Group #22570623 — subgroups

Subgroups — C₂×#16863774 #22570623

Order 8576.

552 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×#11761618	C₂	1	2	yes	maximal
3	C₂×#14524041	C₂	1	2	yes	maximal
4	#16863774	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
8	C₂×Dic₂₆₈	—	2	4	no	—
9	C₂×Dic₂₆₈	C₂²	1	4	yes	—
10	C₂×#11761618	—	2	8	no	—
11	C₂×#11761618	C₂²	1	8	yes	—
12	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#14524041	—	2	4	no	—
14	#14524041	C₂²	1	8	yes	—
15	C₂×Dic₆₇⋊C₄	—	2	4	no	—
16	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₈×Dic₆₇	C₂²	1	4	yes	—
18	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₆₇	—	2	4	no	—
20	C₂×Q₁₆×C₆₇	C₂²	1	4	yes	—
21	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₃₄	—	2	4	no	—
23	C₂×Dic₁₃₄	—	2	4	no	—
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	C₂³	1	4	yes	—
26	C₂²×Dic₆₇	—	2	2	no	—
27	C₂²×Dic₆₇	D₄	1	2	yes	—
28	Dic₂₆₈	—	2	8	no	—
29	#11761618	—	2	16	no	—
30	C₂×C₆₇⋊C₈	D₄	1	4	yes	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	Dic₆₇⋊C₄	—	2	16	no	—
33	C₄×Dic₆₇	D₄	1	2	yes	—
34	C₄×Dic₆₇	C₂³	1	2	yes	—
35	C₂×Q₈×C₆₇	—	2	2	no	—
36	C₂×Q₈×C₆₇	—	2	4	no	—
37	C₂×Q₈×C₆₇	—	2	4	no	—
38	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	Q₁₆×C₆₇	—	2	8	no	—
42	C₂×C₈×C₆₇	D₄	1	4	yes	—
43	C₂×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₆₇	—	2	4	no	—
46	Q₈×C₆₇	—	2	4	no	—
47	Q₈×C₆₇	—	4	4	no	—
48	C₂×C₄×C₆₇	—	2	4	no	—
49	C₂×C₄×C₆₇	—	4	4	no	—
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₆₇	—	2	4	no	—
55	C₂×Dic₆₇	—	4	4	no	—
56	C₂×Dic₆₇	Q₁₆	1	8	yes	complemented
57	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
58	Dic₁₃₄	—	2	4	no	—
59	Dic₁₃₄	—	2	4	no	—
60	Dic₁₃₄	—	4	4	no	—
61	C₈×C₆₇	C₂×D₄	1	4	yes	—
62	C₆₇⋊C₈	C₂×D₄	1	4	yes	—
63	C₂²×C₆₇	#127	1	2	yes	—
64	C₂²×C₆₇	C₂×Q₁₆	1	4	yes	—
65	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₆₇	—	4	4	no	—
67	C₄×C₆₇	#127	1	2	yes	—
68	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₆₇	—	4	4	no	—
71	Dic₆₇	C₂×Q₁₆	1	8	yes	complemented
72	C₂×C₆₇	#494	1	4	yes	complemented
73	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
74	C₂×C₆₇	C₂²×Q₁₆	1	2	yes	—
75	C₂×#494	—	67	1	no	maximal, Sylow
76	C₆₇	C₂×#494	1	1	yes	characteristic, complemented, Sylow
77	C₂²×Q₁₆	—	67	2	no	—
78	C₂²×Q₁₆	—	67	1	no	—
79	C₂²×Q₁₆	D₆₇	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	67	2	no	—
81	#494	—	67	8	no	—
82	C₂×C₄×C₈	—	67	1	no	—
83	C₂²×Q₈	—	67	2	no	—
84	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
85	C₂×Q₁₆	—	2	4	no	—
86	C₂×Q₁₆	—	67	8	no	—
87	C₂×Q₁₆	—	67	4	no	—
88	C₂×Q₁₆	—	134	8	no	—
89	C₂×Q₁₆	—	134	4	no	—
90	C₂×Q₁₆	C₂×D₆₇	1	4	yes	—
91	C₂²×C₈	—	67	1	no	—
92	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₄⋊Q₈	—	67	8	no	—
94	C₄⋊Q₈	—	134	4	no	—
95	C₂×C₄⋊C₄	—	134	4	no	—
96	C₂×C₄²	—	67	1	no	—
97	C₄×C₈	—	67	4	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	2	4	no	—
101	C₂×Q₈	—	67	4	no	—
102	C₂×Q₈	—	134	4	no	—
103	C₂×Q₈	—	134	4	no	—
104	C₂×Q₈	—	134	2	no	—
105	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	67	2	no	—
108	C₂²×C₄	—	134	2	no	—
109	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
110	Q₁₆	—	2	8	no	—
111	Q₁₆	—	134	16	no	—
112	Q₁₆	—	134	8	no	—
113	C₂×C₈	—	67	4	no	—
114	C₂×C₈	—	67	2	no	—
115	C₂×C₈	C₂²⋊D₆₇	1	4	yes	—
116	C₂×C₈	C₂²×D₆₇	1	2	yes	—
117	C₄⋊C₄	—	134	16	no	—
118	C₄²	—	67	2	no	—
119	C₄²	—	67	2	no	—
120	C₂³	D₄×D₆₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	4	no	—
124	Q₈	—	268	4	no	—
125	Q₈	—	134	4	no	—
126	Q₈	—	134	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	268	4	no	—
130	C₂×C₄	—	67	8	no	—
131	C₂×C₄	—	67	4	no	—
132	C₂×C₄	—	134	4	no	—
133	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
134	C₂×C₄	D₄×D₆₇	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
136	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
137	C₈	—	67	4	no	—
138	C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
139	C₂²	#14524039	1	2	yes	—
140	C₂²	Q₁₆×D₆₇	1	4	yes	—
141	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
142	C₄	—	4	4	no	—
143	C₄	—	268	4	no	—
144	C₄	—	67	8	no	—
145	C₄	#14524039	1	2	yes	—
146	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
147	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂	#16863774	1	4	yes	direct product factor
149	C₂	C₂×#14524039	1	1	yes	characteristic
150	C₂	C₂×Q₁₆×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

C₂²×#11761618

C₂

yes

maximal

C₂×#14524041

C₂

yes

maximal

#16863774

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761618

—

C₂×#11761618

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524041

—

#14524041

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

Dic₂₆₈

—

#11761618

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₁₆

yes

complemented

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

—

C₆₇⋊C₈

C₂×D₄

yes

—

C₂²×C₆₇

#127

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#127

yes

—

C₄×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

C₂²×D₄

yes

characteristic

Dic₆₇

—

Dic₆₇

C₂×Q₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

C₂²×Q₁₆

yes

—

C₂×#494

—

maximal, Sylow

C₆₇

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

D₆₇

yes

characteristic

C₂×C₄⋊Q₈

—

#494

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

134

—

103

C₂×Q₈

—

134

—

104

C₂×Q₈

—

134

—

105

C₂×Q₈

C₂²×D₆₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

134

—

109

C₂²×C₄

C₂²×D₆₇

yes

characteristic

110

Q₁₆

—

111

Q₁₆

—

134

—

112

Q₁₆

—

134

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²⋊D₆₇

yes

—

116

C₂×C₈

C₂²×D₆₇

yes

—

117

C₄⋊C₄

—

134

—

118

C₄²

—

119

C₄²

—

120

C₂³

D₄×D₆₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

268

—

125

Q₈

—

134

—

126

Q₈

—

134

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

134

—

133

C₂×C₄

D₄×D₆₇

yes

characteristic

134

C₂×C₄

D₄×D₆₇

yes

—

135

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

136

C₂×C₄

C₂³×D₆₇

yes

characteristic

137

C₈

—

138

C₈

C₂×C₂²⋊D₆₇

yes

—

139

C₂²

#14524039

yes

—

140

C₂²

Q₁₆×D₆₇

yes

—

141

C₂²

C₂×D₄×D₆₇

yes

characteristic

142

C₄

—

143

C₄

—

268

—

144

C₄

—

145

C₄

#14524039

yes

—

146

C₄

C₂×D₄×D₆₇

yes

characteristic

147

C₄

C₂²×C₂²⋊D₆₇

yes

characteristic

148

C₂

#16863774

yes

direct product factor

149

C₂

C₂×#14524039

yes

characteristic

150

C₂

C₂×Q₁₆×D₆₇

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂²×Q₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂²×Q₁₆	D₆₇
hypocenter	C₆₇	C₂×#494
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂²×Q₁₆×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂²×Q₁₆

D₆₇

hypocenter

C₆₇

C₂×#494

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
4. #16863774	C₂	direct (N × H)
56. C₂×Dic₆₇	Q₁₆	semidirect
71. Dic₆₇	C₂×Q₁₆	semidirect
72. C₂×C₆₇	#494	semidirect
76. C₆₇	C₂×#494	semidirect
148. C₂	#16863774	direct (N × H)

H ≅ G/N

type

4. #16863774

C₂

direct (N × H)

56. C₂×Dic₆₇

Q₁₆

semidirect

71. Dic₆₇

C₂×Q₁₆

semidirect

72. C₂×C₆₇

#494

semidirect

76. C₆₇

C₂×#494

semidirect

148. C₂

#16863774

direct (N × H)

Subgroups — C₂×#16863774 #22570623

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863774 #22570623

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863774 #22570623