Group #22560998 — subgroups

Subgroups — #22560998

Order 8576.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#16863445	C₂	1	1	yes	maximal, characteristic
2	#16863444	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523978	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523978	C₂	1	1	yes	maximal, characteristic
5	#16863352	C₂	1	1	yes	maximal, characteristic
6	#16863166	C₂	1	1	yes	maximal, characteristic
7	#344×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	C₂×#11761612	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#14523979	—	2	2	no	—
13	#14523978	—	2	2	no	—
14	#14523978	—	2	1	no	—
15	#14523978	—	2	1	no	—
16	#14523978	C₄	1	4	yes	complemented
17	#14523954	—	2	1	no	—
18	C₄×C₆₇⋊C₈	—	2	1	no	—
19	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#117×C₆₇	C₂²	1	1	yes	characteristic
22	#11761612	—	2	1	no	—
23	#11761612	—	2	1	no	—
24	#11761612	—	2	1	no	—
25	#11761612	—	2	1	no	—
26	#11761612	C₂×C₄	1	2	yes	—
27	#11761612	D₄	1	2	yes	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₆₇	—	2	1	no	—
37	M₁₆×C₆₇	—	2	1	no	—
38	M₁₆×C₆₇	—	2	1	no	—
39	M₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
44	C₄²×C₆₇	—	2	1	no	—
45	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₈×C₆₇	—	2	2	no	—
54	C₈×C₆₇	—	4	1	no	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₆₇⋊C₈	—	2	2	no	—
57	C₆₇⋊C₈	—	4	1	no	—
58	C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₆₇	—	4	1	no	—
61	C₂²×C₆₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₆₇	—	4	2	no	—
65	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₆₇	#121	1	1	yes	characteristic
68	C₄×C₆₇	#125	1	1	yes	characteristic
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	—	2	1	no	—
71	C₂×C₆₇	—	2	1	no	—
72	C₂×C₆₇	#389	1	1	yes	characteristic
73	#1953	—	67	1	no	maximal, Sylow
74	C₆₇	#1953	1	1	yes	characteristic, complemented, Sylow
75	#430	—	67	1	no	—
76	#430	—	67	1	no	—
77	C₂×#108	—	67	1	no	—
78	C₂×#108	—	67	1	no	—
79	#405	—	67	1	no	—
80	#344	—	67	1	no	—
81	#344	D₆₇	1	1	yes	characteristic
82	C₂×M₁₆	—	67	1	no	—
83	C₂×M₁₆	—	67	1	no	—
84	C₂×M₁₆	—	67	1	no	—
85	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
87	C₂²×C₈	—	67	1	no	—
88	#117	C₂×D₆₇	1	1	yes	characteristic
89	#108	—	67	4	no	—
90	#108	—	134	2	no	—
91	#108	—	134	1	no	—
92	#108	—	134	1	no	—
93	#108	—	134	2	no	—
94	#97	—	134	1	no	—
95	C₄×C₈	—	134	1	no	—
96	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	67	2	no	—
101	M₁₆	—	67	2	no	—
102	M₁₆	—	134	1	no	—
103	M₁₆	—	134	1	no	—
104	M₁₆	—	134	1	no	—
105	M₁₆	—	134	1	no	—
106	M₁₆	C₄×D₆₇	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	67	2	no	—
110	C₂×C₈	—	67	1	no	—
111	C₂×C₈	—	67	1	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	—	134	1	no	—
114	C₂×C₈	—	134	1	no	—
115	C₂×C₈	—	134	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₁₃₄	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
122	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
125	C₂×C₄	#11761627	1	1	yes	characteristic
126	C₂×C₄	#11761628	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	268	1	no	—
130	C₈	—	67	2	no	—
131	C₈	—	67	2	no	—
132	C₈	—	134	2	no	—
133	C₈	—	134	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₁₃₄	1	1	yes	characteristic
136	C₂²	#14524020	1	1	yes	characteristic
137	C₂²	#14524042	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
140	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
141	C₄	#14524016	1	1	yes	characteristic
142	C₄	#14524033	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#16863293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863445

C₂

yes

maximal, characteristic

#16863444

C₂

yes

maximal, characteristic

C₂×#14523978

C₂

yes

maximal, characteristic

C₂×#14523978

C₂

yes

maximal, characteristic

#16863352

C₂

yes

maximal, characteristic

#16863166

C₂

yes

maximal, characteristic

#344×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

—

—

—

C₄

yes

complemented

#14523954

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

—

—

—

C₂×C₄

yes

—

#11761612

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

Q₈

yes

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊C₄

yes

—

C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×Q₈

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₆₇

yes

characteristic

#108

—

#108

—

134

—

#108

—

134

—

#108

—

134

—

#108

—

134

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

134

—

103

M₁₆

—

134

—

104

M₁₆

—

134

—

105

M₁₆

—

134

—

106

M₁₆

C₄×D₆₇

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

134

—

114

C₂×C₈

—

134

—

115

C₂×C₈

—

134

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₁₃₄

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₁₃₄

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

122

C₂×C₄

D₄×D₆₇

yes

characteristic

123

C₂×C₄

D₄×D₆₇

yes

characteristic

124

C₂×C₄

Q₈×D₆₇

yes

characteristic

125

C₂×C₄

#11761627

yes

characteristic

126

C₂×C₄

#11761628

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

268

—

130

C₈

—

131

C₈

—

132

C₈

—

134

—

133

C₈

—

134

—

134

C₂²

—

135

C₂²

C₄×Dic₁₃₄

yes

characteristic

136

C₂²

#14524020

yes

characteristic

137

C₂²

#14524042

yes

characteristic

138

C₄

—

139

C₄

C₄⋊C₄×D₆₇

yes

characteristic

140

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

141

C₄

#14524016

yes

characteristic

142

C₄

#14524033

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#16863293

yes

characteristic

	subgroup	G/H
center	C₄	#14524016
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#344×C₆₇	C₂
socle	C₂×C₆₇	#389
hypercenter	#344	D₆₇
hypocenter	C₆₇	#1953
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14524016

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#344×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #14523978	C₄	semidirect
74. C₆₇	#1953	semidirect

H ≅ G/N

type

16. #14523978

C₄

semidirect

74. C₆₇

#1953

semidirect

Subgroups — #22560998

Conjugacy classes of subgroups

Complements