Group #31806023 — subgroups

Subgroups — #31806023

Order 17792.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22581358	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581349	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581325	C₂	1	1	yes	maximal, characteristic
4	#22581305	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581299	C₂	1	1	yes	maximal, characteristic
6	#22580674	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555773	—	2	2	no	—
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984842	C₂²	1	1	yes	characteristic
11	#17984776	—	2	2	no	—
12	#17984773	—	2	2	no	—
13	#17984765	—	2	2	no	—
14	#17984763	—	2	2	no	—
15	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₃₉	—	2	2	no	—
18	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	#127×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₈	—	2	1	no	—
22	C₂×Dic₂₇₈	—	2	1	no	—
23	#14555773	—	4	4	no	—
24	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₃₉	—	2	1	no	—
27	C₄⋊Dic₁₃₉	—	2	1	no	—
28	C₄⋊Dic₁₃₉	—	2	1	no	—
29	C₄⋊Dic₁₃₉	—	2	1	no	—
30	C₂×Q₈×C₁₃₉	—	2	1	no	—
31	C₂×D₄×C₁₃₉	—	2	1	no	—
32	C₂×D₄×C₁₃₉	—	2	1	no	—
33	C₂×D₄×C₁₃₉	—	2	1	no	—
34	SD₁₆×C₁₃₉	—	4	4	no	—
35	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₃₉	—	2	1	no	—
37	C₄⋊C₄×C₁₃₉	—	2	1	no	—
38	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
39	Dic₂₇₈	—	4	2	no	—
40	Dic₂₇₈	—	4	2	no	—
41	C₂×Dic₁₃₉	—	4	1	no	—
42	C₂×Dic₁₃₉	—	4	1	no	—
43	C₁₃₉⋊C₈	—	2	2	no	—
44	C₁₃₉⋊C₈	—	2	2	no	—
45	C₂³×C₁₃₉	—	4	1	no	—
46	Q₈×C₁₃₉	—	4	2	no	—
47	D₄×C₁₃₉	—	4	2	no	—
48	D₄×C₁₃₉	—	4	2	no	—
49	D₄×C₁₃₉	—	4	2	no	—
50	C₂×C₄×C₁₃₉	—	4	1	no	—
51	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₃₉	SD₁₆	1	4	yes	complemented
55	C₂²×C₁₃₉	—	8	1	no	—
56	C₂²×C₁₃₉	—	8	1	no	—
57	C₂²×C₁₃₉	—	8	1	no	—
58	C₂²×C₁₃₉	#127	1	1	yes	characteristic
59	C₄×C₁₃₉	—	8	1	no	—
60	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
61	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₃₉	#137	1	2	yes	—
63	Dic₁₃₉	—	8	1	no	—
64	Dic₁₃₉	—	8	1	no	—
65	C₂×C₁₃₉	—	16	1	no	—
66	C₂×C₁₃₉	#492	1	1	yes	characteristic
67	C₂×C₁₃₉	#496	1	1	yes	characteristic
68	C₂×C₁₃₉	#497	1	1	yes	characteristic
69	C₁₃₉	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	139	1	no	maximal, Sylow
71	#500	—	139	1	no	—
72	#498	—	139	1	no	—
73	#492	—	139	1	no	—
74	#492	D₁₃₉	1	1	yes	characteristic
75	#487	—	139	1	no	—
76	#486	—	139	1	no	—
77	#322	—	139	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	278	2	no	—
80	C₄⋊Q₈	—	139	1	no	—
81	C₄⋊Q₈	—	139	1	no	—
82	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
83	#127	C₂×D₁₃₉	1	1	yes	characteristic
84	#107	—	278	2	no	—
85	#106	—	278	2	no	—
86	#103	—	278	2	no	—
87	#102	—	278	2	no	—
88	C₄×C₈	—	139	1	no	—
89	C₄×C₈	—	139	1	no	—
90	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	278	1	no	—
93	C₂×Q₈	—	278	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	556	4	no	—
99	C₂×C₈	—	139	2	no	—
100	C₂×C₈	—	139	2	no	—
101	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	278	1	no	—
105	C₄⋊C₄	—	278	1	no	—
106	C₄⋊C₄	—	278	1	no	—
107	C₄⋊C₄	—	278	1	no	—
108	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	556	2	no	—
112	Q₈	—	556	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	556	1	no	—
118	C₂×C₄	—	556	1	no	—
119	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
122	C₈	—	278	2	no	—
123	C₈	—	278	2	no	—
124	C₈	#14555773	1	4	yes	complemented
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#17984838	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	1112	1	no	—
131	C₄	—	1112	1	no	—
132	C₄	C₂×#14555773	1	2	yes	—
133	C₄	#17984869	1	2	yes	—
134	C₄	#17984871	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#22581325	1	1	yes	characteristic
137	C₂	#22581339	1	1	yes	characteristic
138	C₂	#22581344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581358

C₂

yes

maximal, characteristic, complemented

#22581349

C₂

yes

maximal, characteristic, complemented

#22581325

C₂

yes

maximal, characteristic

#22581305

C₂

yes

maximal, characteristic, complemented

#22581299

C₂

yes

maximal, characteristic

#22580674

C₂

yes

maximal, characteristic, complemented

#492×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555773

—

#17984842

C₂²

yes

characteristic

#17984842

C₂²

yes

—

—

—

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

#14555773

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

SD₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

SD₁₆

yes

complemented

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₉

#137

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1714

—

139

maximal, Sylow

#500

—

139

—

#498

—

139

—

#492

—

139

—

#492

D₁₃₉

yes

characteristic

#487

—

139

—

#486

—

139

—

#322

—

139

—

C₂×SD₁₆

—

C₂×SD₁₆

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#127

C₂×D₁₃₉

yes

characteristic

#107

—

278

—

#106

—

278

—

#103

—

278

—

#102

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

556

—

C₂×C₈

—

139

—

100

C₂×C₈

—

139

—

101

C₂×C₈

C₂²⋊D₁₃₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

278

—

107

C₄⋊C₄

—

278

—

108

C₄²

C₂²×D₁₃₉

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

556

—

112

Q₈

—

556

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

556

—

118

C₂×C₄

—

556

—

119

C₂×C₄

D₄×D₁₃₉

yes

characteristic

120

C₂×C₄

D₄×D₁₃₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

122

C₈

—

278

—

123

C₈

—

278

—

124

C₈

#14555773

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#17984838

yes

characteristic

129

C₄

—

130

C₄

—

1112

—

131

C₄

—

1112

—

132

C₄

C₂×#14555773

yes

—

133

C₄

#17984869

yes

—

134

C₄

#17984871

yes

—

135

C₂

—

136

C₂

#22581325

yes

characteristic

137

C₂

#22581339

yes

characteristic

138

C₂

#22581344

yes

characteristic

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#492×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
prime radical	#127×C₁₃₉	C₂²
hypercenter	#492	D₁₃₉
hypocenter	C₁₃₉	#1714
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#17984838

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#492×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#127

prime radical

#127×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #22581358	C₂	semidirect
2. #22581349	C₂	semidirect
4. #22581305	C₂	semidirect
6. #22580674	C₂	semidirect
54. C₈×C₁₃₉	SD₁₆	semidirect
69. C₁₃₉	#1714	semidirect
124. C₈	#14555773	semidirect

H ≅ G/N

type

1. #22581358

C₂

semidirect

2. #22581349

C₂

semidirect

4. #22581305

C₂

semidirect

6. #22580674

C₂

semidirect

54. C₈×C₁₃₉

SD₁₆

semidirect

69. C₁₃₉

#1714

semidirect

124. C₈

#14555773

semidirect

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

subgroup	index
#22581358	2
#22581349	2
#22581325	2
#22581305	2
#22581299	2
#22580674	2
#492×C₁₃₉	2
#1714	139

subgroup

index

#492×C₁₃₉

139

Subgroups — #31806023

Conjugacy classes of subgroups

Complements

Maximal subgroups