Group #22559702 — subgroups

Subgroups — #22559702

Order 8576.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863775	C₂	1	1	yes	maximal, characteristic
2	#16863682	C₂	1	1	yes	maximal, characteristic
3	#16863678	C₂	1	1	yes	maximal, characteristic
4	#16863523	C₂	1	1	yes	maximal, characteristic
5	#16863146	C₂	1	1	yes	maximal, characteristic
6	#16863143	C₂	1	1	yes	maximal, characteristic
7	#332×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761618	—	2	1	no	—
9	C₂×#11761618	—	2	1	no	—
10	#14524043	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
12	#14523975	—	2	1	no	—
13	#14523974	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₆₇⋊C₈	—	2	1	no	—
16	#14523968	—	2	1	no	—
17	#14523966	—	2	1	no	—
18	#14523953	—	2	1	no	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
25	#11761618	—	4	2	no	—
26	#11761618	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
33	Dic₆₇⋊C₄	—	2	1	no	—
34	C₄×Dic₆₇	—	2	1	no	—
35	C₂×Q₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	Q₈×C₆₇	—	4	1	no	—
42	Q₈×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	C₂×D₄	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	C₂×Dic₆₇	—	4	1	no	—
50	Dic₁₃₄	—	2	1	no	—
51	Dic₁₃₄	—	2	1	no	—
52	Dic₁₃₄	—	4	1	no	—
53	Dic₁₃₄	—	4	1	no	—
54	C₈×C₆₇	—	4	1	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₆₇⋊C₈	—	2	1	no	—
57	C₆₇⋊C₈	—	4	1	no	—
58	C₆₇⋊C₈	SD₁₆	1	2	yes	—
59	C₂²×C₆₇	#121	1	1	yes	characteristic
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	8	1	no	—
62	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₆₇	AGL(1,ℤ/8)	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	Dic₆₇	—	8	1	no	—
69	C₂×C₆₇	#460	1	1	yes	characteristic
70	C₂×C₆₇	#470	1	1	yes	characteristic
71	C₂×C₆₇	#473	1	1	yes	characteristic
72	#1690	—	67	1	no	maximal, Sylow
73	C₆₇	#1690	1	1	yes	characteristic, complemented, Sylow
74	#494	—	67	1	no	—
75	#475	—	67	1	no	—
76	#475	—	67	1	no	—
77	#446	—	67	1	no	—
78	#334	—	67	1	no	—
79	#332	—	67	1	no	—
80	#332	D₆₇	1	1	yes	characteristic
81	C₂×Q₁₆	—	134	1	no	—
82	C₂×Q₁₆	—	134	1	no	—
83	C₄⋊Q₈	—	67	1	no	—
84	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
85	C₄×Q₈	—	67	1	no	—
86	#106	—	134	1	no	—
87	#106	—	134	1	no	—
88	C₄⋊C₈	—	67	1	no	—
89	C₄⋊C₈	—	134	1	no	—
90	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
92	#103	—	134	1	no	—
93	#103	—	134	1	no	—
94	#97	—	134	1	no	—
95	C₄×C₈	—	67	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	67	1	no	—
98	C₂×Q₈	—	134	1	no	—
99	Q₁₆	—	268	2	no	—
100	Q₁₆	—	268	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	67	1	no	—
104	C₂×C₈	—	67	1	no	—
105	C₂×C₈	—	134	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	67	1	no	—
109	C₄⋊C₄	—	134	1	no	—
110	C₄⋊C₄	—	134	1	no	—
111	C₄⋊C₄	—	134	1	no	—
112	C₄²	—	134	1	no	—
113	C₄²	C₂²×D₆₇	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	268	1	no	—
117	Q₈	—	268	1	no	—
118	Q₈	—	134	1	no	—
119	Q₈	—	134	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	268	1	no	—
122	C₂×C₄	—	134	1	no	—
123	C₂×C₄	—	134	1	no	—
124	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
126	C₂×C₄	#11761628	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	268	1	no	—
130	C₈	—	67	2	no	—
131	C₈	—	134	1	no	—
132	C₂²	#14524016	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	268	1	no	—
136	C₄	—	268	1	no	—
137	C₄	—	536	1	no	—
138	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
139	C₄	#14524067	1	1	yes	characteristic
140	C₄	#14524072	1	1	yes	characteristic
141	C₄	#14524075	1	1	yes	characteristic
142	C₂	#16863595	1	1	yes	characteristic
143	C₂	#16863653	1	1	yes	characteristic
144	C₂	#16863670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863775

C₂

yes

maximal, characteristic

#16863682

C₂

yes

maximal, characteristic

#16863678

C₂

yes

maximal, characteristic

#16863523

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic

#16863143

C₂

yes

maximal, characteristic

#332×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761618

—

C₂×#11761618

—

#14524043

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523975

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523968

—

#14523966

—

#14523953

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

#11761618

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

#106

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#97

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

Q₁₆

—

268

—

100

Q₁₆

—

268

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

134

—

110

C₄⋊C₄

—

134

—

111

C₄⋊C₄

—

134

—

112

C₄²

—

134

—

113

C₄²

C₂²×D₆₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

268

—

117

Q₈

—

268

—

118

Q₈

—

134

—

119

Q₈

—

134

—

120

C₂×C₄

—

121

C₂×C₄

—

268

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

D₄×D₆₇

yes

characteristic

125

C₂×C₄

D₄×D₆₇

yes

characteristic

126

C₂×C₄

#11761628

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

268

—

130

C₈

—

131

C₈

—

134

—

132

C₂²

#14524016

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

268

—

136

C₄

—

268

—

137

C₄

—

536

—

138

C₄

SD₁₆×D₆₇

yes

characteristic

139

C₄

#14524067

yes

characteristic

140

C₄

#14524072

yes

characteristic

141

C₄

#14524075

yes

characteristic

142

C₂

#16863595

yes

characteristic

143

C₂

#16863653

yes

characteristic

144

C₂

#16863670

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#332×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#332	D₆₇
hypocenter	C₆₇	#1690
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#332×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
73. C₆₇	#1690	semidirect

H ≅ G/N

type

73. C₆₇

#1690

semidirect

Subgroups — #22559702

Conjugacy classes of subgroups

Complements