Group #22569129 — subgroups

Subgroups — C₂×#16863520 #22569129

Order 8576.

440 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 253 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×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863520	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₆₇⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₆₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₄	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₃₄	C₄	1	4	yes	—
11	C₄×Dic₁₃₄	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
15	Dic₆₇⋊C₈	C₂²	1	8	yes	—
16	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
17	C₈×Dic₆₇	C₂²	1	8	yes	—
18	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
21	C₂×Dic₁₃₄	C₈	1	8	yes	complemented
22	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	—
23	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₆₇⋊C₈	—	2	2	no	—
26	C₂×C₆₇⋊C₈	Q₈	1	4	yes	complemented
27	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
30	Dic₆₇⋊C₄	C₂×C₄	1	8	yes	—
31	C₄×Dic₆₇	C₂×C₄	1	4	yes	—
32	C₄×Dic₆₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	4	no	—
37	C₂×C₈×C₆₇	C₂³	1	4	yes	—
38	C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₆₇	#41	1	2	yes	—
48	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₆₇	—	2	4	no	—
50	C₂×Dic₆₇	C₂×C₈	1	8	yes	—
51	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
53	Dic₁₃₄	C₂×C₈	1	16	yes	complemented
54	C₈×C₆₇	—	2	4	no	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₆₇⋊C₈	C₂×Q₈	1	4	yes	complemented
57	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₆₇	C₂²×C₈	1	2	yes	—
59	C₂²×C₆₇	#131	1	2	yes	—
60	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₆₇	—	2	2	no	—
62	C₄×C₆₇	C₄×Q₈	1	2	yes	—
63	C₄×C₆₇	C₂²×C₈	1	4	yes	—
64	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
66	Dic₆₇	—	2	4	no	—
67	Dic₆₇	C₂²×C₈	1	8	yes	—
68	C₂×C₆₇	C₈×Q₈	1	4	yes	complemented
69	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₆₇	C₂³×C₈	1	1	yes	characteristic
71	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
72	C₂×C₈×Q₈	—	67	1	no	maximal, Sylow
73	C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	67	1	no	—
75	C₈×Q₈	—	67	8	no	—
76	C₂×C₄⋊C₈	—	67	2	no	—
77	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic
78	C₂×C₄×C₈	—	67	2	no	—
79	C₂×C₄×C₈	—	67	1	no	—
80	C₂²×Q₈	—	67	1	no	—
81	C₂²×C₈	—	67	1	no	—
82	C₂²×C₈	—	67	1	no	—
83	C₂²×C₈	C₂×D₆₇	1	2	yes	—
84	C₄×Q₈	—	67	4	no	—
85	C₄×Q₈	—	67	4	no	—
86	C₂×C₄⋊C₄	—	67	2	no	—
87	C₂×C₄⋊C₄	—	67	1	no	—
88	C₂×C₄²	—	67	2	no	—
89	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
90	C₄⋊C₈	—	67	8	no	—
91	C₄⋊C₈	C₂×D₆₇	1	4	yes	—
92	C₄×C₈	—	67	8	no	—
93	C₄×C₈	—	67	4	no	—
94	C₂×Q₈	—	67	8	no	—
95	C₂×Q₈	—	67	4	no	—
96	C₂²×C₄	—	67	2	no	—
97	C₂²×C₄	—	67	2	no	—
98	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	67	4	no	—
103	C₂×C₈	—	67	2	no	—
104	C₂×C₈	—	67	2	no	—
105	C₂×C₈	—	134	2	no	—
106	C₂×C₈	C₂²×D₆₇	1	4	yes	—
107	C₄⋊C₄	—	67	8	no	—
108	C₄⋊C₄	—	67	4	no	—
109	C₄²	—	67	4	no	—
110	C₄²	—	67	4	no	—
111	C₄²	C₄×D₆₇	1	2	yes	—
112	C₄²	C₂²×D₆₇	1	2	yes	—
113	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
114	Q₈	—	67	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	67	8	no	—
117	C₂×C₄	—	67	4	no	—
118	C₂×C₄	—	67	4	no	—
119	C₂×C₄	—	134	4	no	—
120	C₂×C₄	C₈×D₆₇	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
124	C₂×C₄	Q₈×D₆₇	1	2	yes	—
125	C₂×C₄	#11761627	1	2	yes	—
126	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	67	4	no	—
129	C₈	—	134	2	no	—
130	C₂²	#14524008	1	2	yes	—
131	C₂²	C₂×C₈×D₆₇	1	2	yes	—
132	C₂²	#14524051	1	2	yes	—
133	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	67	8	no	—
136	C₄	—	134	4	no	—
137	C₄	#14524008	1	2	yes	—
138	C₄	C₂×C₈×D₆₇	1	4	yes	—
139	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
140	C₄	C₂×#11761627	1	1	yes	characteristic
141	C₂	#16863520	1	4	yes	direct product factor
142	C₂	C₂×#14524008	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₆₇	1	1	yes	characteristic
144	C₂	C₂×#14524051	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863520

C₂

yes

maximal, direct product factor

C₂×Dic₆₇⋊C₈

C₂

yes

maximal

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₄

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₄

yes

characteristic

C₂×Dic₆₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₈

yes

complemented

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

Dic₁₃₄

C₂×C₈

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₆₇

C₄×Q₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

#131

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

C₂²×C₈

yes

—

C₂×C₆₇

C₈×Q₈

yes

complemented

C₂×C₆₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₇

C₂³×C₈

yes

characteristic

C₂×C₆₇

C₂×#131

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₆₇

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₆₇

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

100

C₂²×C₄

C₂²×D₆₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

C₂²×D₆₇

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₆₇

yes

—

112

C₄²

C₂²×D₆₇

yes

—

113

C₂³

C₂×C₄×D₆₇

yes

characteristic

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

134

—

120

C₂×C₄

C₈×D₆₇

yes

—

121

C₂×C₄

C₂×C₄×D₆₇

yes

—

122

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

123

C₂×C₄

C₂×C₄×D₆₇

yes

—

124

C₂×C₄

Q₈×D₆₇

yes

—

125

C₂×C₄

#11761627

yes

—

126

C₂×C₄

C₂³×D₆₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

134

—

130

C₂²

#14524008

yes

—

131

C₂²

C₂×C₈×D₆₇

yes

—

132

C₂²

#14524051

yes

—

133

C₂²

C₂²×C₄×D₆₇

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

134

—

137

C₄

#14524008

yes

—

138

C₄

C₂×C₈×D₆₇

yes

—

139

C₄

C₂×Q₈×D₆₇

yes

characteristic

140

C₄

C₂×#11761627

yes

characteristic

141

C₂

#16863520

yes

direct product factor

142

C₂

C₂×#14524008

yes

characteristic

143

C₂

C₂²×C₈×D₆₇

yes

characteristic

144

C₂

C₂×#14524051

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	C₂×C₈×Q₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₆₇

commutator

C₂×C₆₇

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₆₇

hypocenter

C₆₇

C₂×C₈×Q₈

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #16863520	C₂	direct (N × H)
21. C₂×Dic₁₃₄	C₈	semidirect
26. C₂×C₆₇⋊C₈	Q₈	semidirect
53. Dic₁₃₄	C₂×C₈	semidirect
56. C₆₇⋊C₈	C₂×Q₈	semidirect
68. C₂×C₆₇	C₈×Q₈	semidirect
73. C₆₇	C₂×C₈×Q₈	semidirect
141. C₂	#16863520	direct (N × H)

H ≅ G/N

type

2. #16863520

C₂

direct (N × H)

21. C₂×Dic₁₃₄

C₈

semidirect

26. C₂×C₆₇⋊C₈

Q₈

semidirect

53. Dic₁₃₄

C₂×C₈

semidirect

56. C₆₇⋊C₈

C₂×Q₈

semidirect

68. C₂×C₆₇

C₈×Q₈

semidirect

73. C₆₇

C₂×C₈×Q₈

semidirect

141. C₂

#16863520

direct (N × H)

Subgroups — C₂×#16863520 #22569129

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863520 #22569129

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863520 #22569129