Group #22559593 — subgroups

Subgroups — #22559593

Order 8576.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863792	C₂	1	1	yes	maximal, characteristic
2	#16863606	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#16863146	C₂	1	1	yes	maximal, characteristic
5	#16863145	C₂	1	1	yes	maximal, characteristic
6	#16863143	C₂	1	1	yes	maximal, characteristic
7	#477×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761618	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14523974	—	2	1	no	—
11	#14523974	—	2	1	no	—
12	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#14523966	—	2	1	no	—
16	C₄×C₆₇⋊C₈	—	2	1	no	—
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#107×C₆₇	—	2	1	no	—
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#103×C₆₇	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	#11761618	—	4	2	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×Q₈×C₆₇	—	2	1	no	—
32	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	1	no	—
34	C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	Q₈×C₆₇	—	2	1	no	—
41	Q₈×C₆₇	—	8	1	no	—
42	Q₈×C₆₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
49	C₂×Dic₆₇	—	4	1	no	—
50	Dic₁₃₄	—	4	1	no	—
51	Dic₁₃₄	—	4	1	no	—
52	C₈×C₆₇	—	4	1	no	—
53	C₆₇⋊C₈	—	2	1	no	—
54	C₆₇⋊C₈	—	4	1	no	—
55	C₆₇⋊C₈	—	4	1	no	—
56	C₆₇⋊C₈	Q₁₆	1	2	yes	complemented
57	C₂²×C₆₇	#121	1	1	yes	characteristic
58	C₄×C₆₇	—	2	2	no	—
59	C₄×C₆₇	—	2	1	no	—
60	C₄×C₆₇	—	4	1	no	—
61	C₄×C₆₇	—	8	1	no	—
62	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₆₇	#135	1	1	yes	characteristic
65	C₄×C₆₇	#137	1	1	yes	characteristic
66	Dic₆₇	—	8	1	no	—
67	C₂×C₆₇	#462	1	1	yes	characteristic
68	C₂×C₆₇	#465	1	1	yes	characteristic
69	C₂×C₆₇	#471	1	1	yes	characteristic
70	#1672	—	67	1	no	maximal, Sylow
71	C₆₇	#1672	1	1	yes	characteristic, complemented, Sylow
72	#498	—	67	1	no	—
73	#477	D₆₇	1	1	yes	characteristic
74	#462	—	67	1	no	—
75	C₈×Q₈	—	67	1	no	—
76	#334	—	67	1	no	—
77	#333	—	67	1	no	—
78	#332	—	67	1	no	—
79	C₂×Q₁₆	—	134	1	no	—
80	C₄⋊Q₈	—	67	1	no	—
81	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
83	#107	—	2	1	no	—
84	#106	—	134	1	no	—
85	#106	—	134	1	no	—
86	C₄⋊C₈	—	67	1	no	—
87	C₄⋊C₈	—	67	1	no	—
88	C₄⋊C₈	—	134	1	no	—
89	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	134	1	no	—
92	C₄×C₈	—	67	1	no	—
93	C₄×C₈	—	134	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	134	1	no	—
96	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
97	Q₁₆	—	268	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	67	1	no	—
100	C₂×C₈	—	67	1	no	—
101	C₂×C₈	—	134	1	no	—
102	C₂×C₈	—	134	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	134	1	no	—
107	C₄⋊C₄	—	134	1	no	—
108	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₆₇	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	268	1	no	—
114	Q₈	—	268	1	no	—
115	Q₈	#11761615	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	268	1	no	—
120	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₄	#11761627	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	268	1	no	—
125	C₈	—	268	1	no	—
126	C₈	—	67	2	no	—
127	C₈	—	134	1	no	—
128	C₂²	#14524015	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	536	1	no	—
134	C₄	C₂×#11761615	1	1	yes	characteristic
135	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
136	C₄	#14524063	1	1	yes	characteristic
137	C₄	#14524075	1	1	yes	characteristic
138	C₂	#16863604	1	1	yes	characteristic
139	C₂	#16863627	1	1	yes	characteristic
140	C₂	#16863657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863792

C₂

yes

maximal, characteristic

#16863606

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic

#16863145

C₂

yes

maximal, characteristic

#16863143

C₂

yes

maximal, characteristic

#477×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761618

—

#14524043

C₂²

yes

characteristic

#14523974

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523966

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂×Dic₁₃₄

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×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

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

Q₁₆

yes

complemented

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#498

—

#477

D₆₇

yes

characteristic

#462

—

C₈×Q₈

—

#334

—

#333

—

#332

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

#106

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

#103

—

134

—

C₄×C₈

—

C₄×C₈

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

Q₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

134

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₆₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

268

—

114

Q₈

—

268

—

115

Q₈

#11761615

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

268

—

120

C₂×C₄

D₄×D₆₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂×C₄

#11761627

yes

characteristic

123

C₈

—

124

C₈

—

268

—

125

C₈

—

268

—

126

C₈

—

127

C₈

—

134

—

128

C₂²

#14524015

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

536

—

134

C₄

C₂×#11761615

yes

characteristic

135

C₄

Q₁₆×D₆₇

yes

characteristic

136

C₄

#14524063

yes

characteristic

137

C₄

#14524075

yes

characteristic

138

C₂

#16863604

yes

characteristic

139

C₂

#16863627

yes

characteristic

140

C₂

#16863657

yes

characteristic

	subgroup	G/H
center	C₂²	#14524015
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#477×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#477	D₆₇
hypocenter	C₆₇	#1672
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14524015

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#477×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
56. C₆₇⋊C₈	Q₁₆	semidirect
71. C₆₇	#1672	semidirect

H ≅ G/N

type

56. C₆₇⋊C₈

Q₁₆

semidirect

71. C₆₇

#1672

semidirect

Subgroups — #22559593

Conjugacy classes of subgroups

Complements