Group #22558639 — subgroups

Subgroups — #22558639

Order 8576.

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

normal subgroups only

This group has 248 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×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863420	C₂	1	1	yes	maximal, characteristic
3	#16863128	C₂	1	4	yes	maximal, complemented
4	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂×#11761612	—	2	1	no	—
7	C₄×Dic₁₃₄	—	2	1	no	—
8	C₄×Dic₁₃₄	—	2	1	no	—
9	C₄×Dic₁₃₄	C₄	1	2	yes	—
10	C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₆₇	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₇	—	2	1	no	—
18	C₂²⋊C₈×C₆₇	—	2	1	no	—
19	C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×Dic₁₃₄	—	2	1	no	—
21	C₂×Dic₁₃₄	—	2	4	no	—
22	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
23	C₂²×Dic₆₇	—	2	1	no	—
24	C₂²×Dic₆₇	—	2	1	no	—
25	#11761612	—	4	2	no	—
26	C₂×C₆₇⋊C₈	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
29	Dic₆₇⋊C₄	—	2	2	no	—
30	Dic₆₇⋊C₄	—	2	2	no	—
31	C₄×Dic₆₇	—	2	2	no	—
32	C₄×Dic₆₇	—	2	2	no	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	2	no	—
37	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	2	2	no	—
54	C₂×Dic₆₇	—	2	2	no	—
55	C₂×Dic₆₇	—	4	2	no	—
56	Dic₁₃₄	—	2	4	no	—
57	Dic₁₃₄	—	2	2	no	—
58	Dic₁₃₄	M₁₆	1	4	yes	complemented
59	C₈×C₆₇	—	4	2	no	—
60	C₆₇⋊C₈	—	4	2	no	—
61	C₂²×C₆₇	—	2	1	no	—
62	C₂²×C₆₇	—	2	1	no	—
63	C₂²×C₆₇	#103	1	2	yes	—
64	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	#103	1	2	yes	—
69	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
70	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₆₇	—	2	4	no	—
73	Dic₆₇	—	4	2	no	—
74	C₂×C₆₇	—	2	2	no	—
75	C₂×C₆₇	#407	1	1	yes	characteristic
76	C₂×C₆₇	C₂×#103	1	1	yes	characteristic
77	C₂×C₆₇	#421	1	1	yes	characteristic
78	#1487	—	67	1	no	maximal, Sylow
79	C₆₇	#1487	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×Q₈	—	67	1	no	—
81	#431	D₆₇	1	1	yes	characteristic
82	#423	—	67	1	no	—
83	Q₈⋊C₈	—	67	4	no	—
84	C₂²×Q₈	—	67	1	no	—
85	C₂×M₁₆	—	134	1	no	—
86	C₄×Q₈	—	67	2	no	—
87	C₄×Q₈	—	67	2	no	—
88	C₄×Q₈	—	134	1	no	—
89	C₄×Q₈	—	134	1	no	—
90	C₂×C₄⋊C₄	—	67	1	no	—
91	C₂×C₄⋊C₄	—	134	1	no	—
92	C₂×C₄²	—	134	1	no	—
93	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	67	2	no	—
96	C₄⋊C₈	—	134	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₄×C₈	C₂×D₆₇	1	2	yes	—
99	C₂×Q₈	—	67	2	no	—
100	C₂×Q₈	—	134	4	no	—
101	C₂×Q₈	—	134	1	no	—
102	C₂²×C₄	—	134	1	no	—
103	C₂²×C₄	—	134	1	no	—
104	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
105	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
107	M₁₆	—	268	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	134	2	no	—
110	C₄⋊C₄	—	67	2	no	—
111	C₄⋊C₄	—	134	1	no	—
112	C₄⋊C₄	—	134	2	no	—
113	C₄⋊C₄	—	134	2	no	—
114	C₄²	—	134	2	no	—
115	C₄²	—	134	2	no	—
116	C₄²	C₄×D₆₇	1	1	yes	characteristic
117	C₄²	D₂₆₈	1	1	yes	characteristic
118	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
119	C₄²	C₂²×D₆₇	1	1	yes	characteristic
120	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
121	Q₈	—	67	4	no	—
122	Q₈	—	134	4	no	—
123	Q₈	—	134	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	268	2	no	—
127	C₂×C₄	—	134	2	no	—
128	C₂×C₄	—	134	4	no	—
129	C₂×C₄	—	134	2	no	—
130	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	#11761617	1	2	yes	—
134	C₂×C₄	Dic₂₆₈	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	268	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#14523968	1	2	yes	—
143	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	268	2	no	—
148	C₄	—	134	4	no	—
149	C₄	#14523968	1	2	yes	—
150	C₄	M₁₆×D₆₇	1	2	yes	—
151	C₄	C₂×#11761617	1	1	yes	characteristic
152	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	#16863359	1	1	yes	characteristic
155	C₂	C₂×#14523968	1	1	yes	characteristic
156	C₂	#16863414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863420

C₂

yes

maximal, characteristic

#16863128

C₂

yes

maximal, complemented

#431×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₄

yes

characteristic

C₂×#11761612

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₄

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₄

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

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

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

M₁₆

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#103

yes

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#103

yes

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

D₆₇

yes

characteristic

#423

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

134

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

134

—

101

C₂×Q₈

—

134

—

102

C₂²×C₄

—

134

—

103

C₂²×C₄

—

134

—

104

C₂²×C₄

C₄×D₆₇

yes

characteristic

105

C₂²×C₄

D₂₆₈

yes

characteristic

106

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

107

M₁₆

—

268

—

108

C₂×C₈

—

109

C₂×C₈

—

134

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

—

134

—

114

C₄²

—

134

—

115

C₄²

—

134

—

116

C₄²

C₄×D₆₇

yes

characteristic

117

C₄²

D₂₆₈

yes

characteristic

118

C₄²

C₂²⋊D₆₇

yes

characteristic

119

C₄²

C₂²×D₆₇

yes

characteristic

120

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

121

Q₈

—

122

Q₈

—

134

—

123

Q₈

—

134

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

—

134

—

130

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

131

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

133

C₂×C₄

#11761617

yes

—

134

C₂×C₄

Dic₂₆₈

yes

—

135

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

137

C₂×C₄

C₂×D₂₆₈

yes

characteristic

138

C₈

—

139

C₈

—

268

—

140

C₂²

—

141

C₂²

—

142

C₂²

#14523968

yes

—

143

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

268

—

148

C₄

—

134

—

149

C₄

#14523968

yes

—

150

C₄

M₁₆×D₆₇

yes

—

151

C₄

C₂×#11761617

yes

characteristic

152

C₄

C₂×Dic₂₆₈

yes

characteristic

153

C₂

—

154

C₂

#16863359

yes

characteristic

155

C₂

C₂×#14523968

yes

characteristic

156

C₂

#16863414

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#431	D₆₇
hypocenter	C₆₇	#1487
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

#431×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16863128	C₂	semidirect
58. Dic₁₃₄	M₁₆	semidirect
79. C₆₇	#1487	semidirect

H ≅ G/N

type

3. #16863128

C₂

semidirect

58. Dic₁₃₄

M₁₆

semidirect

79. C₆₇

#1487

semidirect

Subgroups — #22558639

Conjugacy classes of subgroups

Complements