Group #22559668 — subgroups

Subgroups — #22559668

Order 8576.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16863747	C₂	1	1	yes	maximal, characteristic
2	#16863692	C₂	1	1	yes	maximal, characteristic
3	#16863682	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
5	#16863144	C₂	1	1	yes	maximal, characteristic
6	#16863141	C₂	1	1	yes	maximal, characteristic
7	#335×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	1	yes	characteristic
9	#14524043	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#14523977	—	2	1	no	—
12	#14523975	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₆₇⋊C₈	—	2	1	no	—
15	#14523968	—	2	1	no	—
16	#14523968	—	2	1	no	—
17	#14523968	—	2	1	no	—
18	#14523968	—	2	1	no	—
19	C₈×Dic₆₇	—	2	1	no	—
20	C₄⋊C₈×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	—	2	1	no	—
25	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
32	Dic₆₇⋊C₄	—	2	1	no	—
33	C₄×Dic₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
42	C₂×Dic₆₇	—	2	1	no	—
43	C₂×Dic₆₇	—	2	1	no	—
44	C₂×Dic₆₇	—	4	1	no	—
45	C₂×Dic₆₇	—	4	1	no	—
46	Dic₁₃₄	—	2	1	no	—
47	Dic₁₃₄	—	4	1	no	—
48	Dic₁₃₄	—	4	1	no	—
49	Dic₁₃₄	—	8	1	no	—
50	Dic₁₃₄	Q₁₆	1	2	yes	—
51	C₈×C₆₇	—	2	1	no	—
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₈×C₆₇	—	4	1	no	—
55	C₆₇⋊C₈	—	4	1	no	—
56	C₂²×C₆₇	#121	1	1	yes	characteristic
57	C₄×C₆₇	—	2	1	no	—
58	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₆₇	#135	1	1	yes	characteristic
60	C₄×C₆₇	#135	1	1	yes	characteristic
61	C₄×C₆₇	#137	1	1	yes	characteristic
62	Dic₆₇	—	2	2	no	—
63	Dic₆₇	—	4	1	no	—
64	Dic₆₇	—	8	1	no	—
65	Dic₆₇	—	8	1	no	—
66	C₂×C₆₇	#464	1	1	yes	characteristic
67	C₂×C₆₇	#467	1	1	yes	characteristic
68	C₂×C₆₇	#471	1	1	yes	characteristic
69	#1684	—	67	1	no	maximal, Sylow
70	C₆₇	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	67	1	no	—
72	#477	—	67	1	no	—
73	#475	—	67	1	no	—
74	C₈×Q₈	—	67	1	no	—
75	#335	D₆₇	1	1	yes	characteristic
76	#333	—	67	1	no	—
77	#332	—	67	1	no	—
78	C₄⋊Q₈	—	67	1	no	—
79	C₄⋊Q₈	—	67	1	no	—
80	C₄×Q₈	—	67	1	no	—
81	#107	—	134	1	no	—
82	#106	—	134	1	no	—
83	C₄⋊C₈	—	67	1	no	—
84	C₄⋊C₈	—	134	1	no	—
85	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
87	#103	—	134	1	no	—
88	#103	—	134	1	no	—
89	#103	—	134	1	no	—
90	#103	—	134	1	no	—
91	C₄×C₈	—	134	1	no	—
92	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	67	1	no	—
94	C₂×Q₈	—	134	1	no	—
95	C₂×Q₈	—	134	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	134	1	no	—
99	C₂×C₈	D₂₆₈	1	1	yes	characteristic
100	C₂×C₈	D₂₆₈	1	1	yes	characteristic
101	C₄⋊C₄	—	67	1	no	—
102	C₄⋊C₄	—	134	1	no	—
103	C₄⋊C₄	—	134	1	no	—
104	C₄⋊C₄	—	134	1	no	—
105	C₄⋊C₄	—	134	1	no	—
106	C₄⋊C₄	—	134	1	no	—
107	C₄²	—	134	1	no	—
108	C₄²	C₂²×D₆₇	1	1	yes	characteristic
109	Q₈	—	268	1	no	—
110	Q₈	—	268	1	no	—
111	Q₈	—	536	1	no	—
112	Q₈	—	67	2	no	—
113	Q₈	—	134	1	no	—
114	C₂×C₄	—	268	1	no	—
115	C₂×C₄	—	268	1	no	—
116	C₂×C₄	—	134	1	no	—
117	C₂×C₄	—	134	1	no	—
118	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂×C₄	#11761629	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	268	1	no	—
126	C₂²	#14524018	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	268	1	no	—
129	C₄	—	536	1	no	—
130	C₄	—	536	1	no	—
131	C₄	—	134	2	no	—
132	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
133	C₄	#14524062	1	1	yes	characteristic
134	C₄	#14524065	1	1	yes	characteristic
135	C₄	#14524076	1	1	yes	characteristic
136	C₂	#16863619	1	1	yes	characteristic
137	C₂	#16863638	1	1	yes	characteristic
138	C₂	#16863660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863747

C₂

yes

maximal, characteristic

#16863692

C₂

yes

maximal, characteristic

#16863682

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863144

C₂

yes

maximal, characteristic

#16863141

C₂

yes

maximal, characteristic

#335×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523977

—

#14523975

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

—

—

—

—

C₈×Dic₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

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

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#487

—

#477

—

#475

—

C₈×Q₈

—

#335

D₆₇

yes

characteristic

#333

—

#332

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#103

—

134

—

#103

—

134

—

C₄×C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

characteristic

100

C₂×C₈

D₂₆₈

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄²

—

134

—

108

C₄²

C₂²×D₆₇

yes

characteristic

109

Q₈

—

268

—

110

Q₈

—

268

—

111

Q₈

—

536

—

112

Q₈

—

113

Q₈

—

134

—

114

C₂×C₄

—

268

—

115

C₂×C₄

—

268

—

116

C₂×C₄

—

134

—

117

C₂×C₄

—

134

—

118

C₂×C₄

D₄×D₆₇

yes

characteristic

119

C₂×C₄

C₂×D₂₆₈

yes

characteristic

120

C₂×C₄

#11761629

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

268

—

126

C₂²

#14524018

yes

characteristic

127

C₄

—

128

C₄

—

268

—

129

C₄

—

536

—

130

C₄

—

536

—

131

C₄

—

134

—

132

C₄

Q₁₆×D₆₇

yes

characteristic

133

C₄

#14524062

yes

characteristic

134

C₄

#14524065

yes

characteristic

135

C₄

#14524076

yes

characteristic

136

C₂

#16863619

yes

characteristic

137

C₂

#16863638

yes

characteristic

138

C₂

#16863660

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#335×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#335	D₆₇
hypocenter	C₆₇	#1684
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14524018

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#335×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
70. C₆₇	#1684	semidirect

H ≅ G/N

type

70. C₆₇

#1684

semidirect

Subgroups — #22559668

Conjugacy classes of subgroups

Complements