Group #31807263 — subgroups

Subgroups — #31807263

Order 17792.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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	1	yes	maximal, characteristic
2	C₂×#17984765	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580858	C₂	1	2	yes	maximal
5	C₂×#107×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
7	C₂×Dic₅₅₆	—	2	2	no	—
8	C₂×Dic₅₅₆	—	2	2	no	—
9	C₂×Dic₅₅₆	C₄	1	4	yes	complemented
10	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
12	#17984765	—	2	4	no	—
13	C₈×Dic₁₃₉	—	2	2	no	—
14	C₁₃₉⋊#95	—	2	4	no	—
15	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
17	#107×C₁₃₉	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
24	C₂²×Dic₁₃₉	—	2	2	no	—
25	C₂²×Dic₁₃₉	D₄	1	2	yes	—
26	Dic₅₅₆	—	2	8	no	—
27	C₂×C₁₃₉⋊C₈	—	2	1	no	—
28	C₂×C₁₃₉⋊C₈	—	2	1	no	—
29	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
30	C₄×Dic₁₃₉	—	2	1	no	—
31	C₄×Dic₁₃₉	—	2	1	no	—
32	C₂²×C₄×C₁₃₉	—	2	2	no	—
33	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
36	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₃₉	—	2	2	no	—
38	C₄⋊C₄×C₁₃₉	—	2	2	no	—
39	Dic₂₇₈	—	2	4	no	—
40	Dic₂₇₈	—	2	4	no	—
41	Dic₂₇₈	—	4	2	no	—
42	Dic₂₇₈	—	4	2	no	—
43	C₂×Dic₁₃₉	—	2	2	no	—
44	C₂×Dic₁₃₉	—	2	2	no	—
45	C₂×Dic₁₃₉	—	2	4	no	—
46	C₂×Dic₁₃₉	—	4	2	no	—
47	C₂×Dic₁₃₉	—	4	2	no	—
48	C₂×Dic₁₃₉	Q₁₆	1	4	yes	complemented
49	C₁₃₉⋊C₈	—	2	2	no	—
50	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	—	4	2	no	—
52	C₂×C₄×C₁₃₉	—	4	4	no	—
53	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
59	C₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₃₉	#121	1	1	yes	characteristic
62	C₂²×C₁₃₉	#127	1	1	yes	characteristic
63	C₂²×C₁₃₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₁₃₉	#135	1	2	yes	—
65	C₄×C₁₃₉	—	8	2	no	—
66	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
68	C₄×C₁₃₉	#121	1	1	yes	characteristic
69	C₄×C₁₃₉	#124	1	1	yes	characteristic
70	Dic₁₃₉	—	2	4	no	—
71	Dic₁₃₉	—	4	4	no	—
72	C₂×C₁₃₉	#390	1	1	yes	characteristic
73	C₂×C₁₃₉	C₄×Q₁₆	1	2	yes	—
74	C₂×C₁₃₉	#467	1	2	yes	—
75	C₂×C₁₃₉	#494	1	1	yes	characteristic
76	C₂×C₁₃₉	#495	1	1	yes	characteristic
77	C₁₃₉	#1969	1	1	yes	characteristic, complemented, Sylow
78	#1969	—	139	1	no	maximal, Sylow
79	C₂²×Q₁₆	—	139	1	no	—
80	C₂×#107	D₁₃₉	1	1	yes	characteristic
81	C₂×#103	—	139	2	no	—
82	C₂×C₄×C₈	—	139	1	no	—
83	#391	—	139	2	no	—
84	C₂²×Q₈	—	139	2	no	—
85	C₂×Q₁₆	—	278	2	no	—
86	C₂×Q₁₆	—	278	2	no	—
87	C₂×Q₁₆	—	139	4	no	—
88	C₂²×C₈	—	139	1	no	—
89	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
91	C₂×C₄²	—	139	1	no	—
92	#107	—	2	2	no	—
93	#103	—	278	4	no	—
94	C₄×C₈	—	278	2	no	—
95	#95	—	278	4	no	—
96	C₂×Q₈	—	278	2	no	—
97	C₂×Q₈	—	278	2	no	—
98	C₂×Q₈	—	278	2	no	—
99	C₂×Q₈	—	278	2	no	—
100	C₂×Q₈	—	278	2	no	—
101	C₂×Q₈	—	139	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	278	2	no	—
104	C₂²×C₄	—	139	2	no	—
105	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
106	Q₁₆	—	278	8	no	—
107	C₂×C₈	—	278	1	no	—
108	C₂×C₈	—	278	1	no	—
109	C₂×C₈	—	139	2	no	—
110	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
111	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
112	C₂×C₈	D₅₅₆	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	278	1	no	—
116	C₄²	—	278	1	no	—
117	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
118	Q₈	—	278	4	no	—
119	Q₈	—	278	4	no	—
120	Q₈	—	556	2	no	—
121	Q₈	—	556	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	278	2	no	—
125	C₂×C₄	—	278	2	no	—
126	C₂×C₄	—	278	4	no	—
127	C₂×C₄	—	556	2	no	—
128	C₂×C₄	—	556	2	no	—
129	C₂×C₄	—	139	4	no	—
130	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
134	C₂×C₄	#14555786	1	1	yes	characteristic
135	C₂×C₄	#14555786	1	1	yes	characteristic
136	C₈	—	278	2	no	—
137	C₈	(C₂×D₁₃₉)⋊C₄	1	4	yes	—
138	C₂²	#17984804	1	1	yes	characteristic
139	C₂²	#17984817	1	1	yes	characteristic
140	C₂²	#17984838	1	1	yes	characteristic
141	C₂²	Q₁₆×D₁₃₉	1	2	yes	—
142	C₂²	#17984861	1	2	yes	—
143	C₄	—	8	2	no	—
144	C₄	—	278	4	no	—
145	C₄	—	556	4	no	—
146	C₄	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
147	C₄	#17984804	1	1	yes	characteristic
148	C₄	#17984817	1	1	yes	characteristic
149	C₄	#17984831	1	1	yes	characteristic
150	C₂	#22580854	1	1	yes	characteristic
151	C₂	#22581047	1	2	yes	—
152	C₂	#22581195	1	2	yes	—
153	C₂	#22581331	1	1	yes	characteristic
154	C₂	#22581336	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₅₆

C₂

yes

maximal, characteristic

C₂×#17984765

C₂

yes

maximal

C₂×C₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580858

C₂

yes

maximal

C₂×#107×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

C₄

yes

complemented

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984765

—

C₈×Dic₁₃₉

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

#107×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

D₄

yes

—

Dic₅₅₆

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

Q₁₆

yes

complemented

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

C₂²⋊C₄

yes

—

C₂²×C₁₃₉

C₄×D₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

C₂×Q₁₆

yes

—

C₂²×C₁₃₉

#135

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×D₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

C₄×Q₁₆

yes

—

C₂×C₁₃₉

#467

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1969

—

139

maximal, Sylow

C₂²×Q₁₆

—

139

—

C₂×#107

D₁₃₉

yes

characteristic

C₂×#103

—

139

—

C₂×C₄×C₈

—

139

—

#391

—

139

—

C₂²×Q₈

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

—

139

—

#107

—

#103

—

278

—

C₄×C₈

—

278

—

#95

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

100

C₂×Q₈

—

278

—

101

C₂×Q₈

—

139

—

102

C₂²×C₄

—

103

C₂²×C₄

—

278

—

104

C₂²×C₄

—

139

—

105

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

106

Q₁₆

—

278

—

107

C₂×C₈

—

278

—

108

C₂×C₈

—

278

—

109

C₂×C₈

—

139

—

110

C₂×C₈

C₄×D₁₃₉

yes

—

111

C₂×C₈

C₂²⋊D₁₃₉

yes

—

112

C₂×C₈

D₅₅₆

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

278

—

116

C₄²

—

278

—

117

C₂³

D₄×D₁₃₉

yes

characteristic

118

Q₈

—

278

—

119

Q₈

—

278

—

120

Q₈

—

556

—

121

Q₈

—

556

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

278

—

125

C₂×C₄

—

278

—

126

C₂×C₄

—

278

—

127

C₂×C₄

—

556

—

128

C₂×C₄

—

556

—

129

C₂×C₄

—

139

—

130

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

131

C₂×C₄

D₄×D₁₃₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

133

C₂×C₄

C₂×D₅₅₆

yes

characteristic

134

C₂×C₄

#14555786

yes

characteristic

135

C₂×C₄

#14555786

yes

characteristic

136

C₈

—

278

—

137

C₈

(C₂×D₁₃₉)⋊C₄

yes

—

138

C₂²

#17984804

yes

characteristic

139

C₂²

#17984817

yes

characteristic

140

C₂²

#17984838

yes

characteristic

141

C₂²

Q₁₆×D₁₃₉

yes

—

142

C₂²

#17984861

yes

—

143

C₄

—

144

C₄

—

278

—

145

C₄

—

556

—

146

C₄

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

147

C₄

#17984804

yes

characteristic

148

C₄

#17984817

yes

characteristic

149

C₄

#17984831

yes

characteristic

150

C₂

#22580854

yes

characteristic

151

C₂

#22581047

yes

—

152

C₂

#22581195

yes

—

153

C₂

#22581331

yes

characteristic

154

C₂

#22581336

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#107×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#107	D₁₃₉
hypocenter	C₁₃₉	#1969
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#107×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. C₂×Dic₅₅₆	C₄	semidirect
48. C₂×Dic₁₃₉	Q₁₆	semidirect
77. C₁₃₉	#1969	semidirect

H ≅ G/N

type

9. C₂×Dic₅₅₆

C₄

semidirect

48. C₂×Dic₁₃₉

Q₁₆

semidirect

77. C₁₃₉

#1969

semidirect

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

subgroup	index
C₂²×Dic₅₅₆	2
C₂×#17984765	2
C₂×C₈×Dic₁₃₉	2
#22580858	2
C₂×#107×C₁₃₉	2
#1969	139

subgroup

index

C₂²×Dic₅₅₆

C₂×#17984765

C₂×C₈×Dic₁₃₉

#22580858

C₂×#107×C₁₃₉

#1969

139

Subgroups — #31807263

Conjugacy classes of subgroups

Complements

Maximal subgroups