Group #31808453 — subgroups

Subgroups — #31808453

Order 17792.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₁₁₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984788	C₂	1	1	yes	maximal, characteristic
3	#22581331	C₂	1	1	yes	maximal, characteristic
4	#22581047	C₂	1	1	yes	maximal, characteristic
5	Dic₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#22580756	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
10	C₂×#14555775	—	2	1	no	—
11	#17984840	—	2	1	no	—
12	#17984807	—	2	1	no	—
13	Dic₁₁₁₂	—	2	2	no	—
14	#17984788	—	2	2	no	—
15	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#17984765	—	2	1	no	—
17	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
20	#107×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₈	—	2	1	no	—
22	C₂×Dic₂₇₈	—	2	1	no	—
23	Dic₅₅₆	—	2	1	no	—
24	Dic₅₅₆	—	2	1	no	—
25	Dic₅₅₆	—	2	1	no	—
26	Dic₅₅₆	D₄	1	2	yes	—
27	#14555775	—	4	2	no	—
28	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
29	Dic₁₃₉⋊C₄	—	4	1	no	—
30	Dic₁₃₉⋊C₄	—	4	1	no	—
31	Dic₁₃₉⋊C₄	—	4	1	no	—
32	C₄×Dic₁₃₉	—	4	1	no	—
33	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
34	C₁₃₉⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₁₃₉	—	2	1	no	—
36	Q₁₆×C₁₃₉	—	2	1	no	—
37	Q₁₆×C₁₃₉	—	2	1	no	—
38	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₃₉	—	2	1	no	—
40	C₁₆×C₁₃₉	—	2	1	no	—
41	Dic₂₇₈	—	2	2	no	—
42	Dic₂₇₈	—	4	1	no	—
43	Dic₂₇₈	—	4	1	no	—
44	Dic₂₇₈	—	4	1	no	—
45	C₂×Dic₁₃₉	—	4	1	no	—
46	C₂×Dic₁₃₉	—	4	1	no	—
47	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
48	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
49	C₁₃₉⋊C₈	—	2	1	no	—
50	Q₈×C₁₃₉	—	4	1	no	—
51	Q₈×C₁₃₉	—	4	1	no	—
52	C₂×C₄×C₁₃₉	—	4	1	no	—
53	C₂×C₄×C₁₃₉	—	4	1	no	—
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₃₉	#41	1	1	yes	characteristic
57	C₂²×C₁₃₉	C₂×D₈	1	1	yes	characteristic
58	C₄×C₁₃₉	—	8	1	no	—
59	C₄×C₁₃₉	—	8	1	no	—
60	C₄×C₁₃₉	#121	1	1	yes	characteristic
61	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₃₉	—	2	1	no	—
63	Dic₁₃₉	—	4	2	no	—
64	Dic₁₃₉	—	8	1	no	—
65	Dic₁₃₉	Q₃₂	1	2	yes	complemented
66	C₂×C₁₃₉	#459	1	1	yes	characteristic
67	C₂×C₁₃₉	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₁₃₉	#510	1	1	yes	characteristic
69	C₁₃₉	#2209	1	1	yes	characteristic, complemented, Sylow
70	#2209	—	139	1	no	maximal, Sylow
71	C₂×Q₃₂	—	139	1	no	—
72	C₂×Q₃₂	—	139	1	no	—
73	#494	—	139	1	no	—
74	C₄×Q₁₆	—	139	1	no	—
75	C₄⋊C₁₆	—	139	1	no	—
76	#358	—	139	1	no	—
77	#358	D₁₃₉	1	1	yes	characteristic
78	C₂×Q₁₆	—	278	1	no	—
79	C₂×Q₁₆	—	139	1	no	—
80	C₂×Q₁₆	—	139	1	no	—
81	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
82	C₄⋊Q₈	—	278	1	no	—
83	C₄×Q₈	—	278	1	no	—
84	Q₃₂	—	278	2	no	—
85	Q₃₂	—	278	2	no	—
86	C₂×C₁₆	—	139	1	no	—
87	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
88	#107	C₂×D₁₃₉	1	1	yes	characteristic
89	#103	—	278	1	no	—
90	C₄×C₈	—	139	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	278	1	no	—
93	C₂×Q₈	—	278	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	278	1	no	—
97	Q₁₆	—	278	1	no	—
98	Q₁₆	—	278	1	no	—
99	Q₁₆	—	556	2	no	—
100	Q₁₆	—	139	2	no	—
101	C₂×C₈	—	139	1	no	—
102	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	556	1	no	—
105	C₄⋊C₄	—	556	1	no	—
106	C₄⋊C₄	—	556	1	no	—
107	C₄²	—	556	1	no	—
108	C₄²	—	139	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	278	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	278	2	no	—
114	Q₈	—	556	1	no	—
115	Q₈	—	556	1	no	—
116	Q₈	—	556	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	556	1	no	—
120	C₂×C₄	—	556	1	no	—
121	C₂×C₄	—	139	1	no	—
122	C₂×C₄	—	139	1	no	—
123	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
124	C₈	—	278	1	no	—
125	C₈	D₄×D₁₃₉	1	1	yes	characteristic
126	C₈	#14555785	1	1	yes	characteristic
127	C₂²	D₈×D₁₃₉	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	278	1	no	—
131	C₄	—	556	2	no	—
132	C₄	—	1112	1	no	—
133	C₄	—	139	2	no	—
134	C₄	#17984815	1	1	yes	characteristic
135	C₄	#17984869	1	1	yes	characteristic
136	C₂	#22581147	1	1	yes	characteristic
137	C₂	Q₃₂×D₁₃₉	1	1	yes	characteristic
138	C₂	#22581392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₁₂

C₂

yes

maximal, characteristic

C₂×#17984788

C₂

yes

maximal, characteristic

#22581331

C₂

yes

maximal, characteristic

#22581047

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₁₆

C₂

yes

maximal, characteristic

#22580756

C₂

yes

maximal, characteristic

#358×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

C₂²

yes

characteristic

C₂×Dic₅₅₆

C₂²

yes

—

—

—

Dic₁₁₁₂

—

#17984788

—

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

#17984765

—

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₉

C₂²

yes

characteristic

#107×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

Dic₅₅₆

—

Dic₅₅₆

—

Dic₅₅₆

—

Dic₅₅₆

D₄

yes

—

#14555775

—

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₁₆

—

C₂×Q₈×C₁₃₉

—

Q₁₆×C₁₃₉

—

Q₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₁₆×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₈

yes

characteristic

C₂×Dic₁₃₉

D₈

yes

characteristic

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

#41

yes

characteristic

C₂²×C₁₃₉

C₂×D₈

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

Q₃₂

yes

complemented

C₂×C₁₃₉

#459

yes

characteristic

C₂×C₁₃₉

C₂×Q₃₂

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2209

—

139

maximal, Sylow

C₂×Q₃₂

—

139

—

C₂×Q₃₂

—

139

—

#494

—

139

—

C₄×Q₁₆

—

139

—

C₄⋊C₁₆

—

139

—

#358

—

139

—

#358

D₁₃₉

yes

characteristic

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

139

—

C₂×Q₁₆

—

139

—

C₂×Q₁₆

C₂×D₁₃₉

yes

characteristic

C₄⋊Q₈

—

278

—

C₄×Q₈

—

278

—

Q₃₂

—

278

—

Q₃₂

—

278

—

C₂×C₁₆

—

139

—

C₂×C₁₆

C₂×D₁₃₉

yes

characteristic

#107

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

C₄×C₈

—

139

—

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

278

—

Q₁₆

—

278

—

Q₁₆

—

278

—

Q₁₆

—

556

—

100

Q₁₆

—

139

—

101

C₂×C₈

—

139

—

102

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

556

—

105

C₄⋊C₄

—

556

—

106

C₄⋊C₄

—

556

—

107

C₄²

—

556

—

108

C₄²

—

139

—

109

C₁₆

—

110

C₁₆

—

278

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

278

—

114

Q₈

—

556

—

115

Q₈

—

556

—

116

Q₈

—

556

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

556

—

120

C₂×C₄

—

556

—

121

C₂×C₄

—

139

—

122

C₂×C₄

—

139

—

123

C₂×C₄

D₄×D₁₃₉

yes

characteristic

124

C₈

—

278

—

125

C₈

D₄×D₁₃₉

yes

characteristic

126

C₈

#14555785

yes

characteristic

127

C₂²

D₈×D₁₃₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

278

—

131

C₄

—

556

—

132

C₄

—

1112

—

133

C₄

—

139

—

134

C₄

#17984815

yes

characteristic

135

C₄

#17984869

yes

characteristic

136

C₂

#22581147

yes

characteristic

137

C₂

Q₃₂×D₁₃₉

yes

characteristic

138

C₂

#22581392

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₉
commutator, cosocle	C₂×C₈×C₁₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#358×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	C₂×D₈
hypercenter	#358	D₁₃₉
hypocenter	C₁₃₉	#2209
LCS	G ▷ C₂×C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₃₉

commutator, cosocle

C₂×C₈×C₁₃₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₉

Fitting

#358×C₁₃₉

C₂

socle, prime radical

C₂²×C₁₃₉

C₂×D₈

D₁₃₉

C₁₃₉

G ▷ C₂×C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₁₃₉	Q₃₂	semidirect
69. C₁₃₉	#2209	semidirect

H ≅ G/N

type

65. Dic₁₃₉

Q₃₂

semidirect

69. C₁₃₉

#2209

semidirect

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

subgroup	index
C₂×Dic₁₁₁₂	2
C₂×#17984788	2
#22581331	2
#22581047	2
Dic₁₃₉⋊C₁₆	2
#22580756	2
#358×C₁₃₉	2
#2209	139

subgroup

index

C₂×Dic₁₁₁₂

Dic₁₃₉⋊C₁₆

#358×C₁₃₉

139

Subgroups — #31808453

Conjugacy classes of subgroups

Complements

Maximal subgroups