Group #22559080 — subgroups

Subgroups — #22559080

Order 8576.

152 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16863526	C₂	1	1	yes	maximal, characteristic
2	C₄×#11761618	C₂	1	1	yes	maximal, characteristic
3	#16863146	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#16863126	C₂	1	1	yes	maximal, characteristic
6	#16863118	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761618	C₄	1	1	yes	characteristic
9	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	#14523976	C₄	1	1	yes	characteristic
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₆₇⋊C₈	—	2	1	no	—
13	#14523967	C₄	1	1	yes	characteristic
14	#14523966	C₄	1	1	yes	characteristic
15	#14523953	—	2	1	no	—
16	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₇	—	2	1	no	—
20	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₆₇	—	2	1	no	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
24	#11761618	—	2	2	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
30	Dic₆₇⋊C₄	—	2	1	no	—
31	C₄×Dic₆₇	—	2	1	no	—
32	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	Q₈×C₆₇	—	2	1	no	—
41	Q₈×C₆₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	1	no	—
48	C₂×Dic₆₇	—	2	1	no	—
49	Dic₁₃₄	—	2	1	no	—
50	Dic₁₃₄	—	2	1	no	—
51	C₈×C₆₇	—	2	1	no	—
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₆₇⋊C₈	—	2	2	no	—
55	C₆₇⋊C₈	—	2	1	no	—
56	C₆₇⋊C₈	—	2	1	no	—
57	C₆₇⋊C₈	—	4	1	no	—
58	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₆₇	—	2	2	no	—
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	4	1	no	—
62	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
63	C₄×C₆₇	#131	1	1	yes	characteristic
64	C₄×C₆₇	#137	1	1	yes	characteristic
65	C₄×C₆₇	#137	1	1	yes	characteristic
66	Dic₆₇	—	4	1	no	—
67	Dic₆₇	—	4	1	no	—
68	C₂×C₆₇	#435	1	1	yes	characteristic
69	C₂×C₆₇	#441	1	1	yes	characteristic
70	C₂×C₆₇	#443	1	1	yes	characteristic
71	#1587	—	67	1	no	maximal, Sylow
72	C₆₇	#1587	1	1	yes	characteristic, complemented, Sylow
73	#446	—	67	1	no	—
74	C₈×Q₈	D₆₇	1	1	yes	characteristic
75	C₄×Q₁₆	—	67	1	no	—
76	#334	—	67	1	no	—
77	Q₈⋊C₈	—	67	1	no	—
78	Q₈⋊C₈	—	67	1	no	—
79	#322	—	67	1	no	—
80	C₂×Q₁₆	—	67	1	no	—
81	C₄×Q₈	—	67	1	no	—
82	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
83	#107	—	67	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	67	1	no	—
86	C₄⋊C₈	—	134	1	no	—
87	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
88	#103	—	67	1	no	—
89	#103	—	67	1	no	—
90	#97	—	134	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	67	1	no	—
93	C₄×C₈	—	67	1	no	—
94	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	C₂×Q₈	—	67	1	no	—
96	C₂×Q₈	C₄×D₆₇	1	1	yes	characteristic
97	Q₁₆	—	134	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	67	1	no	—
100	C₂×C₈	—	67	1	no	—
101	C₂×C₈	—	134	1	no	—
102	C₂×C₈	—	134	1	no	—
103	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	67	1	no	—
107	C₄⋊C₄	—	134	1	no	—
108	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	134	1	no	—
111	C₄²	C₂²×D₆₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	134	1	no	—
114	Q₈	—	134	1	no	—
115	Q₈	#11761611	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	134	1	no	—
119	C₂×C₄	—	134	1	no	—
120	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₄	#11761628	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	268	1	no	—
127	C₈	—	134	2	no	—
128	C₈	—	134	1	no	—
129	C₈	—	134	1	no	—
130	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	1	no	—
134	C₄	—	268	1	no	—
135	C₄	—	268	1	no	—
136	C₄	C₂×#11761611	1	1	yes	characteristic
137	C₄	#14524049	1	1	yes	characteristic
138	C₄	#14524073	1	1	yes	characteristic
139	C₄	#14524074	1	1	yes	characteristic
140	C₂	#16863467	1	1	yes	characteristic
141	C₂	#16863501	1	1	yes	characteristic
142	C₂	#16863511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863526

C₂

yes

maximal, characteristic

C₄×#11761618

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic

Q₈⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#16863126

C₂

yes

maximal, characteristic

#16863118

C₂

yes

maximal, characteristic

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761618

C₄

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523976

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523967

C₄

yes

characteristic

#14523966

C₄

yes

characteristic

#14523953

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₂×C₄

yes

characteristic

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₆₇

C₂×C₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#446

—

C₈×Q₈

D₆₇

yes

characteristic

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

—

—

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₆₇

yes

characteristic

Q₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

134

—

102

C₂×C₈

—

134

—

103

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

104

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

C₄×D₆₇

yes

characteristic

109

C₄²

—

110

C₄²

—

134

—

111

C₄²

C₂²×D₆₇

yes

characteristic

112

Q₈

—

113

Q₈

—

134

—

114

Q₈

—

134

—

115

Q₈

#11761611

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

134

—

119

C₂×C₄

—

134

—

120

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂×C₄

#11761628

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

268

—

127

C₈

—

134

—

128

C₈

—

134

—

129

C₈

—

134

—

130

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

268

—

135

C₄

—

268

—

136

C₄

C₂×#11761611

yes

characteristic

137

C₄

#14524049

yes

characteristic

138

C₄

#14524073

yes

characteristic

139

C₄

#14524074

yes

characteristic

140

C₂

#16863467

yes

characteristic

141

C₂

#16863501

yes

characteristic

142

C₂

#16863511

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	#1587
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

hypercenter

C₈×Q₈

D₆₇

hypocenter

C₆₇

#1587

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
72. C₆₇	#1587	semidirect

H ≅ G/N

type

72. C₆₇

#1587

semidirect

Subgroups — #22559080

Conjugacy classes of subgroups

Complements