Group #31816864 — subgroups

Subgroups — #31816864

Order 17792.

552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 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×#17984872	C₂	1	4	yes	maximal, complemented
2	C₂×#17984842	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581348	C₂	1	4	yes	maximal, complemented
4	#22581332	C₂	1	2	yes	maximal, complemented
5	#22581325	C₂	1	2	yes	maximal, complemented
6	C₄×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
9	#17984872	—	2	16	no	—
10	C₂×#14555775	C₂²	1	8	yes	—
11	C₂×#14555773	C₂²	1	8	yes	—
12	C₂×#14555769	C₂²	1	2	yes	—
13	#17984842	—	2	2	no	—
14	#17984842	C₂²	1	2	yes	—
15	#17984842	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
17	#17984753	C₂²	1	2	yes	—
18	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
19	C₂×#41×C₁₃₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	#127×C₁₃₉	C₂²	1	1	yes	characteristic
22	#124×C₁₃₉	C₂²	1	2	yes	—
23	#121×C₁₃₉	—	2	2	no	—
24	C₄×D₄×C₁₃₉	—	2	2	no	—
25	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₂×Dic₂₇₈	—	2	2	no	—
27	C₂×Dic₂₇₈	—	2	8	no	—
28	C₂×Dic₂₇₈	C₂³	1	4	yes	—
29	C₂²×Dic₁₃₉	—	2	2	no	—
30	#14555775	—	2	16	no	—
31	#14555773	—	2	16	no	—
32	#14555769	D₄	1	8	yes	complemented
33	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
34	C₄⋊Dic₁₃₉	—	2	4	no	—
35	C₄⋊Dic₁₃₉	—	2	4	no	—
36	#41×C₁₃₉	—	2	8	no	—
37	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
38	C₂×D₄×C₁₃₉	—	2	2	no	—
39	C₂×D₄×C₁₃₉	—	2	2	no	—
40	C₂×D₄×C₁₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃₉	—	2	2	no	—
42	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
43	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₃₉	—	2	2	no	—
45	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
46	C₄²×C₁₃₉	D₄	1	2	yes	—
47	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
48	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
49	Dic₂₇₈	—	2	8	no	—
50	Dic₂₇₈	—	4	4	no	—
51	C₂×Dic₁₃₉	—	2	4	no	—
52	C₂×Dic₁₃₉	—	4	4	no	—
53	C₁₃₉⋊C₈	C₂×D₄	1	8	yes	—
54	C₂³×C₁₃₉	—	2	2	no	—
55	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₁₃₉	—	2	4	no	—
57	D₄×C₁₃₉	—	2	4	no	—
58	D₄×C₁₃₉	—	4	4	no	—
59	D₄×C₁₃₉	—	4	4	no	—
60	C₂×C₄×C₁₃₉	—	2	4	no	—
61	C₂×C₄×C₁₃₉	—	2	2	no	—
62	C₂×C₄×C₁₃₉	—	4	4	no	—
63	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
68	C₂²×C₁₃₉	—	2	2	no	—
69	C₂²×C₁₃₉	—	4	2	no	—
70	C₂²×C₁₃₉	—	4	4	no	—
71	C₂²×C₁₃₉	#127	1	2	yes	—
72	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₃₉	—	2	2	no	—
74	C₄×C₁₃₉	—	4	2	no	—
75	C₄×C₁₃₉	#127	1	2	yes	—
76	C₄×C₁₃₉	#137	1	4	yes	complemented
77	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
78	Dic₁₃₉	—	4	4	no	—
79	C₂×C₁₃₉	—	2	2	no	—
80	C₂×C₁₃₉	—	8	2	no	—
81	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
82	C₂×C₁₃₉	C₂×#137	1	2	yes	—
83	C₁₃₉	#3152	1	1	yes	characteristic, complemented, Sylow
84	#3152	—	139	1	no	maximal, Sylow
85	C₂×#137	—	139	4	no	—
86	#532	D₁₃₉	1	1	yes	characteristic
87	C₂×C₄⋊Q₈	—	139	1	no	—
88	#497	—	139	4	no	—
89	#494	—	139	2	no	—
90	#492	—	139	2	no	—
91	C₄×M₁₆	—	139	1	no	—
92	C₂×#41	C₂×D₁₃₉	1	2	yes	—
93	C₂²×Q₈	—	139	2	no	—
94	#137	—	278	16	no	—
95	C₂×Q₁₆	—	139	8	no	—
96	C₂×SD₁₆	—	139	8	no	—
97	C₂×M₁₆	—	139	2	no	—
98	C₄⋊Q₈	—	278	2	no	—
99	C₄⋊Q₈	—	139	2	no	—
100	C₄⋊Q₈	—	139	2	no	—
101	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
102	#127	C₂×D₁₃₉	1	1	yes	characteristic
103	#124	C₂×D₁₃₉	1	2	yes	—
104	#121	—	2	2	no	—
105	C₄×D₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	278	2	no	—
107	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
108	#97	—	139	2	no	—
109	C₄×C₈	—	139	2	no	—
110	#41	—	2	8	no	—
111	C₂×Q₈	—	278	2	no	—
112	C₂×Q₈	—	278	8	no	—
113	C₂×Q₈	—	139	4	no	—
114	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂²×D₁₃₉	1	2	yes	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	278	2	no	—
120	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
121	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
122	Q₁₆	—	278	16	no	—
123	SD₁₆	—	278	16	no	—
124	M₁₆	—	139	8	no	—
125	C₂×C₈	—	139	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	278	4	no	—
128	C₄⋊C₄	—	278	4	no	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
131	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
132	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	278	8	no	—
137	Q₈	—	556	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	278	4	no	—
145	C₂×C₄	—	556	4	no	—
146	C₂×C₄	D₄×D₁₃₉	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
150	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
151	C₈	—	139	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#17984838	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	556	4	no	—
160	C₄	#17984838	1	2	yes	—
161	C₄	#17984872	1	4	yes	complemented
162	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#17984838	1	1	yes	characteristic
166	C₂	C₂×#17984872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984872

C₂

yes

maximal, complemented

C₂×#17984842

C₂

yes

maximal, characteristic, complemented

#22581348

C₂

yes

maximal, complemented

#22581332

C₂

yes

maximal, complemented

#22581325

C₂

yes

maximal, complemented

C₄×#14555769

C₂

yes

maximal, characteristic, complemented

#532×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

#17984872

—

C₂×#14555775

C₂²

yes

—

C₂×#14555773

C₂²

yes

—

C₂×#14555769

C₂²

yes

—

#17984842

—

#17984842

C₂²

yes

—

#17984842

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

—

#17984753

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₂×#41×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

#124×C₁₃₉

C₂²

yes

—

#121×C₁₃₉

—

C₄×D₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂²×Dic₁₃₉

—

#14555775

—

#14555773

—

#14555769

D₄

yes

complemented

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

C₂×D₄

yes

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#127

yes

—

C₄×C₁₃₉

#137

yes

complemented

C₄×C₁₃₉

C₂²×D₄

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#127

yes

characteristic

C₂×C₁₃₉

C₂×#137

yes

—

C₁₃₉

#3152

yes

characteristic, complemented, Sylow

#3152

—

139

maximal, Sylow

C₂×#137

—

139

—

#532

D₁₃₉

yes

characteristic

C₂×C₄⋊Q₈

—

139

—

#497

—

139

—

#494

—

139

—

#492

—

139

—

C₄×M₁₆

—

139

—

C₂×#41

C₂×D₁₃₉

yes

—

C₂²×Q₈

—

139

—

#137

—

278

—

C₂×Q₁₆

—

139

—

C₂×SD₁₆

—

139

—

C₂×M₁₆

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

100

C₄⋊Q₈

—

139

—

101

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

102

#127

C₂×D₁₃₉

yes

characteristic

103

#124

C₂×D₁₃₉

yes

—

104

#121

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

278

—

107

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

108

#97

—

139

—

109

C₄×C₈

—

139

—

110

#41

—

111

C₂×Q₈

—

278

—

112

C₂×Q₈

—

278

—

113

C₂×Q₈

—

139

—

114

C₂×Q₈

C₂²×D₁₃₉

yes

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

C₂²×D₁₃₉

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

278

—

120

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

121

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

122

Q₁₆

—

278

—

123

SD₁₆

—

278

—

124

M₁₆

—

139

—

125

C₂×C₈

—

139

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

278

—

128

C₄⋊C₄

—

278

—

129

C₂²⋊C₄

—

130

C₄²

C₂²⋊D₁₃₉

yes

—

131

C₄²

C₂²×D₁₃₉

yes

characteristic

132

C₄²

C₂²×D₁₃₉

yes

characteristic

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

135

Q₈

—

136

Q₈

—

278

—

137

Q₈

—

556

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

278

—

145

C₂×C₄

—

556

—

146

C₂×C₄

D₄×D₁₃₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

148

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

150

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

151

C₈

—

139

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17984838

yes

—

156

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

556

—

160

C₄

#17984838

yes

—

161

C₄

#17984872

yes

complemented

162

C₄

C₂×D₄×D₁₃₉

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂×#17984838

yes

characteristic

166

C₂

C₂×#17984872

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting, prime radical	#532×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#532	D₁₃₉
hypocenter	C₁₃₉	#3152
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting, prime radical

#532×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#17984872	C₂	semidirect
2. C₂×#17984842	C₂	semidirect
3. #22581348	C₂	semidirect
4. #22581332	C₂	semidirect
5. #22581325	C₂	semidirect
6. C₄×#14555769	C₂	semidirect
32. #14555769	D₄	semidirect
76. C₄×C₁₃₉	#137	semidirect
83. C₁₃₉	#3152	semidirect
161. C₄	#17984872	semidirect

H ≅ G/N

type

1. C₂×#17984872

C₂

semidirect

2. C₂×#17984842

C₂

semidirect

3. #22581348

C₂

semidirect

4. #22581332

C₂

semidirect

5. #22581325

C₂

semidirect

6. C₄×#14555769

C₂

semidirect

32. #14555769

D₄

semidirect

76. C₄×C₁₃₉

#137

semidirect

83. C₁₃₉

#3152

semidirect

161. C₄

#17984872

semidirect

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

subgroup	index
C₂×#17984872	2
C₂×#17984842	2
#22581348	2
#22581332	2
#22581325	2
C₄×#14555769	2
#532×C₁₃₉	2
#3152	139

subgroup

index

#532×C₁₃₉

139

Subgroups — #31816864

Conjugacy classes of subgroups

Complements

Maximal subgroups