Group #31806623 — subgroups

Subgroups — #31806623

Order 17792.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22581720	C₂	1	1	yes	maximal, characteristic
2	#22581001	C₂	1	2	yes	maximal, complemented
3	#22580937	C₂	1	1	yes	maximal, characteristic
4	#22580718	C₂	1	2	yes	maximal, complemented
5	#408×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984849	—	2	4	no	—
7	C₂×#14555769	—	2	2	no	—
8	C₂×#14555769	C₂²	1	2	yes	—
9	C₂×#14555769	C₂²	1	2	yes	—
10	#17984777	—	2	4	no	—
11	#17984761	C₄	1	2	yes	—
12	#17984759	C₄	1	2	yes	—
13	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
17	#14555769	—	2	4	no	—
18	#14555769	—	2	2	no	—
19	#14555769	—	2	2	no	—
20	#14555769	C₂×C₄	1	4	yes	—
21	#14555769	D₄	1	4	yes	—
22	C₂×C₁₃₉⋊C₈	—	2	4	no	—
23	C₂×C₁₃₉⋊C₈	—	2	2	no	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	#41×C₁₃₉	—	2	2	no	—
26	#41×C₁₃₉	—	2	2	no	—
27	C₂×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₃₉	Q₈	1	2	yes	—
30	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
31	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₃₉	—	2	2	no	—
33	C₂×C₈×C₁₃₉	—	2	1	no	—
34	C₂×C₈×C₁₃₉	—	2	1	no	—
35	C₂×C₈×C₁₃₉	—	2	2	no	—
36	C₁₃₉⋊C₈	—	2	4	no	—
37	C₁₃₉⋊C₈	—	2	4	no	—
38	C₂³×C₁₃₉	C₄⋊C₄	1	2	yes	—
39	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
40	Q₈×C₁₃₉	—	2	2	no	—
41	D₄×C₁₃₉	—	2	4	no	—
42	D₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	#41	1	2	yes	—
50	C₈×C₁₃₉	—	4	1	no	—
51	C₈×C₁₃₉	—	4	1	no	—
52	C₂²×C₁₃₉	—	2	2	no	—
53	C₂²×C₁₃₉	—	4	1	no	—
54	C₂²×C₁₃₉	—	4	2	no	—
55	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
59	C₄×C₁₃₉	#120	1	1	yes	characteristic
60	C₄×C₁₃₉	#123	1	1	yes	characteristic
61	C₂×C₁₃₉	—	2	1	no	—
62	C₂×C₁₃₉	—	2	2	no	—
63	C₂×C₁₃₉	—	4	2	no	—
64	C₂×C₁₃₉	#385	1	1	yes	characteristic
65	C₁₃₉	#1858	1	1	yes	characteristic, complemented, Sylow
66	#1858	—	139	1	no	maximal, Sylow
67	#568	—	139	1	no	—
68	#430	—	139	2	no	—
69	#413	—	139	1	no	—
70	#408	D₁₃₉	1	1	yes	characteristic
71	#341	—	139	2	no	—
72	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
73	#131	—	278	4	no	—
74	C₂×M₁₆	—	278	2	no	—
75	C₂×M₁₆	—	139	2	no	—
76	C₂×M₁₆	—	139	2	no	—
77	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
79	#108	—	278	4	no	—
80	#101	—	139	2	no	—
81	#100	—	139	2	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₃₉	1	1	yes	characteristic
86	C₂×D₄	C₄×D₁₃₉	1	1	yes	characteristic
87	C₂×D₄	Dic₂₇₈	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	278	4	no	—
92	M₁₆	—	278	2	no	—
93	M₁₆	—	278	2	no	—
94	M₁₆	—	139	4	no	—
95	M₁₆	—	139	4	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	278	4	no	—
100	C₂×C₈	—	278	2	no	—
101	C₂×C₈	—	278	2	no	—
102	C₂³	Dic₁₃₉⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₂₇₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
113	C₂×C₄	#14555784	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	278	4	no	—
117	C₈	—	278	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
122	C₂²	#17984819	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
125	C₄	#17984810	1	1	yes	characteristic
126	C₄	#17984823	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#22580825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581720

C₂

yes

maximal, characteristic

#22581001

C₂

yes

maximal, complemented

#22580937

C₂

yes

maximal, characteristic

#22580718

C₂

yes

maximal, complemented

#408×C₁₃₉

C₂

yes

maximal, characteristic

#17984849

—

C₂×#14555769

—

C₂×#14555769

C₂²

yes

—

C₂×#14555769

C₂²

yes

—

#17984777

—

#17984761

C₄

yes

—

#17984759

C₄

yes

—

C₂×#41×C₁₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₈×C₁₃₉

—

—

—

—

C₂×C₄

yes

—

#14555769

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

#41×C₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₄⋊C₄

yes

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#385

yes

characteristic

C₁₃₉

#1858

yes

characteristic, complemented, Sylow

#1858

—

139

maximal, Sylow

#568

—

139

—

#430

—

139

—

#413

—

139

—

#408

D₁₃₉

yes

characteristic

#341

—

139

—

C₂×#41

C₂×D₁₃₉

yes

characteristic

#131

—

278

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

#101

—

139

—

#100

—

139

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₃₉

yes

characteristic

C₂×D₄

C₄×D₁₃₉

yes

characteristic

C₂×D₄

Dic₂₇₈

yes

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

139

—

M₁₆

—

139

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

100

C₂×C₈

—

278

—

101

C₂×C₈

—

278

—

102

C₂³

Dic₁₃₉⋊C₄

yes

—

103

C₂³

C₂×Dic₂₇₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

111

C₂×C₄

D₄×D₁₃₉

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

113

C₂×C₄

#14555784

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

278

—

117

C₈

—

278

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

122

C₂²

#17984819

yes

—

123

C₄

—

124

C₄

(C₂²⋊D₁₃₉)⋊C₄

yes

—

125

C₄

#17984810

yes

characteristic

126

C₄

#17984823

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#22580825

yes

characteristic

	subgroup	G/H
center	C₄	#17984810
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#408×C₁₃₉	C₂
socle	C₂×C₁₃₉	#385
prime radical	C₂×#41×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#408	D₁₃₉
hypocenter	C₁₃₉	#1858
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#17984810

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#408×C₁₃₉

C₂

socle

C₂×C₁₃₉

#385

prime radical

C₂×#41×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #22581001	C₂	semidirect
4. #22580718	C₂	semidirect
65. C₁₃₉	#1858	semidirect

H ≅ G/N

type

2. #22581001

C₂

semidirect

4. #22580718

C₂

semidirect

65. C₁₃₉

#1858

semidirect

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

subgroup	index
#22581720	2
#22581001	2
#22580937	2
#22580718	2
#408×C₁₃₉	2
#1858	139

subgroup

index

#408×C₁₃₉

139

Subgroups — #31806623

Conjugacy classes of subgroups

Complements

Maximal subgroups