Group #22557504 — subgroups

Subgroups — C₂×#475×C₆₇ #22557504

Order 8576.

416 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 232 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⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#475×C₆₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#106×C₆₇	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#103×C₆₇	C₂	1	2	yes	maximal
7	C₂²×Q₈×C₆₇	—	2	1	no	—
8	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄⋊Q₈×C₆₇	—	2	2	no	—
11	C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₄×Q₈×C₆₇	—	2	2	no	—
13	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₇	—	2	1	no	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#106×C₆₇	C₂²	1	8	yes	—
21	C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
22	#103×C₆₇	C₂²	1	8	yes	—
23	C₂×Q₈×C₆₇	—	2	1	no	—
24	C₂×Q₈×C₆₇	—	2	4	no	—
25	C₂×Q₈×C₆₇	—	2	2	no	—
26	C₂×Q₈×C₆₇	—	4	4	no	—
27	C₂×Q₈×C₆₇	Q₈	1	4	yes	complemented
28	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₆₇	—	2	1	no	—
30	C₂²×C₄×C₆₇	—	2	1	no	—
31	C₂²×C₄×C₆₇	—	2	2	no	—
32	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	4	no	—
36	C₂×C₈×C₆₇	C₂³	1	4	yes	—
37	C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
42	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
43	C₄²×C₆₇	—	2	4	no	—
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	—	2	4	no	—
48	Q₈×C₆₇	—	2	2	no	—
49	Q₈×C₆₇	—	4	8	no	—
50	Q₈×C₆₇	C₂×Q₈	1	4	yes	complemented
51	C₂×C₄×C₆₇	—	2	4	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	4	4	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	SD₁₆	1	4	yes	—
59	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₆₇	#41	1	2	yes	—
64	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₈×C₆₇	—	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	4	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	4	4	no	—
73	C₄×C₆₇	—	4	2	no	—
74	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
76	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
78	C₂×C₆₇	#475	1	4	yes	direct product factor
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	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₆₇	C₂×#475	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
85	C₂×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
86	#475	C₂×C₆₇	1	8	yes	direct product factor
87	C₂×#106	C₂×C₆₇	1	2	yes	—
88	C₂×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
89	C₂×#103	C₂×C₆₇	1	2	yes	—
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
95	C₄×Q₈	—	2	2	no	—
96	C₄×Q₈	C₂²×C₆₇	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	2	1	no	—
99	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
101	C₂×C₄²	—	2	1	no	—
102	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
103	#106	C₂²×C₆₇	1	8	yes	—
104	C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
105	#103	C₂²×C₆₇	1	8	yes	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	4	4	no	—
110	C₂×Q₈	Q₈×C₆₇	1	4	yes	complemented
111	C₂×Q₈	C₂³×C₆₇	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
116	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂³×C₆₇	1	4	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
125	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
126	C₄²	—	2	4	no	—
127	C₄²	D₄×C₆₇	1	2	yes	—
128	C₄²	C₂³×C₆₇	1	2	yes	—
129	C₂³	C₂×D₄×C₆₇	1	1	yes	characteristic
130	Q₈	—	2	4	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	4	8	no	—
133	Q₈	C₂×Q₈×C₆₇	1	4	yes	complemented
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	4	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	SD₁₆×C₆₇	1	4	yes	—
142	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
143	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
145	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
146	C₂×C₄	#41×C₆₇	1	2	yes	—
147	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₂²	C₂²⋊Q₈×C₆₇	1	2	yes	—
150	C₂²	C₂×SD₁₆×C₆₇	1	2	yes	—
151	C₂²	#137×C₆₇	1	2	yes	—
152	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	4	2	no	—
157	C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
158	C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
159	C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
160	C₄	C₂×#41×C₆₇	1	1	yes	characteristic
161	C₂	#475×C₆₇	1	4	yes	direct product factor
162	C₂	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
163	C₂	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
164	C₂	C₂×#137×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

#475×C₆₇

C₂

yes

maximal, direct product factor

C₂×#106×C₆₇

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#103×C₆₇

C₂

yes

maximal

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

—

#103×C₆₇

C₂²

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

complemented

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

C₂×Q₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×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

C₈×C₆₇

—

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

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

C₂×C₆₇

#475

yes

direct product factor

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

C₂×#475

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₆₇

yes

characteristic

C₂×C₄×Q₈

C₂×C₆₇

yes

characteristic

#475

C₂×C₆₇

yes

direct product factor

C₂×#106

C₂×C₆₇

yes

—

C₂×C₄⋊C₈

C₂×C₆₇

yes

characteristic

C₂×#103

C₂×C₆₇

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×C₆₇

yes

characteristic

C₂²×C₈

C₂²×C₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₆₇

yes

characteristic

100

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂²×C₆₇

yes

characteristic

103

#106

C₂²×C₆₇

yes

—

104

C₄⋊C₈

C₂²×C₆₇

yes

—

105

#103

C₂²×C₆₇

yes

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

Q₈×C₆₇

yes

complemented

111

C₂×Q₈

C₂³×C₆₇

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₄×C₆₇

yes

characteristic

116

C₂²×C₄

D₄×C₆₇

yes

characteristic

117

C₂²×C₄

C₂³×C₆₇

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

C₂³×C₆₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂³×C₆₇

yes

—

125

C₄⋊C₄

C₂³×C₆₇

yes

—

126

C₄²

—

127

C₄²

D₄×C₆₇

yes

—

128

C₄²

C₂³×C₆₇

yes

—

129

C₂³

C₂×D₄×C₆₇

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

C₂×Q₈×C₆₇

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

SD₁₆×C₆₇

yes

—

142

C₂×C₄

C₂×D₄×C₆₇

yes

—

143

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

144

C₂×C₄

C₂×D₄×C₆₇

yes

—

145

C₂×C₄

C₂×Q₈×C₆₇

yes

—

146

C₂×C₄

#41×C₆₇

yes

—

147

C₂×C₄

C₂⁴×C₆₇

yes

characteristic

148

C₈

—

149

C₂²

C₂²⋊Q₈×C₆₇

yes

—

150

C₂²

C₂×SD₁₆×C₆₇

yes

—

151

C₂²

#137×C₆₇

yes

—

152

C₂²

C₂²×D₄×C₆₇

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂²⋊Q₈×C₆₇

yes

—

158

C₄

C₂×SD₁₆×C₆₇

yes

—

159

C₄

C₂²×Q₈×C₆₇

yes

characteristic

160

C₄

C₂×#41×C₆₇

yes

characteristic

161

C₂

#475×C₆₇

yes

direct product factor

162

C₂

C₂×C₂²⋊Q₈×C₆₇

yes

characteristic

163

C₂

C₂²×SD₁₆×C₆₇

yes

characteristic

164

C₂

C₂×#137×C₆₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₆₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
3. #475×C₆₇	C₂	direct (N × H)
27. C₂×Q₈×C₆₇	Q₈	semidirect
50. Q₈×C₆₇	C₂×Q₈	semidirect
78. C₂×C₆₇	#475	direct (N × H)
82. C₂×#475	C₆₇	direct (N × H)
83. C₆₇	C₂×#475	direct (N × H)
86. #475	C₂×C₆₇	direct (N × H)
110. C₂×Q₈	Q₈×C₆₇	semidirect
133. Q₈	C₂×Q₈×C₆₇	semidirect
161. C₂	#475×C₆₇	direct (N × H)

H ≅ G/N

type

3. #475×C₆₇

C₂

direct (N × H)

27. C₂×Q₈×C₆₇

Q₈

semidirect

50. Q₈×C₆₇

C₂×Q₈

semidirect

78. C₂×C₆₇

#475

direct (N × H)

82. C₂×#475

C₆₇

direct (N × H)

83. C₆₇

C₂×#475

direct (N × H)

86. #475

C₂×C₆₇

direct (N × H)

110. C₂×Q₈

Q₈×C₆₇

semidirect

133. Q₈

C₂×Q₈×C₆₇

semidirect

161. C₂

#475×C₆₇

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊Q₈×C₆₇	2
C₂×C₄×Q₈×C₆₇	2
#475×C₆₇	2
C₂×#106×C₆₇	2
C₂×C₄⋊C₈×C₆₇	2
C₂×#103×C₆₇	2
C₂×#475	67

subgroup

index

C₂×C₄⋊Q₈×C₆₇

C₂×C₄×Q₈×C₆₇

#475×C₆₇

C₂×#106×C₆₇

C₂×C₄⋊C₈×C₆₇

C₂×#103×C₆₇

C₂×#475

Subgroups — C₂×#475×C₆₇ #22557504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#475×C67 #22557504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#475×C₆₇ #22557504