Group #22561786 — subgroups

Subgroups — #22561786

Order 8576.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863444	C₂	1	1	yes	maximal, characteristic
2	#16863419	C₂	1	1	yes	maximal, characteristic
3	#16863168	C₂	1	1	yes	maximal, characteristic
4	#16863157	C₂	1	1	yes	maximal, characteristic
5	#16863155	C₂	1	1	yes	maximal, characteristic
6	#16863152	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	#14524002	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523978	—	2	2	no	—
13	Dic₆₇⋊C₈	—	2	2	no	—
14	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₇	—	2	1	no	—
18	#11761612	—	2	1	no	—
19	#11761612	—	2	1	no	—
20	#11761612	—	2	1	no	—
21	#11761612	—	2	1	no	—
22	C₂×C₆₇⋊C₈	—	2	1	no	—
23	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
24	C₄⋊Dic₆₇	—	2	1	no	—
25	C₂²⋊Dic₆₇	—	2	1	no	—
26	C₄×Dic₆₇	—	2	1	no	—
27	C₄×Dic₆₇	—	2	1	no	—
28	C₄×Dic₆₇	Q₈	1	1	yes	characteristic
29	C₄×Dic₆₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₆₇	—	2	1	no	—
32	M₁₆×C₆₇	—	2	1	no	—
33	M₁₆×C₆₇	—	2	1	no	—
34	M₁₆×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂²⋊C₄×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	—	2	1	no	—
40	C₂³×C₆₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	4	1	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×Dic₆₇	—	2	1	no	—
49	C₂×Dic₆₇	—	2	1	no	—
50	C₂×Dic₆₇	—	4	2	no	—
51	C₂×Dic₆₇	—	4	1	no	—
52	C₈×C₆₇	—	4	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₆₇⋊C₈	—	2	2	no	—
55	C₆₇⋊C₈	—	4	1	no	—
56	C₂²×C₆₇	—	4	1	no	—
57	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂²×C₆₇	#126	1	1	yes	characteristic
59	C₂²×C₆₇	#126	1	1	yes	characteristic
60	C₄×C₆₇	—	8	1	no	—
61	C₄×C₆₇	#123	1	1	yes	characteristic
62	C₄×C₆₇	#124	1	1	yes	characteristic
63	C₄×C₆₇	#125	1	1	yes	characteristic
64	C₄×C₆₇	#125	1	1	yes	characteristic
65	Dic₆₇	—	2	2	no	—
66	Dic₆₇	—	4	1	no	—
67	Dic₆₇	—	8	1	no	—
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	—	2	1	no	—
71	C₂×C₆₇	#400	1	1	yes	characteristic
72	#2102	—	67	1	no	maximal, Sylow
73	C₆₇	#2102	1	1	yes	characteristic, complemented, Sylow
74	#430	—	67	1	no	—
75	#423	—	67	1	no	—
76	#344	—	67	1	no	—
77	#344	D₆₇	1	1	yes	characteristic
78	#339	—	67	1	no	—
79	#339	—	67	1	no	—
80	H(3,ℤ/4)	—	67	1	no	—
81	C₂×M₁₆	—	67	1	no	—
82	C₂×M₁₆	—	67	1	no	—
83	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
85	#117	—	67	1	no	—
86	#117	C₂×D₆₇	1	1	yes	characteristic
87	C₂×C₄²	—	67	1	no	—
88	#108	—	134	2	no	—
89	C₄⋊C₈	—	134	2	no	—
90	C₂²×C₄	—	134	1	no	—
91	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	134	1	no	—
97	M₁₆	—	134	1	no	—
98	M₁₆	—	134	1	no	—
99	M₁₆	—	134	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	67	2	no	—
103	C₂×C₈	—	134	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	134	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	134	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	67	1	no	—
110	C₄²	—	67	1	no	—
111	C₄²	—	134	1	no	—
112	C₄²	—	134	1	no	—
113	C₂³	#11761628	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	268	2	no	—
116	C₂×C₄	—	268	1	no	—
117	C₂×C₄	—	134	1	no	—
118	C₂×C₄	—	134	1	no	—
119	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
121	C₂×C₄	#11761627	1	1	yes	characteristic
122	C₂×C₄	#11761627	1	1	yes	characteristic
123	C₂×C₄	#11761628	1	1	yes	characteristic
124	C₂×C₄	#11761628	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	268	1	no	—
128	C₈	—	134	2	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#14524021	1	1	yes	characteristic
131	C₂²	#14524036	1	1	yes	characteristic
132	C₂²	#14524037	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	268	1	no	—
135	C₄	—	536	1	no	—
136	C₄	—	134	2	no	—
137	C₄	#14524025	1	1	yes	characteristic
138	C₄	#14524029	1	1	yes	characteristic
139	C₄	#14524033	1	1	yes	characteristic
140	C₄	#14524033	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16863337	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863444

C₂

yes

maximal, characteristic

#16863419

C₂

yes

maximal, characteristic

#16863168

C₂

yes

maximal, characteristic

#16863157

C₂

yes

maximal, characteristic

#16863155

C₂

yes

maximal, characteristic

#16863152

C₂

yes

maximal, characteristic

#344×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523978

—

Dic₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

Q₈

yes

characteristic

C₄×Dic₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

#117

—

#117

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

#108

—

134

—

C₄⋊C₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

134

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

134

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

134

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

134

—

112

C₄²

—

134

—

113

C₂³

#11761628

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

268

—

116

C₂×C₄

—

268

—

117

C₂×C₄

—

134

—

118

C₂×C₄

—

134

—

119

C₂×C₄

D₄×D₆₇

yes

characteristic

120

C₂×C₄

Q₈×D₆₇

yes

characteristic

121

C₂×C₄

#11761627

yes

characteristic

122

C₂×C₄

#11761627

yes

characteristic

123

C₂×C₄

#11761628

yes

characteristic

124

C₂×C₄

#11761628

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

268

—

128

C₈

—

134

—

129

C₂²

—

130

C₂²

#14524021

yes

characteristic

131

C₂²

#14524036

yes

characteristic

132

C₂²

#14524037

yes

characteristic

133

C₄

—

134

C₄

—

268

—

135

C₄

—

536

—

136

C₄

—

134

—

137

C₄

#14524025

yes

characteristic

138

C₄

#14524029

yes

characteristic

139

C₄

#14524033

yes

characteristic

140

C₄

#14524033

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#16863337

yes

characteristic

	subgroup	G/H
center	C₄	#14524029
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#344×C₆₇	C₂
socle	C₂×C₆₇	#400
hypercenter	#344	D₆₇
hypocenter	C₆₇	#2102
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14524029

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#344×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
73. C₆₇	#2102	semidirect

H ≅ G/N

type

73. C₆₇

#2102

semidirect

Subgroups — #22561786

Conjugacy classes of subgroups

Complements