Group #22569971 — subgroups

Subgroups — C₂×#16863678 #22569971

Order 8576.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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	#16863678	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523974	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523966	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₁₃₄	—	2	2	no	—
10	C₄×Dic₁₃₄	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₆₇	—	2	1	no	—
14	#14523974	C₂²	1	8	yes	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
16	#14523966	C₂²	1	8	yes	—
17	C₂²×Q₈×C₆₇	—	2	1	no	—
18	C₄⋊Q₈×C₆₇	—	2	2	no	—
19	C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	Q₈	1	4	yes	complemented
26	C₂×Dic₁₃₄	C₂³	1	2	yes	—
27	C₂²×Dic₆₇	—	2	1	no	—
28	C₂²×Dic₆₇	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	4	no	—
30	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	C₂³	1	2	yes	—
33	Dic₆₇⋊C₄	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₂×Q₈×C₆₇	—	2	4	no	—
36	C₂×Q₈×C₆₇	—	4	4	no	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₄²×C₆₇	D₄	1	2	yes	—
45	C₄²×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₆₇	—	4	8	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	—	4	4	no	—
51	C₂×C₄×C₆₇	SD₁₆	1	4	yes	—
52	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₆₇	#41	1	2	yes	—
57	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	4	no	—
59	C₂×Dic₆₇	—	2	2	no	—
60	C₂×Dic₆₇	—	2	2	no	—
61	C₂×Dic₆₇	—	4	2	no	—
62	Dic₁₃₄	—	2	4	no	—
63	Dic₁₃₄	—	2	2	no	—
64	Dic₁₃₄	C₂×Q₈	1	4	yes	complemented
65	C₆₇⋊C₈	—	2	4	no	—
66	C₂²×C₆₇	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₆₇	#137	1	2	yes	—
69	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	4	4	no	—
72	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
74	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
76	Dic₆₇	—	2	4	no	—
77	Dic₆₇	—	4	2	no	—
78	C₂×C₆₇	#475	1	4	yes	complemented
79	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
81	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
82	C₂×#475	—	67	1	no	maximal, Sylow
83	C₆₇	C₂×#475	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
85	C₂×C₄×Q₈	—	67	1	no	—
86	#475	—	67	8	no	—
87	C₂×#106	—	67	2	no	—
88	C₂×C₄⋊C₈	—	67	1	no	—
89	C₂×#103	—	67	2	no	—
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	67	1	no	—
92	C₂²×C₈	—	67	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂×D₆₇	1	4	yes	—
95	C₄×Q₈	—	67	4	no	—
96	C₄×Q₈	—	134	2	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	67	1	no	—
99	C₂×C₄⋊C₄	—	134	1	no	—
100	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
101	C₂×C₄²	—	134	1	no	—
102	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
103	#106	—	67	8	no	—
104	C₄⋊C₈	—	67	4	no	—
105	#103	—	67	8	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	67	4	no	—
109	C₂×Q₈	—	67	2	no	—
110	C₂×Q₈	—	134	1	no	—
111	C₂×Q₈	—	134	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	134	1	no	—
114	C₂²×C₄	—	134	1	no	—
115	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
118	C₂×C₈	—	67	4	no	—
119	C₂×C₈	—	134	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	67	2	no	—
123	C₄⋊C₄	—	134	1	no	—
124	C₄⋊C₄	—	134	4	no	—
125	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
126	C₄²	—	134	4	no	—
127	C₄²	C₂²⋊D₆₇	1	2	yes	—
128	C₄²	C₂²×D₆₇	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	67	4	no	—
132	Q₈	—	134	4	no	—
133	Q₈	—	134	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	268	2	no	—
138	C₂×C₄	—	134	2	no	—
139	C₂×C₄	—	134	4	no	—
140	C₂×C₄	—	134	2	no	—
141	C₂×C₄	#11761616	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
145	C₂×C₄	Q₈×D₆₇	1	2	yes	—
146	C₂×C₄	#11761629	1	2	yes	—
147	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
148	C₈	—	134	4	no	—
149	C₂²	#14524019	1	2	yes	—
150	C₂²	C₂×#11761616	1	2	yes	—
151	C₂²	#14524074	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	4	4	no	—
155	C₄	—	268	2	no	—
156	C₄	—	134	4	no	—
157	C₄	#14524019	1	2	yes	—
158	C₄	C₂×#11761616	1	4	yes	—
159	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
160	C₄	C₂×#11761629	1	1	yes	characteristic
161	C₂	#16863678	1	4	yes	direct product factor
162	C₂	C₂×#14524019	1	1	yes	characteristic
163	C₂	C₂²×#11761616	1	1	yes	characteristic
164	C₂	C₂×#14524074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863678

C₂

yes

maximal, direct product factor

C₂×#14523974

C₂

yes

maximal

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14523966

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523966

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₈

yes

complemented

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂×Q₈

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#475

yes

complemented

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

characteristic

C₂×C₄×Q₈

—

#475

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

100

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

101

C₂×C₄²

—

134

—

102

C₂×C₄²

C₂×D₆₇

yes

characteristic

103

#106

—

104

C₄⋊C₈

—

105

#103

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

134

—

111

C₂×Q₈

—

134

—

112

C₂²×C₄

—

113

C₂²×C₄

—

134

—

114

C₂²×C₄

—

134

—

115

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

116

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

117

C₂²×C₄

C₂²×D₆₇

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

134

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

134

—

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

C₂²×D₆₇

yes

—

126

C₄²

—

134

—

127

C₄²

C₂²⋊D₆₇

yes

—

128

C₄²

C₂²×D₆₇

yes

—

129

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

134

—

133

Q₈

—

134

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

268

—

138

C₂×C₄

—

134

—

139

C₂×C₄

—

134

—

140

C₂×C₄

—

134

—

141

C₂×C₄

#11761616

yes

—

142

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

145

C₂×C₄

Q₈×D₆₇

yes

—

146

C₂×C₄

#11761629

yes

—

147

C₂×C₄

C₂³×D₆₇

yes

characteristic

148

C₈

—

134

—

149

C₂²

#14524019

yes

—

150

C₂²

C₂×#11761616

yes

—

151

C₂²

#14524074

yes

—

152

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

268

—

156

C₄

—

134

—

157

C₄

#14524019

yes

—

158

C₄

C₂×#11761616

yes

—

159

C₄

C₂×Q₈×D₆₇

yes

characteristic

160

C₄

C₂×#11761629

yes

characteristic

161

C₂

#16863678

yes

direct product factor

162

C₂

C₂×#14524019

yes

characteristic

163

C₂

C₂²×#11761616

yes

characteristic

164

C₂

C₂×#14524074

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	C₂×#475
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

C₂×#475

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16863678	C₂	direct (N × H)
25. C₂×Dic₁₃₄	Q₈	semidirect
64. Dic₁₃₄	C₂×Q₈	semidirect
78. C₂×C₆₇	#475	semidirect
83. C₆₇	C₂×#475	semidirect
161. C₂	#16863678	direct (N × H)

H ≅ G/N

type

2. #16863678

C₂

direct (N × H)

25. C₂×Dic₁₃₄

Q₈

semidirect

64. Dic₁₃₄

C₂×Q₈

semidirect

78. C₂×C₆₇

#475

semidirect

83. C₆₇

C₂×#475

semidirect

161. C₂

#16863678

direct (N × H)

Subgroups — C₂×#16863678 #22569971

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863678 #22569971

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863678 #22569971