Group #22569078 — subgroups

Subgroups — #22569078

Order 8576.

456 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#16863873	C₂	1	2	yes	maximal
3	#16863504	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₆₈	C₂	1	4	yes	maximal, complemented
5	#16863434	C₂	1	1	yes	maximal, characteristic
6	#16863406	C₂	1	2	yes	maximal
7	#405×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
9	C₂×Dic₂₆₈	C₄	1	8	yes	complemented
10	C₂×Dic₂₆₈	C₂²	1	2	yes	—
11	C₂×Dic₂₆₈	C₂²	1	2	yes	—
12	C₄×Dic₁₃₄	C₂²	1	8	yes	—
13	#14524008	—	2	4	no	—
14	#14524002	C₂²	1	2	yes	—
15	#14523999	—	2	4	no	—
16	#14523977	C₂²	1	2	yes	—
17	#14523975	C₂²	1	2	yes	—
18	#14523968	C₂²	1	8	yes	—
19	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#117×C₆₇	C₂²	1	1	yes	characteristic
22	#97×C₆₇	C₂²	1	2	yes	—
23	C₄×C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	—	2	4	no	—
26	C₂×Dic₁₃₄	C₂×C₄	1	8	yes	—
27	C₂×Dic₁₃₄	C₂³	1	4	yes	—
28	C₂²×Dic₆₇	—	2	2	no	—
29	Dic₂₆₈	C₂×C₄	1	8	yes	—
30	Dic₂₆₈	C₂×C₄	1	8	yes	—
31	C₄⋊Dic₆₇	C₂³	1	4	yes	—
32	Dic₆₇⋊C₄	—	2	8	no	—
33	C₂²⋊Dic₆₇	—	2	2	no	—
34	C₄×Dic₆₇	—	2	8	no	—
35	C₄×Dic₆₇	—	2	2	no	—
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
43	C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
44	C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	2	yes	—
50	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₆₇	—	2	8	no	—
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	2	4	no	—
54	Dic₁₃₄	—	2	8	no	—
55	Dic₁₃₄	—	2	4	no	—
56	Dic₁₃₄	C₂²×C₄	1	8	yes	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	C₂²×C₄	1	2	yes	—
59	C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₆₇	—	2	2	no	—
61	C₂²×C₆₇	C₄×D₄	1	2	yes	—
62	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₆₇	—	2	4	no	—
64	C₄×C₆₇	C₄×D₄	1	2	yes	—
65	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
67	Dic₆₇	—	2	8	no	—
68	Dic₆₇	—	4	4	no	—
69	C₂×C₆₇	—	2	2	no	—
70	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₆₇	#578	1	2	yes	—
72	#2950	—	67	1	no	maximal, Sylow
73	C₆₇	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	—	67	1	no	—
75	#519	—	67	2	no	—
76	#441	—	67	4	no	—
77	C₄×Q₁₆	—	67	4	no	—
78	#427	—	67	1	no	—
79	#419	—	67	2	no	—
80	#405	D₆₇	1	1	yes	characteristic
81	C₂²×Q₈	—	67	2	no	—
82	C₂×Q₁₆	—	67	8	no	—
83	C₂×Q₁₆	—	67	2	no	—
84	C₂×Q₁₆	—	67	2	no	—
85	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄×Q₈	—	67	8	no	—
88	C₄×Q₈	—	134	4	no	—
89	#117	—	67	2	no	—
90	#117	—	134	4	no	—
91	#117	C₂×D₆₇	1	1	yes	characteristic
92	#107	—	67	2	no	—
93	#106	—	67	2	no	—
94	#103	—	67	8	no	—
95	#97	C₂×D₆₇	1	2	yes	—
96	C₄×C₈	C₂×D₆₇	1	2	yes	—
97	C₂×Q₈	—	67	8	no	—
98	C₂×Q₈	—	67	4	no	—
99	C₂×Q₈	—	134	2	no	—
100	C₂×Q₈	—	134	4	no	—
101	C₂²×C₄	—	134	2	no	—
102	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
103	Q₁₆	—	67	8	no	—
104	Q₁₆	—	67	8	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₄×D₆₇	1	4	yes	—
107	C₂×C₈	C₂²×D₆₇	1	2	yes	—
108	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
110	C₄⋊C₄	—	67	4	no	—
111	C₄⋊C₄	—	134	8	no	—
112	C₄⋊C₄	D₂₆₈	1	2	yes	—
113	C₂²⋊C₄	—	134	2	no	—
114	C₂²⋊C₄	D₂₆₈	1	2	yes	—
115	C₄²	—	134	2	no	—
116	C₄²	—	134	8	no	—
117	C₄²	C₂²×D₆₇	1	2	yes	—
118	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
119	Q₈	—	67	8	no	—
120	Q₈	—	134	8	no	—
121	Q₈	—	134	4	no	—
122	C₂×C₄	—	134	4	no	—
123	C₂×C₄	—	134	8	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
126	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
127	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
128	C₂×C₄	#11761628	1	2	yes	—
129	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂×C₄×D₆₇	1	2	yes	—
132	C₈	C₂×C₄×D₆₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₂₆₈	1	2	yes	—
135	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	268	4	no	—
138	C₄	—	134	8	no	—
139	C₄	C₄×D₂₆₈	1	2	yes	—
140	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
141	C₄	C₂×#11761628	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₂₆₈	1	1	yes	characteristic
144	C₂	#16864213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

#16863873

C₂

yes

maximal

#16863504

C₂

yes

maximal, complemented

C₄×Dic₂₆₈

C₂

yes

maximal, complemented

#16863434

C₂

yes

maximal, characteristic

#16863406

C₂

yes

maximal

#405×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

C₄

yes

complemented

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524008

—

#14524002

C₂²

yes

—

#14523999

—

#14523977

C₂²

yes

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

C₂×C₄

yes

—

Dic₂₆₈

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂²×C₄

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₄

yes

—

C₈×C₆₇

C₂²×C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂×C₆₇

#578

yes

—

#2950

—

maximal, Sylow

C₆₇

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

#117

—

134

—

#117

C₂×D₆₇

yes

—

—

—

C₂×D₆₇

yes

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

100

C₂×Q₈

—

134

—

101

C₂²×C₄

—

134

—

102

C₂²×C₄

C₂²×D₆₇

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

C₂×C₈

C₄×D₆₇

yes

—

107

C₂×C₈

C₂²×D₆₇

yes

—

108

C₂×C₈

C₂²×D₆₇

yes

characteristic

109

C₂×C₈

C₂²×D₆₇

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

D₂₆₈

yes

—

113

C₂²⋊C₄

—

134

—

114

C₂²⋊C₄

D₂₆₈

yes

—

115

C₄²

—

134

—

116

C₄²

—

134

—

117

C₄²

C₂²×D₆₇

yes

—

118

C₂³

C₂×D₂₆₈

yes

characteristic

119

Q₈

—

120

Q₈

—

134

—

121

Q₈

—

134

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

C₂×C₄×D₆₇

yes

—

126

C₂×C₄

C₂×D₂₆₈

yes

—

127

C₂×C₄

C₂×D₂₆₈

yes

characteristic

128

C₂×C₄

#11761628

yes

—

129

C₂×C₄

C₂³×D₆₇

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₆₇

yes

—

132

C₈

C₂×C₄×D₆₇

yes

—

133

C₂²

—

134

C₂²

C₄×D₂₆₈

yes

—

135

C₂²

C₂²×D₂₆₈

yes

characteristic

136

C₄

—

137

C₄

—

268

—

138

C₄

—

134

—

139

C₄

C₄×D₂₆₈

yes

—

140

C₄

C₂²×C₄×D₆₇

yes

characteristic

141

C₄

C₂×#11761628

yes

characteristic

142

C₂

—

143

C₂

C₂×C₄×D₂₆₈

yes

characteristic

144

C₂

#16864213

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#405	D₆₇
hypocenter	C₆₇	#2950
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₄×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#405×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
3. #16863504	C₂	semidirect
4. C₄×Dic₂₆₈	C₂	semidirect
9. C₂×Dic₂₆₈	C₄	semidirect
73. C₆₇	#2950	semidirect

H ≅ G/N

type

3. #16863504

C₂

semidirect

4. C₄×Dic₂₆₈

C₂

semidirect

9. C₂×Dic₂₆₈

C₄

semidirect

73. C₆₇

#2950

semidirect

Subgroups — #22569078

Conjugacy classes of subgroups

Complements