	C₄×C₂²⋊C₄	—	67	1	no	—
80	#336	—	67	2	no	—
81	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
82	C₂²×C₈	—	67	1	no	—
83	C₂²×C₈	—	67	4	no	—
84	C₂²×C₈	—	67	1	no	—
85	C₂²×C₈	—	134	4	no	—
86	C₂²×C₈	C₂×D₆₇	1	2	yes	—
87	C₂×C₂²⋊C₄	—	67	2	no	—
88	C₂×C₄²	—	67	2	no	—
89	C₂²⋊C₈	Dic₆₇	1	4	yes	complemented
90	C₄×C₈	—	134	4	no	—
91	#95	—	67	2	no	—
92	C₂⁴	C₄×D₆₇	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	67	4	no	—
96	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₆₇	1	2	yes	—
98	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	67	2	no	—
103	C₂×C₈	—	67	4	no	—
104	C₂×C₈	—	67	2	no	—
105	C₂×C₈	—	134	8	no	—
106	C₂×C₈	—	134	2	no	—
107	C₂×C₈	C₂×Dic₆₇	1	4	yes	—
108	C₂²⋊C₄	—	67	8	no	—
109	C₄²	—	134	2	no	—
110	C₄²	—	134	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₄×Dic₆₇	1	2	yes	—
114	C₂³	C₈×D₆₇	1	4	yes	—
115	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	67	8	no	—
121	C₂×C₄	—	134	4	no	—
122	C₂×C₄	—	134	4	no	—
123	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₆₇	1	2	yes	—
127	C₂×C₄	#11761627	1	2	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	67	4	no	—
130	C₈	—	134	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	8	no	—
134	C₂²	C₈×Dic₆₇	1	4	yes	complemented
135	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
136	C₂²	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
137	C₂²	C₂×C₈×D₆₇	1	2	yes	—
138	C₂²	#14524051	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	134	8	no	—
141	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
142	C₄	#14523999	1	1	yes	characteristic
143	C₄	D₄×Dic₆₇	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
146	C₂	C₂×C₈×Dic₆₇	1	1	yes	characteristic
147	C₂	#16863354	1	1	yes	characteristic
148	C₂	(C₂²⋊D₆₇)⋊C₈	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₆₇

C₂

yes

maximal, characteristic

#16863150

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₂²⋊Dic₆₇

C₄

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₈×Dic₆₇

—

C₆₇⋊#95

C₄

yes

—

C₂³×C₄×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

—

C₂²⋊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

—

C₂²⋊Dic₆₇

C₈

yes

complemented

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂⁴×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₄²

yes

—

C₂³×C₆₇

C₂×C₈

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₈

yes

complemented

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

—

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

#131

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×C₆₇

#405

yes

characteristic

C₂×C₆₇

C₈×D₄

yes

—

C₈×C₂²⋊C₄

—

maximal, Sylow

C₆₇

C₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×C₂²⋊C₈

D₆₇

yes

characteristic

C₂×C₄×C₈

—

C₄×C₂²⋊C₄

—

#336

—

C₂³×C₄

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

134

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

Dic₆₇

yes

complemented

C₄×C₈

—

134

—

#95

—

C₂⁴

C₄×D₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

—

134

—

107

C₂×C₈

C₂×Dic₆₇

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

134

—

110

C₄²

—

134

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₄×Dic₆₇

yes

—

114

C₂³

C₈×D₆₇

yes

—

115

C₂³

C₂×C₄×D₆₇

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

134

—

122

C₂×C₄

—

134

—

123

C₂×C₄

C₄×Dic₆₇

yes

—

124

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

125

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

126

C₂×C₄

D₄×D₆₇

yes

—

127

C₂×C₄

#11761627

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

134

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₈×Dic₆₇

yes

complemented

135

C₂²

C₂×C₄×Dic₆₇

yes

characteristic

136

C₂²

(C₂²⋊D₆₇)⋊C₄

yes

—

137

C₂²

C₂×C₈×D₆₇

yes

—

138

C₂²

#14524051

yes

—

139

C₄

—

140

C₄

—

134

—

141

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

142

C₄

#14523999

yes

characteristic

143

C₄

D₄×Dic₆₇

yes

—

144

C₂

—

145

C₂

C₂²⋊C₄×Dic₆₇

yes

characteristic

146

C₂

C₂×C₈×Dic₆₇

yes

characteristic

147

C₂

#16863354

yes

characteristic

148

C₂

(C₂²⋊D₆₇)⋊C₈

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₇

commutator

C₂×C₆₇

C₂×C₄×C₈

Fitting

C₂×C₂²⋊C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²×C₄

hypercenter

C₂×C₂²⋊C₈

D₆₇

hypocenter

C₆₇

C₈×C₂²⋊C₄

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×C₈×Dic₆₇	C₂	semidirect
4. #16863150	C₂	semidirect
7. C₂²×C₆₇⋊C₈	C₄	semidirect
16. C₂²⋊C₈×C₆₇	C₄	semidirect
23. C₂²⋊Dic₆₇	C₈	semidirect
52. C₂×Dic₆₇	C₂×C₈	semidirect
55. C₆₇⋊C₈	C₂²⋊C₄	semidirect
59. C₂²×C₆₇	C₄×C₈	semidirect
75. C₆₇	C₈×C₂²⋊C₄	semidirect
89. C₂²⋊C₈	Dic₆₇	semidirect
134. C₂²	C₈×Dic₆₇	semidirect

H ≅ G/N

type

2. C₂×C₈×Dic₆₇

C₂

semidirect

4. #16863150

C₂

semidirect

7. C₂²×C₆₇⋊C₈

C₄

semidirect

16. C₂²⋊C₈×C₆₇

C₄

semidirect

23. C₂²⋊Dic₆₇

C₈

semidirect

52. C₂×Dic₆₇

C₂×C₈

semidirect

55. C₆₇⋊C₈

C₂²⋊C₄

semidirect

59. C₂²×C₆₇

C₄×C₈

semidirect

75. C₆₇

C₈×C₂²⋊C₄

semidirect

89. C₂²⋊C₈

Dic₆₇

semidirect

134. C₂²

C₈×Dic₆₇

semidirect

Subgroups — #22559983

Conjugacy classes of subgroups

Complements