Group #22559108 — subgroups

Subgroups — #22559108

Order 8576.

162 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16863523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11761616	C₂	1	1	yes	maximal, characteristic
3	#16863474	C₂	1	1	yes	maximal, characteristic
4	#16863148	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863128	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863118	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761616	C₄	1	1	yes	characteristic
9	C₂×#11761612	—	2	1	no	—
10	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#14523974	C₄	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₆₇⋊C₈	—	2	1	no	—
14	#14523966	C₄	1	1	yes	characteristic
15	#14523964	C₄	1	1	yes	characteristic
16	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
17	#14523953	—	2	1	no	—
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×D₄×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	C₂×C₄	1	1	yes	characteristic
24	#11761616	—	2	2	no	—
25	#11761612	—	4	2	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
31	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	Dic₆₇⋊C₄	—	2	1	no	—
33	C₄×Dic₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	—	2	1	no	—
42	D₄×C₆₇	—	2	1	no	—
43	D₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	4	1	no	—
47	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	1	no	—
51	C₂×Dic₆₇	—	2	1	no	—
52	Dic₁₃₄	—	2	1	no	—
53	Dic₁₃₄	—	2	1	no	—
54	C₈×C₆₇	—	4	1	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₆₇⋊C₈	—	2	1	no	—
57	C₆₇⋊C₈	—	2	1	no	—
58	C₆₇⋊C₈	—	2	1	no	—
59	C₆₇⋊C₈	—	4	1	no	—
60	C₆₇⋊C₈	M₁₆	1	2	yes	complemented
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	4	1	no	—
67	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₆₇	#131	1	1	yes	characteristic
69	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₆₇	#137	1	1	yes	characteristic
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	4	1	no	—
73	C₂×C₆₇	—	8	1	no	—
74	C₂×C₆₇	#436	1	1	yes	characteristic
75	C₂×C₆₇	#440	1	1	yes	characteristic
76	C₂×C₆₇	#444	1	1	yes	characteristic
77	#1592	—	67	1	no	maximal, Sylow
78	C₆₇	#1592	1	1	yes	characteristic, complemented, Sylow
79	#446	—	67	1	no	—
80	C₄×SD₁₆	—	67	1	no	—
81	#436	—	67	1	no	—
82	#335	—	67	1	no	—
83	Q₈⋊C₈	—	67	1	no	—
84	D₄⋊C₈	D₆₇	1	1	yes	characteristic
85	#322	—	67	1	no	—
86	C₂×SD₁₆	—	67	1	no	—
87	C₂×M₁₆	—	134	1	no	—
88	C₄×Q₈	—	67	1	no	—
89	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
90	#106	—	67	1	no	—
91	C₄⋊C₈	—	67	1	no	—
92	C₄⋊C₈	—	134	1	no	—
93	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
94	#103	—	67	1	no	—
95	#102	—	67	1	no	—
96	C₂²⋊C₈	—	134	1	no	—
97	#97	—	134	1	no	—
98	C₄×C₈	—	67	1	no	—
99	C₄×C₈	—	67	1	no	—
100	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	C₂×Q₈	—	67	1	no	—
102	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	134	2	no	—
105	M₁₆	—	268	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	67	1	no	—
109	C₂×C₈	—	67	1	no	—
110	C₂×C₈	—	67	1	no	—
111	C₂×C₈	—	67	1	no	—
112	C₂×C₈	—	134	1	no	—
113	C₄⋊C₄	—	67	1	no	—
114	C₄⋊C₄	—	134	1	no	—
115	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	134	1	no	—
118	C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	134	1	no	—
121	Q₈	—	134	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	134	1	no	—
128	C₂×C₄	—	134	1	no	—
129	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
131	C₂×C₄	#11761627	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	268	1	no	—
135	C₈	—	67	2	no	—
136	C₈	—	134	1	no	—
137	C₈	—	134	1	no	—
138	C₈	—	134	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	268	1	no	—
146	C₄	—	268	1	no	—
147	C₄	M₁₆×D₆₇	1	1	yes	characteristic
148	C₄	#14524049	1	1	yes	characteristic
149	C₄	#14524067	1	1	yes	characteristic
150	C₄	#14524072	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#16863471	1	1	yes	characteristic
153	C₂	#16863493	1	1	yes	characteristic
154	C₂	#16863515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863523

C₂

yes

maximal, characteristic, complemented

C₄×#11761616

C₂

yes

maximal, characteristic

#16863474

C₂

yes

maximal, characteristic

#16863148

C₂

yes

maximal, characteristic, complemented

#16863128

C₂

yes

maximal, characteristic, complemented

#16863118

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

C₄

yes

characteristic

C₂×#11761612

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523974

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523966

C₄

yes

characteristic

#14523964

C₄

yes

characteristic

C₂²⋊(C₆₇⋊C₈)

—

#14523953

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₂×C₄

yes

characteristic

#11761616

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂×C₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

M₁₆

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#446

—

C₄×SD₁₆

—

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

D₆₇

yes

characteristic

#322

—

C₂×SD₁₆

—

C₂×M₁₆

—

134

—

C₄×Q₈

—

C₄×D₄

C₂×D₆₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

134

—

#97

—

134

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₆₇

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

C₄×D₆₇

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

134

—

105

M₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

134

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

134

—

115

C₄⋊C₄

C₄×D₆₇

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

134

—

118

C₄²

C₂²×D₆₇

yes

characteristic

119

C₂³

—

120

Q₈

—

134

—

121

Q₈

—

134

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

130

C₂×C₄

D₄×D₆₇

yes

characteristic

131

C₂×C₄

#11761627

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

268

—

135

C₈

—

136

C₈

—

134

—

137

C₈

—

134

—

138

C₈

—

134

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

268

—

146

C₄

—

268

—

147

C₄

M₁₆×D₆₇

yes

characteristic

148

C₄

#14524049

yes

characteristic

149

C₄

#14524067

yes

characteristic

150

C₄

#14524072

yes

characteristic

151

C₂

—

152

C₂

#16863471

yes

characteristic

153

C₂

#16863493

yes

characteristic

154

C₂

#16863515

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	D₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	D₄⋊C₈	D₆₇
hypocenter	C₆₇	#1592
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ D₄⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

D₄⋊C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

hypercenter

D₄⋊C₈

D₆₇

hypocenter

C₆₇

#1592

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ D₄⋊C₈

N	H ≅ G/N	type
1. #16863523	C₂	semidirect
4. #16863148	C₂	semidirect
5. #16863128	C₂	semidirect
6. #16863118	C₂	semidirect
60. C₆₇⋊C₈	M₁₆	semidirect
78. C₆₇	#1592	semidirect

H ≅ G/N

type

1. #16863523

C₂

semidirect

4. #16863148

C₂

semidirect

5. #16863128

C₂

semidirect

6. #16863118

C₂

semidirect

60. C₆₇⋊C₈

M₁₆

semidirect

78. C₆₇

#1592

semidirect

Subgroups — #22559108

Conjugacy classes of subgroups

Complements