Group #22558882 — subgroups

Subgroups — #22558882

Order 8576.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14524043	C₂	1	1	yes	maximal, characteristic
2	#16863449	C₂	1	1	yes	maximal, characteristic
3	#16863143	C₂	1	4	yes	maximal, complemented
4	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	—	2	1	no	—
6	#14524043	—	2	1	no	—
7	#14524043	—	2	1	no	—
8	#14524043	C₄	1	2	yes	—
9	#14524043	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	—	2	1	no	—
11	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
14	C₄×C₆₇⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₆₇	—	2	1	no	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₇	—	2	1	no	—
18	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	4	2	no	—
22	C₂×Dic₁₃₄	—	4	2	no	—
23	C₂²×Dic₆₇	—	2	2	no	—
24	C₂×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	2	no	—
28	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
32	M₁₆×C₆₇	—	4	2	no	—
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	D₈	1	2	yes	—
45	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₇	—	2	4	no	—
50	C₂×Dic₆₇	—	4	4	no	—
51	Dic₁₃₄	—	4	4	no	—
52	Dic₁₃₄	—	4	2	no	—
53	Dic₁₃₄	—	4	2	no	—
54	C₈×C₆₇	—	4	2	no	—
55	C₆₇⋊C₈	—	4	2	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	#102	1	2	yes	—
59	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	#101	1	2	yes	—
63	C₄×C₆₇	#102	1	2	yes	—
64	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₆₇	#137	1	2	yes	—
67	Dic₆₇	—	4	4	no	—
68	C₂×C₆₇	—	2	2	no	—
69	C₂×C₆₇	C₂×#101	1	1	yes	characteristic
70	C₂×C₆₇	C₂×#102	1	1	yes	characteristic
71	C₂×C₆₇	#419	1	1	yes	characteristic
72	#1546	—	67	1	no	maximal, Sylow
73	C₆₇	#1546	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	67	1	no	—
75	#431	—	67	1	no	—
76	#423	D₆₇	1	1	yes	characteristic
77	#332	—	67	4	no	—
78	C₂²×Q₈	—	134	1	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	67	2	no	—
81	C₄⋊Q₈	—	67	2	no	—
82	C₄⋊Q₈	—	134	1	no	—
83	C₄⋊Q₈	—	134	1	no	—
84	C₂×C₄⋊C₄	—	67	2	no	—
85	C₂×C₄⋊C₄	—	134	1	no	—
86	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	67	2	no	—
89	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
90	C₂²⋊C₈	—	134	1	no	—
91	C₄×C₈	—	134	1	no	—
92	C₂×Q₈	—	268	2	no	—
93	C₂×Q₈	—	268	2	no	—
94	C₂×Q₈	—	134	2	no	—
95	C₂×Q₈	—	134	2	no	—
96	C₂²×C₄	—	134	2	no	—
97	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
98	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	134	2	no	—
103	C₄⋊C₄	—	67	4	no	—
104	C₄⋊C₄	—	134	2	no	—
105	C₄⋊C₄	—	134	2	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₄²	C₄×D₆₇	1	1	yes	characteristic
108	C₄²	D₂₆₈	1	1	yes	characteristic
109	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₄²	C₂²×D₆₇	1	1	yes	characteristic
111	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
112	Q₈	—	268	4	no	—
113	Q₈	—	268	2	no	—
114	Q₈	—	268	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	268	4	no	—
118	C₂×C₄	—	134	4	no	—
119	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	#11761613	1	2	yes	—
123	C₂×C₄	#11761615	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	268	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#14523963	1	2	yes	—
132	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	268	4	no	—
136	C₄	#14523961	1	2	yes	—
137	C₄	#14523963	1	2	yes	—
138	C₄	C₂×#11761613	1	1	yes	characteristic
139	C₄	C₂×#11761615	1	1	yes	characteristic
140	C₄	#14524076	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#14523961	1	1	yes	characteristic
143	C₂	C₂×#14523963	1	1	yes	characteristic
144	C₂	#16863406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

#16863449

C₂

yes

maximal, characteristic

#16863143

C₂

yes

maximal, complemented

#423×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

#14524043

—

#14524043

—

#14524043

C₄

yes

—

#14524043

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₄

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#102

yes

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#101

yes

—

C₄×C₆₇

#102

yes

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

#137

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

—

#423

D₆₇

yes

characteristic

#332

—

C₂²×Q₈

—

134

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

—

C₂²⋊C₈

—

134

—

C₄×C₈

—

134

—

C₂×Q₈

—

268

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄²

C₄×D₆₇

yes

characteristic

108

C₄²

D₂₆₈

yes

characteristic

109

C₄²

C₂²⋊D₆₇

yes

characteristic

110

C₄²

C₂²×D₆₇

yes

characteristic

111

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

112

Q₈

—

268

—

113

Q₈

—

268

—

114

Q₈

—

268

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

134

—

119

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

122

C₂×C₄

#11761613

yes

—

123

C₂×C₄

#11761615

yes

—

124

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

126

C₂×C₄

C₂×D₂₆₈

yes

characteristic

127

C₈

—

128

C₈

—

268

—

129

C₂²

—

130

C₂²

—

131

C₂²

#14523963

yes

—

132

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

268

—

136

C₄

#14523961

yes

—

137

C₄

#14523963

yes

—

138

C₄

C₂×#11761613

yes

characteristic

139

C₄

C₂×#11761615

yes

characteristic

140

C₄

#14524076

yes

—

141

C₂

—

142

C₂

C₂×#14523961

yes

characteristic

143

C₂

C₂×#14523963

yes

characteristic

144

C₂

#16863406

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#423	D₆₇
hypocenter	C₆₇	#1546
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

#423×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16863143	C₂	semidirect
73. C₆₇	#1546	semidirect

H ≅ G/N

type

3. #16863143

C₂

semidirect

73. C₆₇

#1546

semidirect

Subgroups — #22558882

Conjugacy classes of subgroups

Complements