Group #22570945 — subgroups

Subgroups — #22570945

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	#16863944	C₂	1	2	yes	maximal, complemented
2	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863742	C₂	1	2	yes	maximal, complemented
4	#16863646	C₂	1	4	yes	maximal, complemented
5	#16863508	C₂	1	4	yes	maximal, complemented
6	C₄×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761627	—	2	2	no	—
9	C₂×#11761611	C₂²	1	2	yes	—
10	#14524043	C₂²	1	2	yes	—
11	#14524030	—	2	2	no	—
12	#14524014	—	2	2	no	—
13	#14524014	C₂²	1	4	yes	—
14	D₄×Dic₆₇	—	2	2	no	—
15	D₄×Dic₆₇	C₂²	1	4	yes	—
16	C₄²×D₆₇	C₂²	1	1	yes	characteristic
17	#14523975	C₂²	1	4	yes	—
18	#14523964	C₂²	1	8	yes	—
19	#14523953	C₂²	1	2	yes	—
20	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₆₇	C₂²	1	4	yes	—
22	#127×C₆₇	C₂²	1	2	yes	—
23	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#11761627	—	4	8	no	—
25	C₂×Dic₁₃₄	—	2	2	no	—
26	C₂×C₂²⋊D₆₇	—	2	4	no	—
27	C₂²×Dic₆₇	—	2	4	no	—
28	C₂×C₄×D₆₇	D₄	1	2	yes	—
29	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
30	#11761611	—	2	4	no	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂²⋊Dic₆₇	—	2	4	no	—
35	C₂²⋊Dic₆₇	—	2	4	no	—
36	C₄×Dic₆₇	D₄	1	2	yes	—
37	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₆₇	—	2	2	no	—
39	C₂×D₄×C₆₇	—	2	2	no	—
40	C₂×D₄×C₆₇	C₂³	1	4	yes	—
41	D₈×C₆₇	—	2	8	no	—
42	C₂×C₈×C₆₇	C₂³	1	2	yes	—
43	C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₆₇	—	2	4	no	—
45	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
46	D₄×C₆₇	—	2	8	no	—
47	D₄×C₆₇	—	4	4	no	—
48	D₄×C₆₇	—	4	4	no	—
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₆₇	—	4	8	no	—
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	4	8	no	—
54	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
56	C₄×D₆₇	—	2	4	no	—
57	C₄×D₆₇	—	2	2	no	—
58	Dic₁₃₄	—	4	4	no	—
59	C₈×C₆₇	#41	1	4	yes	complemented
60	C₆₇⋊C₈	—	2	2	no	—
61	C₂²×C₆₇	—	4	4	no	—
62	C₂²×C₆₇	—	4	4	no	—
63	C₂²×C₆₇	—	4	4	no	—
64	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₆₇	—	2	1	no	—
66	C₂×D₆₇	—	2	2	no	—
67	C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₆₇	C₂×#41	1	2	yes	—
69	Dic₆₇	—	2	1	no	—
70	Dic₆₇	—	2	2	no	—
71	Dic₆₇	—	4	4	no	—
72	C₂×C₆₇	—	8	4	no	—
73	C₂×C₆₇	#532	1	1	yes	characteristic
74	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	D₆₇	—	4	1	no	—
76	#3184	—	67	1	no	maximal, Sylow
77	C₆₇	#3184	1	1	yes	characteristic, complemented, Sylow
78	#532	—	67	2	no	—
79	#498	—	67	1	no	—
80	#493	D₆₇	1	1	yes	characteristic, complemented
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	C₂×D₆₇	1	4	yes	—
87	C₂×M₁₆	—	67	2	no	—
88	C₄⋊Q₈	—	67	2	no	—
89	#127	C₂×D₆₇	1	2	yes	—
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	—	67	4	no	—
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	—	134	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	134	4	no	—
106	C₂×D₄	C₂²×D₆₇	1	4	yes	—
107	C₂²×C₄	—	67	2	no	—
108	C₂²×C₄	—	67	1	no	—
109	C₂²×C₄	—	134	4	no	—
110	D₈	—	2	8	no	—
111	M₁₆	—	134	4	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	C₂²×D₆₇	1	2	yes	—
114	C₄⋊C₄	—	67	4	no	—
115	C₄⋊C₄	—	134	2	no	—
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	—	2	4	no	—
122	C₂³	—	67	1	no	—
123	Q₈	—	268	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	268	8	no	—
128	C₂×C₄	—	268	8	no	—
129	C₂×C₄	—	67	1	no	—
130	C₂×C₄	—	67	2	no	—
131	C₂×C₄	—	134	4	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	#11761627	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	134	2	no	—
142	C₂²	—	134	1	no	—
143	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
144	C₄	—	268	4	no	—
145	C₄	—	134	2	no	—
146	C₄	—	134	1	no	—
147	C₄	#14524067	1	4	yes	—
148	C₄	C₂×#11761627	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	268	1	no	—
151	C₂	#16863944	1	1	yes	characteristic
152	C₂	C₂×#14524067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863944

C₂

yes

maximal, complemented

#16863794

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, complemented

#16863646

C₂

yes

maximal, complemented

#16863508

C₂

yes

maximal, complemented

C₄×#11761611

C₂

yes

maximal, characteristic, complemented

#493×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761627

—

C₂×#11761611

C₂²

yes

—

#14524043

C₂²

yes

—

#14524030

—

#14524014

—

#14524014

C₂²

yes

—

D₄×Dic₆₇

—

D₄×Dic₆₇

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523975

C₂²

yes

—

#14523964

C₂²

yes

—

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₂²

yes

—

#127×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761627

—

C₂×Dic₁₃₄

—

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

#11761611

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂³

yes

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

C₂×D₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

#41

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#532

yes

characteristic

C₂×C₆₇

C₂×AGL(1,ℤ/8)

yes

—

D₆₇

—

#3184

—

maximal, Sylow

C₆₇

#3184

yes

characteristic, complemented, Sylow

#532

—

#498

—

#493

D₆₇

yes

characteristic, complemented

#486

—

#469

—

#442

—

C₄×M₁₆

—

C₂×#41

—

134

—

C₂×D₈

C₂×D₆₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

C₂×D₆₇

yes

—

#124

—

134

—

#121

—

#121

—

134

—

C₄×D₄

—

C₄×D₄

—

134

—

C₂×C₄²

—

#106

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₆₇

yes

characteristic

101

#41

—

268

—

102

C₂×Q₈

—

134

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

134

—

106

C₂×D₄

C₂²×D₆₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

134

—

110

D₈

—

111

M₁₆

—

134

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₆₇

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

134

—

116

C₂²⋊C₄

—

134

—

117

C₂²⋊C₄

—

134

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×D₆₇

yes

characteristic

121

C₂³

—

122

C₂³

—

123

Q₈

—

268

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

268

—

128

C₂×C₄

—

268

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

134

—

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

#11761627

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

134

—

142

C₂²

—

134

—

143

C₂²

C₂×D₄×D₆₇

yes

characteristic

144

C₄

—

268

—

145

C₄

—

134

—

146

C₄

—

134

—

147

C₄

#14524067

yes

—

148

C₄

C₂×#11761627

yes

—

149

C₂

—

150

C₂

—

268

—

151

C₂

#16863944

yes

characteristic

152

C₂

C₂×#14524067

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#493×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#493	D₆₇
hypocenter	C₆₇	#3184
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#493×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16863944	C₂	semidirect
2. #16863794	C₂	semidirect
3. #16863742	C₂	semidirect
4. #16863646	C₂	semidirect
5. #16863508	C₂	semidirect
6. C₄×#11761611	C₂	semidirect
7. #493×C₆₇	C₂	semidirect
59. C₈×C₆₇	#41	semidirect
77. C₆₇	#3184	semidirect
80. #493	D₆₇	semidirect
137. C₈	#11761627	semidirect

H ≅ G/N

type

1. #16863944

C₂

semidirect

2. #16863794

C₂

semidirect

3. #16863742

C₂

semidirect

4. #16863646

C₂

semidirect

5. #16863508

C₂

semidirect

6. C₄×#11761611

C₂

semidirect

7. #493×C₆₇

C₂

semidirect

59. C₈×C₆₇

#41

semidirect

77. C₆₇

#3184

semidirect

80. #493

D₆₇

semidirect

137. C₈

#11761627

semidirect

Subgroups — #22570945

Conjugacy classes of subgroups

Complements