Group #22573307 — subgroups

Subgroups — C₂×#16863812 #22573307

Order 8576.

368 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#14524049	C₂	1	1	yes	maximal, characteristic
2	#16863812	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523982	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×D₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11761628	C₄	1	1	yes	characteristic
8	#14524049	C₄	1	4	yes	—
9	#14524049	C₂²	1	4	yes	—
10	C₂×#11761612	C₄	1	1	yes	characteristic
11	C₂×#11761611	C₄	1	2	yes	—
12	C₂×C₈×D₆₇	C₂²	1	2	yes	—
13	#14523983	C₂²	1	4	yes	complemented
14	#14523982	C₂²	1	8	yes	complemented
15	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	C₁₆×D₆₇	C₂²	1	8	yes	complemented
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₆₇	C₂²	1	4	yes	complemented
21	#11761628	C₈	1	4	yes	—
22	#11761628	C₂×C₄	1	4	yes	—
23	C₂×Dic₁₃₄	C₈	1	1	yes	characteristic
24	C₂×D₂₆₈	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊D₆₇	C₈	1	2	yes	—
26	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
27	#11761612	C₂×C₄	1	4	yes	—
28	#11761611	C₂×C₄	1	8	yes	—
29	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₈×D₆₇	C₂×C₄	1	4	yes	—
31	C₈×D₆₇	C₂³	1	4	yes	—
32	C₆₇⋊C₁₆	C₂³	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂³	1	2	yes	—
38	C₁₆×C₆₇	C₂³	1	2	yes	—
39	C₁₆×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₆₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
46	C₂²⋊D₆₇	C₂×C₈	1	8	yes	—
47	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
48	D₂₆₈	C₂×C₈	1	4	yes	—
49	C₄×D₆₇	C₂×C₈	1	4	yes	—
50	C₄×D₆₇	C₂²×C₄	1	4	yes	—
51	Dic₁₃₄	C₂×C₈	1	4	yes	—
52	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₆₇	C₂²×C₄	1	2	yes	—
54	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
55	C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
56	C₂²×C₆₇	—	2	2	no	—
57	C₂²×C₆₇	C₂²×C₈	1	2	yes	—
58	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₆₇	—	2	4	no	—
60	C₂×D₆₇	C₂²×C₈	1	4	yes	—
61	C₄×C₆₇	C₂²×C₈	1	2	yes	—
62	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
64	Dic₆₇	C₂²×C₈	1	4	yes	—
65	C₂×C₆₇	—	2	2	no	—
66	C₂×C₆₇	#504	1	2	yes	complemented
67	C₂×C₆₇	C₂³×C₈	1	1	yes	characteristic
68	D₆₇	—	2	4	no	—
69	C₂×#504	—	67	1	no	maximal, Sylow
70	C₆₇	C₂×#504	1	1	yes	characteristic, complemented, Sylow
71	C₂×#131	—	67	1	no	—
72	#504	—	67	8	no	—
73	C₂×M₃₂	—	67	1	no	—
74	C₂×M₃₂	—	67	2	no	—
75	C₂²×C₁₆	—	67	2	no	—
76	C₂²×C₁₆	D₆₇	1	1	yes	characteristic, complemented
77	C₂×#41	—	67	1	no	—
78	#131	—	67	4	no	—
79	#131	—	67	4	no	—
80	C₂×M₁₆	—	67	1	no	—
81	C₂×M₁₆	—	67	2	no	—
82	C₂²×C₈	—	67	2	no	—
83	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
84	M₃₂	—	67	4	no	—
85	M₃₂	—	67	8	no	—
86	C₂×C₁₆	—	67	2	no	—
87	C₂×C₁₆	—	67	8	no	—
88	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
89	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
90	C₂×C₁₆	C₂×D₆₇	1	4	yes	complemented
91	#41	—	67	4	no	—
92	#41	—	67	4	no	—
93	C₂×Q₈	—	67	1	no	—
94	C₂×D₄	—	67	2	no	—
95	C₂×D₄	—	67	1	no	—
96	C₂²×C₄	—	67	2	no	—
97	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
98	M₁₆	—	67	4	no	—
99	M₁₆	—	67	8	no	—
100	C₂×C₈	—	67	2	no	—
101	C₂×C₈	—	67	4	no	—
102	C₂×C₈	—	67	4	no	—
103	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
104	C₂×C₈	C₄×D₆₇	1	2	yes	—
105	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
106	C₂×C₈	C₂²×D₆₇	1	2	yes	—
107	C₁₆	—	67	4	no	—
108	C₁₆	C₂²×D₆₇	1	2	yes	—
109	C₁₆	C₂²×D₆₇	1	2	yes	—
110	C₂³	—	67	2	no	—
111	C₂³	C₈×D₆₇	1	1	yes	characteristic
112	Q₈	—	67	4	no	—
113	D₄	—	67	8	no	—
114	D₄	—	67	4	no	—
115	C₂×C₄	—	67	2	no	—
116	C₂×C₄	—	67	4	no	—
117	C₂×C₄	—	67	4	no	—
118	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
119	C₂×C₄	C₈×D₆₇	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
122	C₈	—	67	4	no	—
123	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
124	C₈	C₂×C₄×D₆₇	1	2	yes	—
125	C₈	C₂³×D₆₇	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	—	67	4	no	—
128	C₂²	—	134	4	no	—
129	C₂²	C₂×C₈×D₆₇	1	2	yes	—
130	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
131	C₄	—	67	4	no	—
132	C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
133	C₄	C₂×C₈×D₆₇	1	2	yes	—
134	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	134	4	no	—
137	C₂	#16863812	1	2	yes	direct product factor
138	C₂	C₂²×C₈×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524049

C₂

yes

maximal, characteristic

#16863812

C₂

yes

maximal, direct product factor

C₂×#14523983

C₂

yes

maximal, characteristic, complemented

C₂×#14523982

C₂

yes

maximal, complemented

C₂×C₁₆×D₆₇

C₂

yes

maximal, complemented

C₂²×C₁₆×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₄

yes

characteristic

#14524049

C₄

yes

—

#14524049

C₂²

yes

—

C₂×#11761612

C₄

yes

characteristic

C₂×#11761611

C₄

yes

—

C₂×C₈×D₆₇

C₂²

yes

—

#14523983

C₂²

yes

complemented

#14523982

C₂²

yes

complemented

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₁₆×D₆₇

C₂²

yes

complemented

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

complemented

#11761628

C₈

yes

—

#11761628

C₂×C₄

yes

—

C₂×Dic₁₃₄

C₈

yes

characteristic

C₂×D₂₆₈

C₈

yes

characteristic

C₂×C₂²⋊D₆₇

C₈

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

#11761612

C₂×C₄

yes

—

#11761611

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₈×D₆₇

C₂×C₄

yes

—

C₈×D₆₇

C₂³

yes

—

C₆₇⋊C₁₆

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₁₆×C₆₇

C₂³

yes

—

C₁₆×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂²×D₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂²⋊D₆₇

C₂×C₈

yes

—

C₂×Dic₆₇

C₂×C₈

yes

—

D₂₆₈

C₂×C₈

yes

—

C₄×D₆₇

C₂×C₈

yes

—

C₄×D₆₇

C₂²×C₄

yes

—

Dic₁₃₄

C₂×C₈

yes

—

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂²×C₄

yes

—

C₈×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

C₂³×C₄

yes

characteristic

Dic₆₇

C₂²×C₈

yes

—

C₂×C₆₇

—

C₂×C₆₇

#504

yes

complemented

C₂×C₆₇

C₂³×C₈

yes

characteristic

D₆₇

—

C₂×#504

—

maximal, Sylow

C₆₇

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₆₇

yes

characteristic, complemented

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×C₁₆

C₂×D₆₇

yes

complemented

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄×D₆₇

yes

characteristic

104

C₂×C₈

C₄×D₆₇

yes

—

105

C₂×C₈

C₂²×D₆₇

yes

characteristic

106

C₂×C₈

C₂²×D₆₇

yes

—

107

C₁₆

—

108

C₁₆

C₂²×D₆₇

yes

—

109

C₁₆

C₂²×D₆₇

yes

—

110

C₂³

—

111

C₂³

C₈×D₆₇

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₈×D₆₇

yes

characteristic

119

C₂×C₄

C₈×D₆₇

yes

—

120

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

121

C₂×C₄

C₂×C₄×D₆₇

yes

—

122

C₈

—

123

C₈

C₂×C₄×D₆₇

yes

characteristic

124

C₈

C₂×C₄×D₆₇

yes

—

125

C₈

C₂³×D₆₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

134

—

129

C₂²

C₂×C₈×D₆₇

yes

—

130

C₂²

C₂×C₈×D₆₇

yes

characteristic

131

C₄

—

132

C₄

C₂×C₈×D₆₇

yes

characteristic

133

C₄

C₂×C₈×D₆₇

yes

—

134

C₄

C₂²×C₄×D₆₇

yes

characteristic

135

C₂

—

136

C₂

—

134

—

137

C₂

#16863812

yes

direct product factor

138

C₂

C₂²×C₈×D₆₇

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₆₇
commutator	C₂×C₆₇	C₂³×C₈
Frattini	C₈	C₂³×D₆₇
Fitting	C₂²×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₈
hypercenter	C₂²×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#504
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₆₇

commutator

C₂×C₆₇

C₂³×C₈

Frattini

C₈

C₂³×D₆₇

Fitting

C₂²×C₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×C₈

hypercenter

C₂²×C₁₆

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #16863812	C₂	direct (N × H)
3. C₂×#14523983	C₂	semidirect
4. C₂×#14523982	C₂	semidirect
5. C₂×C₁₆×D₆₇	C₂	semidirect
6. C₂²×C₁₆×C₆₇	C₂	semidirect
13. #14523983	C₂²	semidirect
14. #14523982	C₂²	semidirect
16. C₁₆×D₆₇	C₂²	semidirect
20. C₂×C₁₆×C₆₇	C₂²	semidirect
66. C₂×C₆₇	#504	semidirect
70. C₆₇	C₂×#504	semidirect
76. C₂²×C₁₆	D₆₇	semidirect
90. C₂×C₁₆	C₂×D₆₇	semidirect
137. C₂	#16863812	direct (N × H)

H ≅ G/N

type

2. #16863812

C₂

direct (N × H)

3. C₂×#14523983

C₂

semidirect

4. C₂×#14523982

C₂

semidirect

5. C₂×C₁₆×D₆₇

C₂

semidirect

6. C₂²×C₁₆×C₆₇

C₂

semidirect

13. #14523983

C₂²

semidirect

14. #14523982

C₂²

semidirect

16. C₁₆×D₆₇

C₂²

semidirect

20. C₂×C₁₆×C₆₇

C₂²

semidirect

66. C₂×C₆₇

#504

semidirect

70. C₆₇

C₂×#504

semidirect

76. C₂²×C₁₆

D₆₇

semidirect

90. C₂×C₁₆

C₂×D₆₇

semidirect

137. C₂

#16863812

direct (N × H)

Subgroups — C₂×#16863812 #22573307

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863812 #22573307

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863812 #22573307