Group #31808648 — subgroups

Subgroups — #31808648

Order 17792.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22581354	C₂	1	1	yes	maximal, characteristic
2	#22581331	C₂	1	1	yes	maximal, characteristic
3	#22580758	C₂	1	1	yes	maximal, characteristic
4	#22580757	C₂	1	1	yes	maximal, characteristic
5	#22580756	C₂	1	1	yes	maximal, characteristic
6	#22580728	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×#14555775	—	2	1	no	—
10	#17984840	—	2	1	no	—
11	#17984833	—	2	1	no	—
12	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#17984776	C₂²	1	1	yes	characteristic
14	#17984773	—	2	1	no	—
15	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	#107×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₈	—	2	1	no	—
20	Dic₅₅₆	—	2	1	no	—
21	Dic₅₅₆	—	2	1	no	—
22	#14555775	—	4	2	no	—
23	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₃₉	—	2	1	no	—
25	C₄⋊Dic₁₃₉	—	4	1	no	—
26	Dic₁₃₉⋊C₄	—	4	1	no	—
27	Dic₁₃₉⋊C₄	—	4	1	no	—
28	Dic₁₃₉⋊C₄	—	4	1	no	—
29	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
30	C₁₃₉⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₃₉	—	2	1	no	—
32	Q₁₆×C₁₃₉	—	2	1	no	—
33	Q₁₆×C₁₃₉	—	2	1	no	—
34	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₉	—	2	1	no	—
36	C₁₆×C₁₃₉	—	2	1	no	—
37	Dic₂₇₈	—	4	1	no	—
38	Dic₂₇₈	—	4	1	no	—
39	C₂×Dic₁₃₉	—	4	1	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
42	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
43	C₁₃₉⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₃₉	—	4	1	no	—
45	Q₈×C₁₃₉	—	4	1	no	—
46	C₂×C₄×C₁₃₉	—	4	1	no	—
47	C₂×C₄×C₁₃₉	—	4	1	no	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₉	#41	1	1	yes	characteristic
50	C₈×C₁₃₉	#41	1	1	yes	characteristic
51	C₂²×C₁₃₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₃₉	—	8	1	no	—
53	C₄×C₁₃₉	—	8	1	no	—
54	C₄×C₁₃₉	#124	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₃₉	—	2	1	no	—
57	Dic₁₃₉	—	2	1	no	—
58	Dic₁₃₉	—	8	1	no	—
59	Dic₁₃₉	—	8	1	no	—
60	C₂×C₁₃₉	#486	1	1	yes	characteristic
61	C₂×C₁₃₉	#510	1	1	yes	characteristic
62	C₂×C₁₃₉	#510	1	1	yes	characteristic
63	C₁₃₉	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	139	1	no	maximal, Sylow
65	#499	—	139	1	no	—
66	#494	—	139	1	no	—
67	#358	—	139	1	no	—
68	#358	—	139	1	no	—
69	#358	—	139	1	no	—
70	#358	D₁₃₉	1	1	yes	characteristic
71	#346	—	139	1	no	—
72	C₂×Q₁₆	—	278	1	no	—
73	C₂×Q₁₆	—	139	1	no	—
74	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
75	C₄⋊Q₈	—	278	1	no	—
76	#125	—	278	1	no	—
77	C₂×C₁₆	—	139	1	no	—
78	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
79	#107	—	139	1	no	—
80	#107	C₂×D₁₃₉	1	1	yes	characteristic
81	#106	—	278	1	no	—
82	C₄×C₈	—	139	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	278	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	278	1	no	—
88	Q₁₆	—	278	1	no	—
89	Q₁₆	—	556	2	no	—
90	C₂×C₈	—	139	1	no	—
91	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	278	1	no	—
94	C₄⋊C₄	—	556	1	no	—
95	C₄⋊C₄	—	556	1	no	—
96	C₄⋊C₄	—	556	1	no	—
97	C₄⋊C₄	—	556	1	no	—
98	C₄²	—	139	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	278	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	556	1	no	—
104	Q₈	—	556	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	556	1	no	—
108	C₂×C₄	—	556	1	no	—
109	C₂×C₄	—	139	1	no	—
110	C₂×C₄	—	139	1	no	—
111	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
112	C₈	—	139	2	no	—
113	C₈	#14555784	1	1	yes	characteristic
114	C₈	#14555785	1	1	yes	characteristic
115	C₂²	D₈×D₁₃₉	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	278	1	no	—
119	C₄	—	278	1	no	—
120	C₄	—	1112	1	no	—
121	C₄	—	1112	1	no	—
122	C₄	#17984828	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₃₉	1	1	yes	characteristic
124	C₂	#22581298	1	1	yes	characteristic
125	C₂	#22581392	1	1	yes	characteristic
126	C₂	#22581395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581354

C₂

yes

maximal, characteristic

#22581331

C₂

yes

maximal, characteristic

#22580758

C₂

yes

maximal, characteristic

#22580757

C₂

yes

maximal, characteristic

#22580756

C₂

yes

maximal, characteristic

#22580728

C₂

yes

maximal, characteristic

#358×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

C₂²

yes

—

—

—

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

#17984776

C₂²

yes

characteristic

#17984773

—

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₉

C₂²

yes

characteristic

#107×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

Dic₅₅₆

—

Dic₅₅₆

—

#14555775

—

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

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

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₈

yes

characteristic

C₂×Dic₁₃₉

D₈

yes

characteristic

C₁₃₉⋊C₈

SD₁₆

yes

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

C₈×C₁₃₉

yes

C₈×C₁₃₉

yes

C₂²×C₁₃₉

C₂×D₈

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#124

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2245

—

139

maximal, Sylow

#499

—

139

—

#494

—

139

—

#358

—

139

—

#358

—

139

—

#358

—

139

—

#358

D₁₃₉

yes

characteristic

#346

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

139

—

C₂×Q₁₆

C₂×D₁₃₉

yes

characteristic

C₄⋊Q₈

—

278

—

#125

—

278

—

C₂×C₁₆

—

139

—

C₂×C₁₆

C₂×D₁₃₉

yes

characteristic

#107

—

139

—

#107

C₂×D₁₃₉

yes

characteristic

#106

—

278

—

C₄×C₈

—

139

—

C₂×Q₈

—

C₂×Q₈

—

278

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

278

—

Q₁₆

—

278

—

Q₁₆

—

556

—

C₂×C₈

—

139

—

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

556

—

C₄⋊C₄

—

556

—

C₄⋊C₄

—

556

—

C₄⋊C₄

—

556

—

C₄²

—

139

—

C₁₆

—

100

C₁₆

—

278

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

556

—

104

Q₈

—

556

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

556

—

108

C₂×C₄

—

556

—

109

C₂×C₄

—

139

—

110

C₂×C₄

—

139

—

111

C₂×C₄

D₄×D₁₃₉

yes

characteristic

112

C₈

—

139

—

113

C₈

#14555784

yes

characteristic

114

C₈

#14555785

yes

characteristic

115

C₂²

D₈×D₁₃₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

278

—

119

C₄

—

278

—

120

C₄

—

1112

—

121

C₄

—

1112

—

122

C₄

#17984828

yes

characteristic

123

C₄

SD₁₆×D₁₃₉

yes

characteristic

124

C₂

#22581298

yes

characteristic

125

C₂

#22581392

yes

characteristic

126

C₂

#22581395

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	#2245
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
63. C₁₃₉	#2245	semidirect

H ≅ G/N

type

63. C₁₃₉

#2245

semidirect

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

subgroup	index
#22581354	2
#22581331	2
#22580758	2
#22580757	2
#22580756	2
#22580728	2
#358×C₁₃₉	2
#2245	139

subgroup

index

#358×C₁₃₉

139

Subgroups — #31808648

Conjugacy classes of subgroups

Complements

Maximal subgroups