Group #31805456 — subgroups

Subgroups — #31805456

Order 17792.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22581588	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581145	C₂	1	2	yes	maximal, complemented
3	#22580697	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580682	C₂	1	2	yes	maximal, complemented
5	#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₃₉	—	2	2	no	—
7	C₂×#14555770	—	2	2	no	—
8	#17984813	—	2	2	no	—
9	#17984810	—	2	2	no	—
10	C₄×D₅₅₆	C₂²	1	2	yes	—
11	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	complemented
12	#17984763	—	2	2	no	—
13	C₂×D₈×C₁₃₉	—	2	2	no	—
14	#127×C₁₃₉	C₂²	1	1	yes	characteristic
15	#127×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₁₃₉	—	2	2	no	—
18	C₂×D₅₅₆	D₄	1	2	yes	—
19	C₂×C₂²⋊D₁₃₉	—	2	4	no	—
20	D₄×D₁₃₉	—	4	4	no	—
21	D₄×D₁₃₉	—	4	4	no	—
22	C₂×C₄×D₁₃₉	—	2	2	no	—
23	#14555770	—	4	4	no	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
26	C₂²⋊Dic₁₃₉	—	2	2	no	—
27	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
28	C₂×D₄×C₁₃₉	—	2	2	no	—
29	C₂×D₄×C₁₃₉	—	2	2	no	—
30	C₂×D₄×C₁₃₉	—	2	1	no	—
31	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
32	D₈×C₁₃₉	—	4	4	no	—
33	C₂×C₈×C₁₃₉	—	2	1	no	—
34	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₂²×D₁₃₉	—	2	2	no	—
36	C₂²×D₁₃₉	—	4	8	no	—
37	C₂²×D₁₃₉	—	8	2	no	—
38	D₅₅₆	—	2	2	no	—
39	D₅₅₆	D₈	1	4	yes	complemented
40	C₂²⋊D₁₃₉	—	4	8	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₄×D₁₃₉	—	4	2	no	—
43	C₄×D₁₃₉	—	4	2	no	—
44	C₁₃₉⋊C₈	—	4	2	no	—
45	C₂³×C₁₃₉	—	2	2	no	—
46	C₂³×C₁₃₉	—	4	1	no	—
47	D₄×C₁₃₉	—	4	2	no	—
48	D₄×C₁₃₉	—	4	2	no	—
49	D₄×C₁₃₉	—	4	2	no	—
50	D₄×C₁₃₉	—	4	2	no	—
51	D₄×C₁₃₉	—	4	2	no	—
52	D₄×C₁₃₉	—	4	2	no	—
53	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₃₉	—	2	2	no	—
56	C₂²×C₁₃₉	—	4	4	no	—
57	C₂²×C₁₃₉	—	4	2	no	—
58	C₂²×C₁₃₉	—	8	2	no	—
59	C₂²×C₁₃₉	—	8	1	no	—
60	C₂²×C₁₃₉	#120	1	1	yes	characteristic
61	C₂×D₁₃₉	—	2	4	no	—
62	C₂×D₁₃₉	—	4	2	no	—
63	C₂×D₁₃₉	—	4	2	no	—
64	C₂×D₁₃₉	—	8	8	no	—
65	C₄×C₁₃₉	—	2	1	no	—
66	C₄×C₁₃₉	C₂×D₈	1	2	yes	—
67	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₁₃₉	—	4	2	no	—
69	C₂×C₁₃₉	—	8	2	no	—
70	C₂×C₁₃₉	—	16	1	no	—
71	C₂×C₁₃₉	#447	1	2	yes	—
72	C₂×C₁₃₉	#453	1	1	yes	characteristic
73	D₁₃₉	—	4	4	no	—
74	C₁₃₉	#1619	1	1	yes	characteristic, complemented, Sylow
75	#1619	—	139	1	no	maximal, Sylow
76	D₄²	—	139	1	no	—
77	#493	D₁₃₉	1	1	yes	characteristic, complemented
78	#459	—	139	2	no	—
79	#331	—	139	1	no	—
80	D₄⋊C₈	—	139	2	no	—
81	C₂²×D₄	—	278	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	278	2	no	—
84	#127	C₂×D₁₃₉	1	1	yes	characteristic
85	#127	C₂×D₁₃₉	1	1	yes	characteristic, complemented
86	#121	—	278	2	no	—
87	#120	—	278	2	no	—
88	C₄×D₄	—	139	2	no	—
89	C₄⋊C₈	—	139	2	no	—
90	#102	—	278	2	no	—
91	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
92	C₂⁴	—	278	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	278	4	no	—
97	C₂×D₄	—	556	4	no	—
98	C₂×D₄	—	556	4	no	—
99	C₂×D₄	—	139	2	no	—
100	C₂×D₄	C₂²⋊D₁₃₉	1	2	yes	—
101	C₂²×C₄	—	278	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	556	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	278	2	no	—
106	C₄⋊C₄	—	139	2	no	—
107	C₂²⋊C₄	—	278	2	no	—
108	C₂²⋊C₄	—	278	2	no	—
109	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	278	2	no	—
113	C₂³	—	556	8	no	—
114	C₂³	—	1112	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	278	2	no	—
122	D₄	—	556	8	no	—
123	D₄	—	139	4	no	—
124	C₂×C₄	—	278	2	no	—
125	C₂×C₄	—	556	2	no	—
126	C₂×C₄	—	556	2	no	—
127	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	556	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	278	4	no	—
136	C₂²	—	556	2	no	—
137	C₂²	—	556	2	no	—
138	C₂²	—	1112	8	no	—
139	C₂²	#17984810	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	556	2	no	—
142	C₄	D₈×D₁₃₉	1	2	yes	—
143	C₄	#17984866	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	556	4	no	—
147	C₂	#22581085	1	2	yes	—
148	C₂	#22581120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581588

C₂

yes

maximal, characteristic, complemented

#22581145

C₂

yes

maximal, complemented

#22580697

C₂

yes

maximal, characteristic, complemented

#22580682

C₂

yes

maximal, complemented

#493×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₃₉

—

C₂×#14555770

—

#17984813

—

#17984810

—

C₄×D₅₅₆

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

complemented

#17984763

—

C₂×D₈×C₁₃₉

—

#127×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic, complemented

C₄×C₈×C₁₃₉

C₂²

yes

characteristic, complemented

C₂³×D₁₃₉

—

C₂×D₅₅₆

D₄

yes

—

C₂×C₂²⋊D₁₃₉

—

D₄×D₁₃₉

—

D₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

#14555770

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₂²⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

D₄

yes

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

D₈

yes

complemented

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₈

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#447

yes

—

C₂×C₁₃₉

#453

yes

characteristic

D₁₃₉

—

C₁₃₉

#1619

yes

characteristic, complemented, Sylow

#1619

—

139

maximal, Sylow

D₄²

—

139

—

#493

D₁₃₉

yes

characteristic, complemented

#459

—

139

—

#331

—

139

—

D₄⋊C₈

—

139

—

C₂²×D₄

—

278

—

C₂×D₈

—

C₂×D₈

—

278

—

#127

C₂×D₁₃₉

yes

characteristic

#127

C₂×D₁₃₉

yes

characteristic, complemented

#121

—

278

—

#120

—

278

—

C₄×D₄

—

139

—

C₄⋊C₈

—

139

—

#102

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic, complemented

C₂⁴

—

278

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

278

—

C₂×D₄

—

556

—

C₂×D₄

—

556

—

C₂×D₄

—

139

—

100

C₂×D₄

C₂²⋊D₁₃₉

yes

—

101

C₂²×C₄

—

278

—

102

D₈

—

103

D₈

—

556

—

104

C₂×C₈

—

105

C₂×C₈

—

278

—

106

C₄⋊C₄

—

139

—

107

C₂²⋊C₄

—

278

—

108

C₂²⋊C₄

—

278

—

109

C₄²

C₂²×D₁₃₉

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

278

—

113

C₂³

—

556

—

114

C₂³

—

1112

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

278

—

122

D₄

—

556

—

123

D₄

—

139

—

124

C₂×C₄

—

278

—

125

C₂×C₄

—

556

—

126

C₂×C₄

—

556

—

127

C₂×C₄

D₄×D₁₃₉

yes

—

128

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

129

C₈

—

130

C₈

—

556

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

278

—

136

C₂²

—

556

—

137

C₂²

—

556

—

138

C₂²

—

1112

—

139

C₂²

#17984810

yes

characteristic

140

C₄

—

141

C₄

—

556

—

142

C₄

D₈×D₁₃₉

yes

—

143

C₄

#17984866

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

556

—

147

C₂

#22581085

yes

—

148

C₂

#22581120

yes

characteristic

	subgroup	G/H
center	C₂²	#17984810
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#493×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
hypercenter	#493	D₁₃₉
hypocenter	C₁₃₉	#1619
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#17984810

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#493×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22581588	C₂	semidirect
2. #22581145	C₂	semidirect
3. #22580697	C₂	semidirect
4. #22580682	C₂	semidirect
5. #493×C₁₃₉	C₂	semidirect
11. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
15. #127×C₁₃₉	C₂²	semidirect
16. C₄×C₈×C₁₃₉	C₂²	semidirect
39. D₅₅₆	D₈	semidirect
74. C₁₃₉	#1619	semidirect
77. #493	D₁₃₉	semidirect
85. #127	C₂×D₁₃₉	semidirect
91. C₄×C₈	C₂×D₁₃₉	semidirect

H ≅ G/N

type

1. #22581588

C₂

semidirect

2. #22581145

C₂

semidirect

3. #22580697

C₂

semidirect

4. #22580682

C₂

semidirect

5. #493×C₁₃₉

C₂

semidirect

11. C₄⋊(C₁₃₉⋊C₈)

C₂²

semidirect

15. #127×C₁₃₉

C₂²

semidirect

16. C₄×C₈×C₁₃₉

C₂²

semidirect

39. D₅₅₆

D₈

semidirect

74. C₁₃₉

#1619

semidirect

77. #493

D₁₃₉

semidirect

85. #127

C₂×D₁₃₉

semidirect

91. C₄×C₈

C₂×D₁₃₉

semidirect

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

subgroup	index
#22581588	2
#22581145	2
#22580697	2
#22580682	2
#493×C₁₃₉	2
#1619	139

subgroup

index

#493×C₁₃₉

139

Subgroups — #31805456

Conjugacy classes of subgroups

Complements

Maximal subgroups