Group #31807506 — subgroups

Subgroups — #31807506

Order 17792.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×Dic₅₅₆	C₂	1	2	yes	maximal
2	C₂×#17984842	C₂	1	1	yes	maximal, characteristic
3	C₂×#17984767	C₂	1	2	yes	maximal
4	#22580859	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
8	C₂×Dic₅₅₆	—	2	4	no	—
9	C₂×Dic₅₅₆	—	2	4	no	—
10	C₂×Dic₅₅₆	—	2	4	no	—
11	#17984842	—	2	4	no	—
12	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
13	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
14	#17984767	—	2	4	no	—
15	C₁₃₉⋊#95	—	2	2	no	—
16	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₇₈	—	2	2	no	—
19	C₂×Dic₂₇₈	—	2	2	no	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	—	2	2	no	—
22	C₂×Dic₂₇₈	—	2	2	no	—
23	C₂×Dic₂₇₈	—	2	2	no	—
24	C₂×Dic₂₇₈	—	2	1	no	—
25	C₂×Dic₂₇₈	—	2	2	no	—
26	C₂×Dic₂₇₈	D₄	1	4	yes	—
27	C₂×Dic₂₇₈	D₄	1	2	yes	—
28	C₂²×Dic₁₃₉	—	2	2	no	—
29	C₂²×Dic₁₃₉	—	2	1	no	—
30	C₂²×Dic₁₃₉	—	2	1	no	—
31	Dic₅₅₆	—	2	16	no	—
32	C₄⋊Dic₁₃₉	—	2	8	no	—
33	C₄⋊Dic₁₃₉	—	2	2	no	—
34	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₃₉	—	2	2	no	—
37	C₂×C₈×C₁₃₉	—	2	2	no	—
38	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
39	C₄²×C₁₃₉	—	2	2	no	—
40	Dic₂₇₈	—	2	4	no	—
41	Dic₂₇₈	—	2	4	no	—
42	Dic₂₇₈	—	2	4	no	—
43	Dic₂₇₈	—	4	2	no	—
44	Dic₂₇₈	—	4	2	no	—
45	Dic₂₇₈	—	4	2	no	—
46	C₂×Dic₁₃₉	—	2	4	no	—
47	C₂×Dic₁₃₉	—	2	2	no	—
48	C₂×Dic₁₃₉	—	4	1	no	—
49	C₂×Dic₁₃₉	—	4	2	no	—
50	C₂×Dic₁₃₉	—	4	2	no	—
51	C₂×Dic₁₃₉	—	4	1	no	—
52	C₂×Dic₁₃₉	—	4	1	no	—
53	C₂×Dic₁₃₉	—	4	2	no	—
54	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	—	2	2	no	—
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	Q₁₆	1	4	yes	complemented
58	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
62	C₈×C₁₃₉	—	2	4	no	—
63	C₂²×C₁₃₉	#120	1	1	yes	characteristic
64	C₂²×C₁₃₉	#121	1	1	yes	characteristic
65	C₂²×C₁₃₉	#127	1	1	yes	characteristic
66	C₂²×C₁₃₉	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₃₉	#137	1	2	yes	—
68	C₄×C₁₃₉	—	2	4	no	—
69	C₄×C₁₃₉	#120	1	2	yes	—
70	C₄×C₁₃₉	#121	1	2	yes	—
71	Dic₁₃₉	—	4	4	no	—
72	Dic₁₃₉	—	4	2	no	—
73	Dic₁₃₉	—	8	1	no	—
74	C₂×C₁₃₉	#392	1	1	yes	characteristic
75	C₂×C₁₃₉	#451	1	2	yes	—
76	C₂×C₁₃₉	#462	1	2	yes	—
77	C₂×C₁₃₉	#494	1	1	yes	characteristic
78	C₂×C₁₃₉	#497	1	1	yes	characteristic
79	C₁₃₉	#2016	1	1	yes	characteristic, complemented, Sylow
80	#2016	—	139	1	no	maximal, Sylow
81	C₂²×Q₁₆	—	139	2	no	—
82	C₂×C₄⋊Q₈	—	139	1	no	—
83	C₂×#103	—	139	2	no	—
84	#391	—	139	1	no	—
85	#336	D₁₃₉	1	1	yes	characteristic
86	C₂²×Q₈	—	139	2	no	—
87	C₂²×Q₈	—	139	1	no	—
88	C₂×Q₁₆	—	278	4	no	—
89	C₂×Q₁₆	—	278	4	no	—
90	C₂×Q₁₆	—	278	4	no	—
91	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
92	C₄⋊Q₈	—	278	4	no	—
93	C₂×C₄⋊C₄	—	278	2	no	—
94	C₂×C₄⋊C₄	—	139	1	no	—
95	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
96	#103	—	278	4	no	—
97	#95	—	278	2	no	—
98	C₂×Q₈	—	278	2	no	—
99	C₂×Q₈	—	278	1	no	—
100	C₂×Q₈	—	278	2	no	—
101	C₂×Q₈	—	278	2	no	—
102	C₂×Q₈	—	278	2	no	—
103	C₂×Q₈	—	278	2	no	—
104	C₂×Q₈	—	278	2	no	—
105	C₂×Q₈	—	278	2	no	—
106	C₂×Q₈	—	139	4	no	—
107	C₂×Q₈	—	139	2	no	—
108	C₂²×C₄	—	278	1	no	—
109	C₂²×C₄	—	278	2	no	—
110	C₂²×C₄	—	278	1	no	—
111	C₂²×C₄	D₅₅₆	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
113	Q₁₆	—	278	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₅₅₆	1	4	yes	—
117	C₄⋊C₄	—	278	8	no	—
118	C₄⋊C₄	—	278	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
121	Q₈	—	278	4	no	—
122	Q₈	—	278	4	no	—
123	Q₈	—	278	4	no	—
124	Q₈	—	556	2	no	—
125	Q₈	—	556	2	no	—
126	Q₈	—	556	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	278	4	no	—
130	C₂×C₄	—	278	2	no	—
131	C₂×C₄	—	556	1	no	—
132	C₂×C₄	—	556	2	no	—
133	C₂×C₄	—	556	1	no	—
134	C₂×C₄	—	556	2	no	—
135	C₂×C₄	—	556	1	no	—
136	C₂×C₄	—	556	2	no	—
137	C₂×C₄	Dic₅₅₆	1	4	yes	complemented
138	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
140	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
141	C₂×C₄	#14555786	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#17984812	1	1	yes	characteristic
144	C₂²	#17984817	1	1	yes	characteristic
145	C₂²	#17984839	1	1	yes	characteristic
146	C₂²	C₂×Dic₅₅₆	1	2	yes	—
147	C₂²	#17984875	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	556	4	no	—
150	C₄	—	556	2	no	—
151	C₄	—	1112	1	no	—
152	C₄	#17984812	1	2	yes	—
153	C₄	#17984817	1	2	yes	—
154	C₂	#22580863	1	1	yes	characteristic
155	C₂	#22581111	1	2	yes	—
156	C₂	#22581164	1	2	yes	—
157	C₂	#22581333	1	1	yes	characteristic
158	C₂	#22581347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₅₆

C₂

yes

maximal

C₂×#17984842

C₂

yes

maximal, characteristic

C₂×#17984767

C₂

yes

maximal

#22580859

C₂

yes

maximal, characteristic

#336×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

#17984842

—

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

#17984767

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

—

C₂×Dic₂₇₈

D₄

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

Dic₅₅₆

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

Q₁₆

yes

complemented

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

C₂×Q₁₆

yes

—

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#120

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#392

yes

characteristic

C₂×C₁₃₉

#451

yes

—

C₂×C₁₃₉

#462

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2016

—

139

maximal, Sylow

C₂²×Q₁₆

—

139

—

C₂×C₄⋊Q₈

—

139

—

C₂×#103

—

139

—

#391

—

139

—

#336

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂²×Q₈

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

278

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

#95

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

100

C₂×Q₈

—

278

—

101

C₂×Q₈

—

278

—

102

C₂×Q₈

—

278

—

103

C₂×Q₈

—

278

—

104

C₂×Q₈

—

278

—

105

C₂×Q₈

—

278

—

106

C₂×Q₈

—

139

—

107

C₂×Q₈

—

139

—

108

C₂²×C₄

—

278

—

109

C₂²×C₄

—

278

—

110

C₂²×C₄

—

278

—

111

C₂²×C₄

D₅₅₆

yes

—

112

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

113

Q₁₆

—

278

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₅₅₆

yes

—

117

C₄⋊C₄

—

278

—

118

C₄⋊C₄

—

278

—

119

C₄²

—

120

C₂³

C₂×D₅₅₆

yes

characteristic

121

Q₈

—

278

—

122

Q₈

—

278

—

123

Q₈

—

278

—

124

Q₈

—

556

—

125

Q₈

—

556

—

126

Q₈

—

556

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

278

—

130

C₂×C₄

—

278

—

131

C₂×C₄

—

556

—

132

C₂×C₄

—

556

—

133

C₂×C₄

—

556

—

134

C₂×C₄

—

556

—

135

C₂×C₄

—

556

—

136

C₂×C₄

—

556

—

137

C₂×C₄

Dic₅₅₆

yes

complemented

138

C₂×C₄

D₄×D₁₃₉

yes

characteristic

139

C₂×C₄

D₄×D₁₃₉

yes

—

140

C₂×C₄

C₂×D₅₅₆

yes

—

141

C₂×C₄

#14555786

yes

characteristic

142

C₈

—

143

C₂²

#17984812

yes

characteristic

144

C₂²

#17984817

yes

characteristic

145

C₂²

#17984839

yes

characteristic

146

C₂²

C₂×Dic₅₅₆

yes

—

147

C₂²

#17984875

yes

—

148

C₄

—

149

C₄

—

556

—

150

C₄

—

556

—

151

C₄

—

1112

—

152

C₄

#17984812

yes

—

153

C₄

#17984817

yes

—

154

C₂

#22580863

yes

characteristic

155

C₂

#22581111

yes

—

156

C₂

#22581164

yes

—

157

C₂

#22581333

yes

characteristic

158

C₂

#22581347

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#336×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#336	D₁₃₉
hypocenter	C₁₃₉	#2016
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#336×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
57. C₂×C₄×C₁₃₉	Q₁₆	semidirect
79. C₁₃₉	#2016	semidirect
137. C₂×C₄	Dic₅₅₆	semidirect

H ≅ G/N

type

57. C₂×C₄×C₁₃₉

Q₁₆

semidirect

79. C₁₃₉

#2016

semidirect

137. C₂×C₄

Dic₅₅₆

semidirect

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

subgroup	index
C₂²×Dic₅₅₆	2
C₂×#17984842	2
C₂×#17984767	2
#22580859	2
#336×C₁₃₉	2
#2016	139

subgroup

index

C₂²×Dic₅₅₆

#336×C₁₃₉

139

Subgroups — #31807506

Conjugacy classes of subgroups

Complements

Maximal subgroups