Group #22570948 — subgroups

Subgroups — #22570948

Order 8576.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16863947	C₂	1	2	yes	maximal, complemented
2	#16863773	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863744	C₂	1	2	yes	maximal, complemented
4	#16863650	C₂	1	4	yes	maximal, complemented
5	#16863507	C₂	1	4	yes	maximal, complemented
6	C₄×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761629	—	2	2	no	—
9	C₂×D₅₃₆	C₂²	1	4	yes	—
10	C₂×#11761611	C₂²	1	2	yes	—
11	#14524040	C₂²	1	2	yes	—
12	#14524032	—	2	2	no	—
13	#14524018	—	2	2	no	—
14	#14524018	C₂²	1	4	yes	—
15	#14524005	—	2	2	no	—
16	#14524005	C₂²	1	4	yes	—
17	C₄²×D₆₇	C₂²	1	1	yes	characteristic
18	#14523963	C₂²	1	8	yes	—
19	#14523953	C₂²	1	2	yes	—
20	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#106×C₆₇	C₂²	1	4	yes	—
23	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#11761629	—	4	8	no	—
25	C₂×D₂₆₈	—	2	4	no	—
26	C₂×D₂₆₈	—	2	2	no	—
27	C₂×D₂₆₈	—	2	2	no	—
28	C₂×D₂₆₈	C₂³	1	4	yes	—
29	C₂×C₄×D₆₇	—	2	4	no	—
30	C₂×C₄×D₆₇	D₄	1	2	yes	—
31	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
32	D₅₃₆	—	2	8	no	—
33	#11761611	—	2	4	no	—
34	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
35	(C₂×D₆₇)⋊C₄	—	2	4	no	—
36	(C₂×D₆₇)⋊C₄	—	2	4	no	—
37	C₄×Dic₆₇	D₄	1	2	yes	—
38	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	C₂³	1	2	yes	—
41	C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₆₇	—	2	4	no	—
45	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₆₇	—	4	4	no	—
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
52	D₂₆₈	—	2	8	no	—
53	D₂₆₈	—	4	8	no	—
54	D₂₆₈	—	4	4	no	—
55	D₂₆₈	—	4	4	no	—
56	C₄×D₆₇	—	2	4	no	—
57	C₄×D₆₇	—	2	2	no	—
58	C₄×D₆₇	—	4	8	no	—
59	C₈×C₆₇	#41	1	4	yes	complemented
60	C₆₇⋊C₈	—	2	2	no	—
61	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₆₇	—	2	1	no	—
63	C₂×D₆₇	—	2	2	no	—
64	C₂×D₆₇	—	4	4	no	—
65	C₂×D₆₇	—	4	4	no	—
66	C₂×D₆₇	—	4	4	no	—
67	C₄×C₆₇	—	4	4	no	—
68	C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₆₇	C₂×#41	1	2	yes	—
70	Dic₆₇	—	2	1	no	—
71	Dic₆₇	—	2	2	no	—
72	C₂×C₆₇	#532	1	1	yes	characteristic
73	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₆₇	—	4	1	no	—
75	D₆₇	—	8	4	no	—
76	#3184	—	67	1	no	maximal, Sylow
77	C₆₇	#3184	1	1	yes	characteristic, complemented, Sylow
78	#532	—	67	2	no	—
79	#498	D₆₇	1	1	yes	characteristic, complemented
80	#493	—	67	1	no	—
81	#486	—	67	2	no	—
82	#469	—	67	4	no	—
83	#442	—	67	4	no	—
84	C₄×M₁₆	—	67	1	no	—
85	C₂×#41	—	134	2	no	—
86	C₂×D₈	—	67	4	no	—
87	C₂×M₁₆	—	67	2	no	—
88	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
89	#127	—	67	2	no	—
90	#124	—	134	2	no	—
91	#121	—	67	4	no	—
92	#121	—	134	2	no	—
93	C₄×D₄	—	67	4	no	—
94	C₄×D₄	—	134	2	no	—
95	C₂×C₄²	—	67	1	no	—
96	#106	C₂×D₆₇	1	4	yes	—
97	#102	—	67	8	no	—
98	#97	—	67	2	no	—
99	C₄×C₈	—	67	1	no	—
100	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	#41	—	268	8	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×D₄	—	67	4	no	—
104	C₂×D₄	—	134	4	no	—
105	C₂×D₄	—	134	2	no	—
106	C₂×D₄	—	134	2	no	—
107	C₂²×C₄	—	67	2	no	—
108	C₂²×C₄	—	67	1	no	—
109	C₂²×C₄	—	134	4	no	—
110	D₈	—	134	8	no	—
111	M₁₆	—	134	4	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	C₂²×D₆₇	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
116	C₂²⋊C₄	—	134	4	no	—
117	C₂²⋊C₄	—	134	4	no	—
118	C₄²	—	67	2	no	—
119	C₄²	—	67	1	no	—
120	C₄²	C₂²×D₆₇	1	1	yes	characteristic
121	C₂³	—	67	1	no	—
122	C₂³	—	134	4	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	268	8	no	—
125	D₄	—	268	4	no	—
126	D₄	—	268	4	no	—
127	D₄	—	134	8	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	268	8	no	—
130	C₂×C₄	—	67	1	no	—
131	C₂×C₄	—	67	2	no	—
132	C₂×C₄	—	134	4	no	—
133	C₂×C₄	—	134	2	no	—
134	C₂×C₄	D₄×D₆₇	1	2	yes	—
135	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
136	C₈	—	134	2	no	—
137	C₈	#11761629	1	4	yes	complemented
138	C₂²	—	268	4	no	—
139	C₂²	—	268	4	no	—
140	C₂²	—	268	4	no	—
141	C₂²	—	134	2	no	—
142	C₂²	—	134	1	no	—
143	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	134	2	no	—
146	C₄	—	134	1	no	—
147	C₄	#14524070	1	4	yes	—
148	C₄	C₂×#11761629	1	2	yes	—
149	C₂	—	268	1	no	—
150	C₂	—	536	4	no	—
151	C₂	#16863947	1	1	yes	characteristic
152	C₂	C₂×#14524070	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863947

C₂

yes

maximal, complemented

#16863773

C₂

yes

maximal, characteristic, complemented

#16863744

C₂

yes

maximal, complemented

#16863650

C₂

yes

maximal, complemented

#16863507

C₂

yes

maximal, complemented

C₄×#11761611

C₂

yes

maximal, characteristic, complemented

#498×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761629

—

C₂×D₅₃₆

C₂²

yes

—

C₂×#11761611

C₂²

yes

—

#14524040

C₂²

yes

—

#14524032

—

#14524018

—

#14524018

C₂²

yes

—

#14524005

—

#14524005

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523963

C₂²

yes

—

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

—

#106×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761629

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

D₅₃₆

—

#11761611

—

C₂×C₆₇⋊C₈

C₂³

yes

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

C₂×D₄

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

#41

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#532

yes

characteristic

C₂×C₆₇

C₂×AGL(1,ℤ/8)

yes

—

D₆₇

—

D₆₇

—

#3184

—

maximal, Sylow

C₆₇

#3184

yes

characteristic, complemented, Sylow

#532

—

#498

D₆₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×#41

—

134

—

C₂×D₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₆₇

yes

—

#127

—

#124

—

134

—

#121

—

#121

—

134

—

C₄×D₄

—

C₄×D₄

—

134

—

C₂×C₄²

—

#106

C₂×D₆₇

yes

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₆₇

yes

characteristic

101

#41

—

268

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

134

—

105

C₂×D₄

—

134

—

106

C₂×D₄

—

134

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

134

—

110

D₈

—

134

—

111

M₁₆

—

134

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₆₇

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²×D₆₇

yes

—

116

C₂²⋊C₄

—

134

—

117

C₂²⋊C₄

—

134

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×D₆₇

yes

characteristic

121

C₂³

—

122

C₂³

—

134

—

123

Q₈

—

124

D₄

—

268

—

125

D₄

—

268

—

126

D₄

—

268

—

127

D₄

—

134

—

128

C₂×C₄

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

134

—

133

C₂×C₄

—

134

—

134

C₂×C₄

D₄×D₆₇

yes

—

135

C₂×C₄

C₂³×D₆₇

yes

characteristic

136

C₈

—

134

—

137

C₈

#11761629

yes

complemented

138

C₂²

—

268

—

139

C₂²

—

268

—

140

C₂²

—

268

—

141

C₂²

—

134

—

142

C₂²

—

134

—

143

C₂²

C₂×D₄×D₆₇

yes

characteristic

144

C₄

—

145

C₄

—

134

—

146

C₄

—

134

—

147

C₄

#14524070

yes

—

148

C₄

C₂×#11761629

yes

—

149

C₂

—

268

—

150

C₂

—

536

—

151

C₂

#16863947

yes

characteristic

152

C₂

C₂×#14524070

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#498×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#498	D₆₇
hypocenter	C₆₇	#3184
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#498×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #16863947	C₂	semidirect
2. #16863773	C₂	semidirect
3. #16863744	C₂	semidirect
4. #16863650	C₂	semidirect
5. #16863507	C₂	semidirect
6. C₄×#11761611	C₂	semidirect
7. #498×C₆₇	C₂	semidirect
59. C₈×C₆₇	#41	semidirect
77. C₆₇	#3184	semidirect
79. #498	D₆₇	semidirect
137. C₈	#11761629	semidirect

H ≅ G/N

type

1. #16863947

C₂

semidirect

2. #16863773

C₂

semidirect

3. #16863744

C₂

semidirect

4. #16863650

C₂

semidirect

5. #16863507

C₂

semidirect

6. C₄×#11761611

C₂

semidirect

7. #498×C₆₇

C₂

semidirect

59. C₈×C₆₇

#41

semidirect

77. C₆₇

#3184

semidirect

79. #498

D₆₇

semidirect

137. C₈

#11761629

semidirect

Subgroups — #22570948

Conjugacy classes of subgroups

Complements