Group #31809317 — subgroups

Subgroups — Q₈×Dic₁₃₉⋊C₄ #31809317

Order 17792.

568 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 287 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×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22580851	C₂	1	3	yes	maximal
4	#22580849	C₂	1	3	yes	maximal
5	#22580822	C₂	1	3	yes	maximal
6	C₄×Dic₁₃₉⋊C₄	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₃₉	—	2	4	no	—
9	Q₈×Dic₁₃₉	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
12	C₂×Dic₁₃₉⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₁₃₉⋊C₄	C₂²	1	6	yes	—
14	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	—
16	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	—
17	C₁₃₉⋊#95	C₂²	1	3	yes	—
18	C₁₃₉⋊#95	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
22	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
23	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
24	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
25	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₁₃₉	—	2	6	no	—
28	C₄⋊Dic₁₃₉	C₂×C₄	1	12	yes	—
29	Dic₁₃₉⋊C₄	—	2	6	no	—
30	Dic₁₃₉⋊C₄	—	4	6	no	—
31	Dic₁₃₉⋊C₄	Q₈	1	4	yes	direct product factor
32	C₄×Dic₁₃₉	—	2	6	no	—
33	C₄×Dic₁₃₉	C₂×C₄	1	12	yes	—
34	C₂×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
35	C₂×Q₈×C₁₃₉	D₄	1	4	yes	—
36	C₂×Q₈×C₁₃₉	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
38	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
40	C₄⋊C₄×C₁₃₉	—	2	6	no	—
41	C₄²×C₁₃₉	—	2	6	no	—
42	C₂×Dic₁₃₉	—	2	1	no	—
43	C₂×Dic₁₃₉	—	2	3	no	—
44	C₂×Dic₁₃₉	—	2	3	no	—
45	C₂×Dic₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	2	3	no	—
47	C₂×Dic₁₃₉	—	2	3	no	—
48	C₂×Dic₁₃₉	—	2	3	no	—
49	C₂×Dic₁₃₉	C₂²×C₄	1	6	yes	—
50	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
51	C₂×Dic₁₃₉	C₂×Q₈	1	2	yes	—
52	C₂×Dic₁₃₉	C₂×Q₈	1	2	yes	—
53	C₂×Dic₁₃₉	#41	1	2	yes	—
54	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
55	Q₈×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
56	C₂×C₄×C₁₃₉	—	2	1	no	—
57	C₂×C₄×C₁₃₉	—	2	1	no	—
58	C₂×C₄×C₁₃₉	—	2	3	no	—
59	C₂×C₄×C₁₃₉	—	2	3	no	—
60	C₂×C₄×C₁₃₉	—	2	3	no	—
61	C₂×C₄×C₁₃₉	C₂²×C₄	1	6	yes	—
62	C₂×C₄×C₁₃₉	C₂×D₄	1	6	yes	—
63	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃₉	C₂×Q₈	1	6	yes	—
65	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
66	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
68	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
70	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
72	C₄×C₁₃₉	—	2	2	no	—
73	C₄×C₁₃₉	—	4	3	no	—
74	C₄×C₁₃₉	C₂×C₄⋊C₄	1	12	yes	—
75	Dic₁₃₉	—	2	6	no	—
76	Dic₁₃₉	—	2	2	no	—
77	Dic₁₃₉	—	4	3	no	—
78	Dic₁₃₉	C₄×Q₈	1	4	yes	complemented
79	C₂×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
81	C₂×C₁₃₉	#520	1	1	yes	characteristic
82	C₂×C₁₃₉	D₄×Q₈	1	1	yes	characteristic
83	C₂×C₁₃₉	#550	1	1	yes	characteristic
84	C₂×C₁₃₉	#557	1	1	yes	characteristic
85	C₂×C₁₃₉	Q₈²	1	1	yes	characteristic
86	C₁₃₉	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
87	Q₈×C₄⋊C₄	—	139	1	no	maximal, Sylow
88	C₂×C₄×Q₈	—	139	1	no	—
89	C₂×C₄×Q₈	—	139	1	no	—
90	C₂×C₄×Q₈	D₁₃₉	1	1	yes	characteristic
91	#389	—	139	3	no	—
92	#389	—	139	3	no	—
93	#384	—	139	3	no	—
94	C₄×C₄⋊C₄	—	139	3	no	—
95	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
96	C₄×Q₈	—	2	4	no	—
97	C₄×Q₈	—	278	4	no	—
98	C₄×Q₈	—	139	8	no	—
99	C₂×C₄⋊C₄	—	139	3	no	—
100	C₂×C₄⋊C₄	—	139	3	no	—
101	C₂×C₄⋊C₄	—	139	6	no	—
102	C₂×C₄⋊C₄	—	139	3	no	—
103	C₂×C₄⋊C₄	—	139	1	no	—
104	C₂×C₄⋊C₄	C₂×D₁₃₉	1	3	yes	—
105	C₂×C₄²	—	139	3	no	—
106	C₂×C₄²	—	139	3	no	—
107	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
108	#95	—	139	3	no	—
109	#95	—	139	3	no	—
110	C₂×Q₈	C₄×D₁₃₉	1	4	yes	—
111	C₂×Q₈	C₂²⋊D₁₃₉	1	4	yes	—
112	C₂×Q₈	Dic₂₇₈	1	4	yes	—
113	C₂²×C₄	—	139	3	no	—
114	C₂²×C₄	—	139	1	no	—
115	C₂²×C₄	—	139	3	no	—
116	C₂²×C₄	—	139	1	no	—
117	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
118	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
120	C₄⋊C₄	—	2	6	no	—
121	C₄⋊C₄	—	278	6	no	—
122	C₄⋊C₄	—	278	6	no	—
123	C₄⋊C₄	—	556	6	no	—
124	C₄⋊C₄	—	139	12	no	—
125	C₄⋊C₄	—	139	4	no	—
126	C₄²	—	2	6	no	—
127	C₄²	—	278	6	no	—
128	C₄²	—	139	12	no	—
129	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
130	Q₈	Dic₁₃₉⋊C₄	1	16	yes	direct product factor
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	278	1	no	—
137	C₂×C₄	—	278	3	no	—
138	C₂×C₄	—	278	1	no	—
139	C₂×C₄	—	278	3	no	—
140	C₂×C₄	—	278	3	no	—
141	C₂×C₄	—	278	3	no	—
142	C₂×C₄	—	278	3	no	—
143	C₂×C₄	—	139	6	no	—
144	C₂×C₄	—	139	2	no	—
145	C₂×C₄	—	139	2	no	—
146	C₂×C₄	—	139	2	no	—
147	C₂×C₄	—	139	2	no	—
148	C₂×C₄	C₂×C₄×D₁₃₉	1	6	yes	—
149	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	6	yes	—
150	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
151	C₂×C₄	C₂×Dic₂₇₈	1	6	yes	—
152	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
155	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
156	C₂²	#41×D₁₃₉	1	1	yes	characteristic
157	C₂²	#17984888	1	1	yes	characteristic
158	C₂²	#17984890	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	—	278	2	no	—
162	C₄	—	278	6	no	—
163	C₄	—	556	3	no	—
164	C₄	—	139	4	no	—
165	C₄	C₂×Dic₁₃₉⋊C₄	1	12	yes	—
166	C₂	C₂²×Dic₁₃₉⋊C₄	1	1	yes	characteristic
167	C₂	C₄×Q₈×D₁₃₉	1	1	yes	characteristic
168	C₂	#22581435	1	1	yes	characteristic
169	C₂	Q₈×C₂²⋊D₁₃₉	1	1	yes	characteristic
170	C₂	#22581622	1	1	yes	characteristic
171	C₂	#22581679	1	1	yes	characteristic
172	C₂	Q₈×Dic₂₇₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580851

C₂

yes

maximal

#22580849

C₂

yes

maximal

#22580822

C₂

yes

maximal

C₄×Dic₁₃₉⋊C₄

C₂

yes

maximal

C₂×C₄×Q₈×C₁₃₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₉

—

Q₈×Dic₁₃₉

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄×Q₈×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

characteristic

C₂²×Dic₁₃₉

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

Q₈

yes

direct product factor

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

D₄

yes

—

C₂×Q₈×C₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂²×C₄

yes

—

C₂×Dic₁₃₉

C₂²×C₄

yes

—

C₂×Dic₁₃₉

C₂×Q₈

yes

—

C₂×Dic₁₃₉

C₂×Q₈

yes

—

C₂×Dic₁₃₉

#41

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

Q₈×C₁₃₉

C₄⋊C₄

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂³×C₄

yes

characteristic

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×#41

yes

characteristic

C₂²×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃₉

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₄×Q₈

yes

complemented

C₂×C₁₃₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₃₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₃₉

#520

yes

characteristic

C₂×C₁₃₉

D₄×Q₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

Q₈²

yes

characteristic

C₁₃₉

Q₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

Q₈×C₄⋊C₄

—

139

maximal, Sylow

C₂×C₄×Q₈

—

139

—

C₂×C₄×Q₈

—

139

—

C₂×C₄×Q₈

D₁₃₉

yes

characteristic

#389

—

139

—

#389

—

139

—

#384

—

139

—

C₄×C₄⋊C₄

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

C₂×C₄⋊C₄

—

139

—

100

C₂×C₄⋊C₄

—

139

—

101

C₂×C₄⋊C₄

—

139

—

102

C₂×C₄⋊C₄

—

139

—

103

C₂×C₄⋊C₄

—

139

—

104

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

105

C₂×C₄²

—

139

—

106

C₂×C₄²

—

139

—

107

C₂×C₄²

C₂×D₁₃₉

yes

—

108

#95

—

139

—

109

#95

—

139

—

110

C₂×Q₈

C₄×D₁₃₉

yes

—

111

C₂×Q₈

C₂²⋊D₁₃₉

yes

—

112

C₂×Q₈

Dic₂₇₈

yes

—

113

C₂²×C₄

—

139

—

114

C₂²×C₄

—

139

—

115

C₂²×C₄

—

139

—

116

C₂²×C₄

—

139

—

117

C₂²×C₄

C₂²×D₁₃₉

yes

—

118

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

119

C₂²×C₄

C₂²×D₁₃₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

278

—

122

C₄⋊C₄

—

278

—

123

C₄⋊C₄

—

556

—

124

C₄⋊C₄

—

139

—

125

C₄⋊C₄

—

139

—

126

C₄²

—

127

C₄²

—

278

—

128

C₄²

—

139

—

129

C₂³

C₂³×D₁₃₉

yes

characteristic

130

Q₈

Dic₁₃₉⋊C₄

yes

direct product factor

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

278

—

137

C₂×C₄

—

278

—

138

C₂×C₄

—

278

—

139

C₂×C₄

—

278

—

140

C₂×C₄

—

278

—

141

C₂×C₄

—

278

—

142

C₂×C₄

—

278

—

143

C₂×C₄

—

139

—

144

C₂×C₄

—

139

—

145

C₂×C₄

—

139

—

146

C₂×C₄

—

139

—

147

C₂×C₄

—

139

—

148

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

149

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

150

C₂×C₄

Q₈×D₁₃₉

yes

—

151

C₂×C₄

C₂×Dic₂₇₈

yes

—

152

C₂²

C₂²×C₄×D₁₃₉

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

154

C₂²

C₂×Q₈×D₁₃₉

yes

characteristic

155

C₂²

C₂²×Dic₂₇₈

yes

characteristic

156

C₂²

#41×D₁₃₉

yes

characteristic

157

C₂²

#17984888

yes

characteristic

158

C₂²

#17984890

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

278

—

162

C₄

—

278

—

163

C₄

—

556

—

164

C₄

—

139

—

165

C₄

C₂×Dic₁₃₉⋊C₄

yes

—

166

C₂

C₂²×Dic₁₃₉⋊C₄

yes

characteristic

167

C₂

C₄×Q₈×D₁₃₉

yes

characteristic

168

C₂

#22581435

yes

characteristic

169

C₂

Q₈×C₂²⋊D₁₃₉

yes

characteristic

170

C₂

#22581622

yes

characteristic

171

C₂

#22581679

yes

characteristic

172

C₂

Q₈×Dic₂₇₈

yes

characteristic

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

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator

C₂²×C₁₃₉

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₃₉

C₂

socle, prime radical, cosocle

C₂³×C₁₃₉

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₃₉

hypocenter

C₁₃₉

Q₈×C₄⋊C₄

LCS

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
9. Q₈×Dic₁₃₉	C₄	semidirect
31. Dic₁₃₉⋊C₄	Q₈	direct (N × H)
55. Q₈×C₁₃₉	C₄⋊C₄	semidirect
78. Dic₁₃₉	C₄×Q₈	semidirect
86. C₁₃₉	Q₈×C₄⋊C₄	semidirect
130. Q₈	Dic₁₃₉⋊C₄	direct (N × H)

H ≅ G/N

type

9. Q₈×Dic₁₃₉

C₄

semidirect

31. Dic₁₃₉⋊C₄

Q₈

direct (N × H)

55. Q₈×C₁₃₉

C₄⋊C₄

semidirect

78. Dic₁₃₉

C₄×Q₈

semidirect

86. C₁₃₉

Q₈×C₄⋊C₄

semidirect

130. Q₈

Dic₁₃₉⋊C₄

direct (N × H)

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

subgroup	index
C₂×Q₈×Dic₁₃₉	2
#22580851	2
#22580849	2
#22580822	2
C₄×Dic₁₃₉⋊C₄	2
C₂×C₄×Q₈×C₁₃₉	2
Q₈×C₄⋊C₄	139

subgroup

index

C₂×Q₈×Dic₁₃₉

#22580851

#22580849

#22580822

C₄×Dic₁₃₉⋊C₄

C₂×C₄×Q₈×C₁₃₉

Q₈×C₄⋊C₄

139

Subgroups — Q₈×Dic₁₃₉⋊C₄ #31809317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Dic139⋊C4 #31809317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₁₃₉⋊C₄ #31809317