Group #31816166 — subgroups

Subgroups — C₂×#22581235 #31816166

Order 17792.

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×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	#22581235	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984773	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×#17984765	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₂₇₈	—	2	2	no	—
10	C₄×Dic₂₇₈	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₁₃₉	—	2	1	no	—
14	#17984773	C₂²	1	8	yes	—
15	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	—
16	#17984765	C₂²	1	8	yes	—
17	C₂²×Q₈×C₁₃₉	—	2	1	no	—
18	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
19	C₄⋊Q₈×C₁₃₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₇₈	—	2	1	no	—
24	C₂×Dic₂₇₈	—	2	2	no	—
25	C₂×Dic₂₇₈	Q₈	1	4	yes	complemented
26	C₂×Dic₂₇₈	C₂³	1	2	yes	—
27	C₂²×Dic₁₃₉	—	2	1	no	—
28	C₂²×Dic₁₃₉	—	2	1	no	—
29	C₂×C₁₃₉⋊C₈	—	2	4	no	—
30	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₁₃₉	—	2	1	no	—
32	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
33	Dic₁₃₉⋊C₄	—	2	4	no	—
34	C₄×Dic₁₃₉	—	2	4	no	—
35	C₂×Q₈×C₁₃₉	—	2	4	no	—
36	C₂×Q₈×C₁₃₉	—	4	4	no	—
37	C₂²×C₄×C₁₃₉	—	2	2	no	—
38	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₃₉	—	2	4	no	—
42	C₄⋊C₄×C₁₃₉	—	2	2	no	—
43	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
44	C₄²×C₁₃₉	D₄	1	2	yes	—
45	C₄²×C₁₃₉	C₂³	1	2	yes	—
46	Dic₂₇₈	—	2	4	no	—
47	Dic₂₇₈	—	2	2	no	—
48	Dic₂₇₈	C₂×Q₈	1	4	yes	complemented
49	C₂×Dic₁₃₉	—	2	4	no	—
50	C₂×Dic₁₃₉	—	2	2	no	—
51	C₂×Dic₁₃₉	—	2	2	no	—
52	C₂×Dic₁₃₉	—	4	2	no	—
53	C₁₃₉⋊C₈	—	2	4	no	—
54	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₃₉	—	4	8	no	—
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	—	2	4	no	—
58	C₂×C₄×C₁₃₉	—	4	4	no	—
59	C₂×C₄×C₁₃₉	SD₁₆	1	4	yes	—
60	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃₉	#41	1	2	yes	—
65	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₁₃₉	#137	1	2	yes	—
69	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₁₃₉	—	2	2	no	—
71	C₄×C₁₃₉	—	4	4	no	—
72	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₁₃₉	C₂×SD₁₆	1	4	yes	—
74	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
76	Dic₁₃₉	—	2	4	no	—
77	Dic₁₃₉	—	4	2	no	—
78	C₂×C₁₃₉	#475	1	4	yes	complemented
79	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₃₉	C₂²×SD₁₆	1	1	yes	characteristic
81	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
82	C₁₃₉	C₂×#475	1	1	yes	characteristic, complemented, Sylow
83	C₂×#475	—	139	1	no	maximal, Sylow
84	C₂×C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic
85	C₂×C₄×Q₈	—	139	1	no	—
86	#475	—	139	8	no	—
87	C₂×#106	—	139	2	no	—
88	C₂×C₄⋊C₈	—	139	1	no	—
89	C₂×#103	—	139	2	no	—
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	139	1	no	—
92	C₂²×C₈	—	139	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂×D₁₃₉	1	4	yes	—
95	C₄×Q₈	—	278	2	no	—
96	C₄×Q₈	—	139	4	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	278	1	no	—
99	C₂×C₄⋊C₄	—	139	1	no	—
100	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
101	C₂×C₄²	—	278	1	no	—
102	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
103	#106	—	139	8	no	—
104	C₄⋊C₈	—	139	4	no	—
105	#103	—	139	8	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	278	1	no	—
109	C₂×Q₈	—	278	2	no	—
110	C₂×Q₈	—	139	4	no	—
111	C₂×Q₈	—	139	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	278	1	no	—
114	C₂²×C₄	—	278	1	no	—
115	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
118	C₂×C₈	—	278	4	no	—
119	C₂×C₈	—	139	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	278	4	no	—
123	C₄⋊C₄	—	278	1	no	—
124	C₄⋊C₄	—	139	2	no	—
125	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
126	C₄²	—	278	4	no	—
127	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
128	C₄²	C₂²×D₁₃₉	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	278	4	no	—
132	Q₈	—	278	2	no	—
133	Q₈	—	139	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	278	4	no	—
138	C₂×C₄	—	278	2	no	—
139	C₂×C₄	—	278	2	no	—
140	C₂×C₄	—	556	2	no	—
141	C₂×C₄	#14555773	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
145	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
146	C₂×C₄	#14555786	1	2	yes	—
147	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
148	C₈	—	278	4	no	—
149	C₂²	#17984818	1	2	yes	—
150	C₂²	C₂×#14555773	1	2	yes	—
151	C₂²	#17984873	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	4	4	no	—
155	C₄	—	278	4	no	—
156	C₄	—	556	2	no	—
157	C₄	#17984818	1	2	yes	—
158	C₄	C₂×#14555773	1	4	yes	—
159	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
160	C₄	C₂×#14555786	1	1	yes	characteristic
161	C₂	#22581235	1	4	yes	direct product factor
162	C₂	C₂×#17984818	1	1	yes	characteristic
163	C₂	C₂²×#14555773	1	1	yes	characteristic
164	C₂	C₂×#17984873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₇₈

C₂

yes

maximal, characteristic

#22581235

C₂

yes

maximal, direct product factor

C₂×#17984773

C₂

yes

maximal

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#17984765

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₄×Dic₂₇₈

—

C₄×Dic₂₇₈

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×Dic₁₃₉⋊C₄

—

C₂×C₄×Dic₁₃₉

—

#17984773

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984765

C₂²

yes

—

C₂²×Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

Q₈

yes

complemented

C₂×Dic₂₇₈

C₂³

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

C₂×Q₈

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

#137

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

C₄×C₁₃₉

C₂×#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#475

yes

complemented

C₂×C₁₃₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂²×SD₁₆

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

C₂×#475

—

139

maximal, Sylow

C₂×C₄⋊Q₈

D₁₃₉

yes

characteristic

C₂×C₄×Q₈

—

139

—

#475

—

139

—

C₂×#106

—

139

—

C₂×C₄⋊C₈

—

139

—

C₂×#103

—

139

—

C₂²×Q₈

—

C₂²×Q₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

100

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

101

C₂×C₄²

—

278

—

102

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

103

#106

—

139

—

104

C₄⋊C₈

—

139

—

105

#103

—

139

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

278

—

109

C₂×Q₈

—

278

—

110

C₂×Q₈

—

139

—

111

C₂×Q₈

—

139

—

112

C₂²×C₄

—

113

C₂²×C₄

—

278

—

114

C₂²×C₄

—

278

—

115

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

116

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

117

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

118

C₂×C₈

—

278

—

119

C₂×C₈

—

139

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

278

—

123

C₄⋊C₄

—

278

—

124

C₄⋊C₄

—

139

—

125

C₄⋊C₄

C₂²×D₁₃₉

yes

—

126

C₄²

—

278

—

127

C₄²

C₂²⋊D₁₃₉

yes

—

128

C₄²

C₂²×D₁₃₉

yes

—

129

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

130

Q₈

—

131

Q₈

—

278

—

132

Q₈

—

278

—

133

Q₈

—

139

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

278

—

138

C₂×C₄

—

278

—

139

C₂×C₄

—

278

—

140

C₂×C₄

—

556

—

141

C₂×C₄

#14555773

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

145

C₂×C₄

Q₈×D₁₃₉

yes

—

146

C₂×C₄

#14555786

yes

—

147

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

148

C₈

—

278

—

149

C₂²

#17984818

yes

—

150

C₂²

C₂×#14555773

yes

—

151

C₂²

#17984873

yes

—

152

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

278

—

156

C₄

—

556

—

157

C₄

#17984818

yes

—

158

C₄

C₂×#14555773

yes

—

159

C₄

C₂×Q₈×D₁₃₉

yes

characteristic

160

C₄

C₂×#14555786

yes

characteristic

161

C₂

#22581235

yes

direct product factor

162

C₂

C₂×#17984818

yes

characteristic

163

C₂

C₂²×#14555773

yes

characteristic

164

C₂

C₂×#17984873

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×C₄⋊Q₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	C₂×#475
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂×C₄⋊Q₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₃₉

hypocenter

C₁₃₉

C₂×#475

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #22581235	C₂	direct (N × H)
25. C₂×Dic₂₇₈	Q₈	semidirect
48. Dic₂₇₈	C₂×Q₈	semidirect
78. C₂×C₁₃₉	#475	semidirect
82. C₁₃₉	C₂×#475	semidirect
161. C₂	#22581235	direct (N × H)

H ≅ G/N

type

2. #22581235

C₂

direct (N × H)

25. C₂×Dic₂₇₈

Q₈

semidirect

48. Dic₂₇₈

C₂×Q₈

semidirect

78. C₂×C₁₃₉

#475

semidirect

82. C₁₃₉

C₂×#475

semidirect

161. C₂

#22581235

direct (N × H)

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

subgroup	index
C₂×C₄×Dic₂₇₈	2
#22581235	2
C₂×#17984773	2
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×#17984765	2
C₂×C₄⋊Q₈×C₁₃₉	2
C₂×#475	139

subgroup

index

C₂×C₄×Dic₂₇₈

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂×C₄⋊Q₈×C₁₃₉

139

Subgroups — C₂×#22581235 #31816166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581235 #31816166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581235 #31816166