Group #22558613 — subgroups

Subgroups — #22558613

Order 8576.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16863953	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₈)	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×C₆₇⋊C₈	—	2	1	no	—
6	#14524043	C₄	1	1	yes	characteristic
7	#14524043	C₄	1	1	yes	characteristic
8	#14524035	—	2	1	no	—
9	#14524022	—	2	1	no	—
10	#14524022	C₄	1	2	yes	—
11	C₄×Dic₁₃₄	—	2	1	no	—
12	C₄×Dic₁₃₄	C₂²	1	2	yes	—
13	#14524002	—	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	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₈	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	4	2	no	—
24	C₄⋊Dic₆₇	—	2	1	no	—
25	C₄⋊Dic₆₇	—	2	1	no	—
26	C₄⋊Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
28	Dic₆₇⋊C₄	—	2	2	no	—
29	Dic₆₇⋊C₄	—	2	2	no	—
30	C₂²⋊Dic₆₇	—	2	2	no	—
31	C₄×Dic₆₇	—	2	2	no	—
32	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	2	no	—
36	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
47	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
48	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₇	—	2	2	no	—
52	C₂×Dic₆₇	—	2	2	no	—
53	Dic₁₃₄	—	2	4	no	—
54	Dic₁₃₄	—	4	2	no	—
55	C₈×C₆₇	—	4	2	no	—
56	C₆₇⋊C₈	—	4	2	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	—	2	1	no	—
59	C₂²×C₆₇	#103	1	2	yes	—
60	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	—	2	1	no	—
63	C₄×C₆₇	—	2	1	no	—
64	C₄×C₆₇	#103	1	2	yes	—
65	C₄×C₆₇	#131	1	2	yes	—
66	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
68	Dic₆₇	—	4	2	no	—
69	Dic₆₇	—	4	2	no	—
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	#408	1	1	yes	characteristic
72	C₂×C₆₇	C₂×#103	1	1	yes	characteristic
73	C₂×C₆₇	#421	1	1	yes	characteristic
74	#1481	—	67	1	no	maximal, Sylow
75	C₆₇	#1481	1	1	yes	characteristic, complemented, Sylow
76	#533	—	67	1	no	—
77	#431	D₆₇	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	67	1	no	—
79	Q₈⋊C₈	—	67	4	no	—
80	C₂²×C₈	—	134	1	no	—
81	C₄⋊Q₈	—	67	1	no	—
82	C₄⋊Q₈	—	67	1	no	—
83	#125	—	134	1	no	—
84	C₂²⋊Q₈	—	67	2	no	—
85	C₂²⋊Q₈	—	134	1	no	—
86	C₄×Q₈	—	67	2	no	—
87	C₄×Q₈	—	134	1	no	—
88	#117	—	134	1	no	—
89	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	67	2	no	—
92	C₄⋊C₈	—	134	1	no	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₄×C₈	C₂×D₆₇	1	2	yes	—
95	C₂×Q₈	—	67	2	no	—
96	C₂×Q₈	—	134	1	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	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	268	2	no	—
102	C₂×C₈	—	134	2	no	—
103	C₄⋊C₄	—	67	2	no	—
104	C₄⋊C₄	—	134	1	no	—
105	C₄⋊C₄	—	134	1	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₄⋊C₄	—	134	1	no	—
108	C₄⋊C₄	—	134	2	no	—
109	C₂²⋊C₄	—	134	2	no	—
110	C₄²	—	134	2	no	—
111	C₄²	C₄×D₆₇	1	1	yes	characteristic
112	C₄²	D₂₆₈	1	1	yes	characteristic
113	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
114	C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
116	Q₈	—	268	2	no	—
117	Q₈	—	134	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	134	2	no	—
121	C₂×C₄	—	134	2	no	—
122	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#11761617	1	2	yes	—
126	C₂×C₄	Dic₂₆₈	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	268	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#14523968	1	2	yes	—
135	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	268	2	no	—
140	C₄	—	268	2	no	—
141	C₄	#14523968	1	2	yes	—
142	C₄	#14524051	1	2	yes	—
143	C₄	C₂×#11761617	1	1	yes	characteristic
144	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#16863364	1	1	yes	characteristic
147	C₂	C₂×#14523968	1	1	yes	characteristic
148	C₂	#16863414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#16863128

C₂

yes

maximal, complemented

#431×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

—

#14524043

C₄

yes

characteristic

#14524043

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

—

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×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊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

—

Dic₁₃₄

—

Dic₁₃₄

—

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

#131

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

#533

—

#431

D₆₇

yes

characteristic

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

C₂²×C₈

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

134

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

268

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

—

134

—

109

C₂²⋊C₄

—

134

—

110

C₄²

—

134

—

111

C₄²

C₄×D₆₇

yes

characteristic

112

C₄²

D₂₆₈

yes

characteristic

113

C₄²

C₂²⋊D₆₇

yes

characteristic

114

C₄²

C₂²×D₆₇

yes

characteristic

115

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

116

Q₈

—

268

—

117

Q₈

—

134

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

124

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

125

C₂×C₄

#11761617

yes

—

126

C₂×C₄

Dic₂₆₈

yes

—

127

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

129

C₂×C₄

C₂×D₂₆₈

yes

characteristic

130

C₈

—

131

C₈

—

268

—

132

C₂²

—

133

C₂²

—

134

C₂²

#14523968

yes

—

135

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

268

—

140

C₄

—

268

—

141

C₄

#14523968

yes

—

142

C₄

#14524051

yes

—

143

C₄

C₂×#11761617

yes

characteristic

144

C₄

C₂×Dic₂₆₈

yes

characteristic

145

C₂

—

146

C₂

#16863364

yes

characteristic

147

C₂

C₂×#14523968

yes

characteristic

148

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	#1481
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
75. C₆₇	#1481	semidirect

H ≅ G/N

type

3. #16863128

C₂

semidirect

75. C₆₇

#1481

semidirect

Subgroups — #22558613

Conjugacy classes of subgroups

Complements