Group #22559660 — subgroups

Subgroups — #22559660

Order 8576.

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

normal subgroups only

This group has 160 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	#16863747	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863686	C₂	1	1	yes	maximal, characteristic
3	#16863676	C₂	1	1	yes	maximal, characteristic
4	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863144	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863143	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	1	yes	characteristic
9	#14524043	C₂²	1	1	yes	characteristic
10	#14523976	—	2	1	no	—
11	#14523974	—	2	1	no	—
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#14523968	—	2	1	no	—
15	#14523968	—	2	1	no	—
16	#14523964	—	2	1	no	—
17	#14523964	—	2	1	no	—
18	C₂²×C₈×C₆₇	—	2	1	no	—
19	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₆₇	—	2	1	no	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
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	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	—	2	1	no	—
33	C₂×C₈×C₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	—	4	2	no	—
35	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₆₇	—	2	1	no	—
41	D₄×C₆₇	—	2	1	no	—
42	D₄×C₆₇	Q₁₆	1	2	yes	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	4	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	C₂×Dic₆₇	—	4	1	no	—
51	Dic₁₃₄	—	4	1	no	—
52	Dic₁₃₄	—	4	1	no	—
53	Dic₁₃₄	—	8	1	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	—	4	1	no	—
57	C₆₇⋊C₈	—	4	1	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₂²×C₆₇	—	2	2	no	—
60	C₂²×C₆₇	—	4	1	no	—
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	#121	1	1	yes	characteristic
63	C₄×C₆₇	—	2	1	no	—
64	C₄×C₆₇	—	4	1	no	—
65	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₆₇	#135	1	1	yes	characteristic
67	C₄×C₆₇	#135	1	1	yes	characteristic
68	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₆₇	—	8	1	no	—
70	Dic₆₇	—	8	1	no	—
71	C₂×C₆₇	—	4	2	no	—
72	C₂×C₆₇	#463	1	1	yes	characteristic
73	C₂×C₆₇	#467	1	1	yes	characteristic
74	C₂×C₆₇	#471	1	1	yes	characteristic
75	#1683	—	67	1	no	maximal, Sylow
76	C₆₇	#1683	1	1	yes	characteristic, complemented, Sylow
77	#487	—	67	1	no	—
78	#476	—	67	1	no	—
79	#474	—	67	1	no	—
80	C₈×D₄	D₆₇	1	1	yes	characteristic
81	#335	—	67	1	no	—
82	#333	—	67	1	no	—
83	#332	—	67	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₄⋊Q₈	—	67	1	no	—
86	C₄⋊Q₈	—	67	1	no	—
87	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
88	#107	—	134	1	no	—
89	#106	—	134	1	no	—
90	C₄⋊C₈	—	67	1	no	—
91	C₄⋊C₈	—	67	1	no	—
92	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
93	#103	—	134	1	no	—
94	#103	—	134	1	no	—
95	#102	—	134	1	no	—
96	#102	—	134	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	C₂×Q₈	—	134	1	no	—
100	C₂×Q₈	—	134	1	no	—
101	C₂×D₄	D₂₆₈	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	134	1	no	—
106	C₂×C₈	—	134	1	no	—
107	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
109	C₄⋊C₄	—	134	1	no	—
110	C₄⋊C₄	—	134	1	no	—
111	C₄⋊C₄	—	134	1	no	—
112	C₄⋊C₄	—	134	1	no	—
113	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	C₂³	—	2	1	no	—
117	Q₈	—	268	1	no	—
118	Q₈	—	268	1	no	—
119	Q₈	—	536	1	no	—
120	D₄	—	2	1	no	—
121	D₄	Dic₂₆₈	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	268	1	no	—
126	C₂×C₄	—	268	1	no	—
127	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
129	C₂×C₄	#11761628	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	268	1	no	—
134	C₈	—	268	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#14524017	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	536	1	no	—
142	C₄	—	536	1	no	—
143	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
144	C₄	#14524063	1	1	yes	characteristic
145	C₄	#14524065	1	1	yes	characteristic
146	C₄	#14524069	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	#16863611	1	1	yes	characteristic
149	C₂	#16863637	1	1	yes	characteristic
150	C₂	#16863659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863747

C₂

yes

maximal, characteristic, complemented

#16863686

C₂

yes

maximal, characteristic

#16863676

C₂

yes

maximal, characteristic

#16863149

C₂

yes

maximal, characteristic, complemented

#16863144

C₂

yes

maximal, characteristic, complemented

#16863143

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14523976

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

C₂²×C₈×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

Q₁₆

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

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

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

#487

—

#476

—

#474

—

C₈×D₄

D₆₇

yes

—

—

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₆₇

yes

characteristic

#107

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#102

—

134

—

#102

—

134

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

100

C₂×Q₈

—

134

—

101

C₂×D₄

D₂₆₈

yes

characteristic

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

—

134

—

107

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

108

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

109

C₄⋊C₄

—

134

—

110

C₄⋊C₄

—

134

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

D₂₆₈

yes

characteristic

114

C₂²⋊C₄

—

115

C₄²

C₂²×D₆₇

yes

characteristic

116

C₂³

—

117

Q₈

—

268

—

118

Q₈

—

268

—

119

Q₈

—

536

—

120

D₄

—

121

D₄

Dic₂₆₈

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

268

—

126

C₂×C₄

—

268

—

127

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

128

C₂×C₄

C₂×D₂₆₈

yes

characteristic

129

C₂×C₄

#11761628

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

268

—

134

C₈

—

268

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14524017

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

536

—

142

C₄

—

536

—

143

C₄

C₂×Dic₂₆₈

yes

characteristic

144

C₄

#14524063

yes

characteristic

145

C₄

#14524065

yes

characteristic

146

C₄

#14524069

yes

characteristic

147

C₂

—

148

C₂

#16863611

yes

characteristic

149

C₂

#16863637

yes

characteristic

150

C₂

#16863659

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈×D₄×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#1683
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14524017

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

C₈×D₄×C₆₇

C₂

socle

C₂²×C₆₇

#121

hypercenter

C₈×D₄

D₆₇

hypocenter

C₆₇

#1683

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #16863747	C₂	semidirect
4. #16863149	C₂	semidirect
5. #16863144	C₂	semidirect
6. #16863143	C₂	semidirect
76. C₆₇	#1683	semidirect

H ≅ G/N

type

1. #16863747

C₂

semidirect

4. #16863149

C₂

semidirect

5. #16863144

C₂

semidirect

6. #16863143

C₂

semidirect

76. C₆₇

#1683

semidirect

Subgroups — #22559660

Conjugacy classes of subgroups

Complements