Group #31817798 — subgroups

Subgroups — #31817798

Order 17792.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#22581510	C₂	1	2	yes	maximal
2	#22581239	C₂	1	4	yes	maximal, complemented
3	#22581211	C₂	1	4	yes	maximal
4	#22581164	C₂	1	4	yes	maximal, complemented
5	#423×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₆	C₂²	1	4	yes	—
7	#17984842	C₂²	1	2	yes	—
8	#17984842	C₂²	1	2	yes	—
9	#17984834	—	2	2	no	—
10	#17984821	—	2	2	no	—
11	#17984821	C₂²	1	4	yes	—
12	C₄×Dic₂₇₈	—	2	2	no	—
13	C₄×Dic₂₇₈	C₂²	1	4	yes	—
14	#17984801	—	2	2	no	—
15	#17984774	C₂²	1	4	yes	—
16	#17984767	C₂²	1	8	yes	—
17	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	C₂³	1	4	yes	—
22	Dic₅₅₆	—	4	4	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₄⋊Dic₁₃₉	—	2	2	no	—
25	C₄⋊Dic₁₃₉	—	2	2	no	—
26	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
27	Dic₁₃₉⋊C₄	—	2	4	no	—
28	Dic₁₃₉⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₃₉	—	2	4	no	—
30	C₄×Dic₁₃₉	—	2	4	no	—
31	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₃₉	—	4	2	no	—
34	C₂×C₈×C₁₃₉	C₂³	1	4	yes	—
35	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₃₉	C₂³	1	2	yes	—
38	Dic₂₇₈	—	2	4	no	—
39	Dic₂₇₈	—	2	4	no	—
40	Dic₂₇₈	—	4	4	no	—
41	C₂×Dic₁₃₉	—	2	4	no	—
42	C₂×Dic₁₃₉	—	2	4	no	—
43	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	—	2	4	no	—
45	C₂×C₄×C₁₃₉	—	2	1	no	—
46	C₂×C₄×C₁₃₉	—	2	1	no	—
47	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₃₉	—	2	4	no	—
52	C₂²×C₁₃₉	—	2	1	no	—
53	C₂²×C₁₃₉	—	2	2	no	—
54	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	—	2	2	no	—
57	C₄×C₁₃₉	#137	1	4	yes	—
58	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₃₉	—	4	4	no	—
61	Dic₁₃₉	—	4	4	no	—
62	C₂×C₁₃₉	—	4	1	no	—
63	C₂×C₁₃₉	#537	1	1	yes	characteristic
64	C₂×C₁₃₉	C₂×#137	1	2	yes	—
65	C₁₃₉	#3255	1	1	yes	characteristic, complemented, Sylow
66	#3255	—	139	1	no	maximal, Sylow
67	#533	—	139	2	no	—
68	#475	—	139	4	no	—
69	#470	—	139	4	no	—
70	#462	—	139	4	no	—
71	#423	D₁₃₉	1	1	yes	characteristic
72	C₂×Q₁₆	—	139	4	no	—
73	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
74	C₄⋊Q₈	—	139	2	no	—
75	C₄⋊Q₈	—	139	2	no	—
76	#125	—	278	2	no	—
77	C₂²⋊Q₈	—	278	2	no	—
78	C₂²⋊Q₈	—	139	4	no	—
79	C₄×Q₈	—	278	2	no	—
80	C₄×Q₈	—	139	4	no	—
81	#117	—	278	2	no	—
82	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
83	#106	—	139	4	no	—
84	C₄⋊C₈	C₂×D₁₃₉	1	4	yes	—
85	#103	—	139	8	no	—
86	C₂×Q₈	—	278	2	no	—
87	C₂×Q₈	—	139	4	no	—
88	C₂²×C₄	D₅₅₆	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
90	Q₁₆	—	556	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₁₃₉	1	4	yes	—
93	C₄⋊C₄	—	278	4	no	—
94	C₄⋊C₄	—	278	4	no	—
95	C₄⋊C₄	—	278	2	no	—
96	C₄⋊C₄	—	278	2	no	—
97	C₄⋊C₄	—	278	2	no	—
98	C₄⋊C₄	—	139	4	no	—
99	C₂²⋊C₄	—	278	4	no	—
100	C₄²	—	278	4	no	—
101	C₄²	D₅₅₆	1	1	yes	characteristic
102	C₄²	D₅₅₆	1	1	yes	characteristic
103	C₄²	C₂²×D₁₃₉	1	2	yes	—
104	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
105	Q₈	—	278	4	no	—
106	Q₈	—	278	4	no	—
107	Q₈	—	556	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	278	4	no	—
112	C₂×C₄	—	278	4	no	—
113	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
114	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
115	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
116	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	556	4	no	—
124	C₄	—	556	4	no	—
125	C₄	#17984875	1	4	yes	—
126	C₄	#17984887	1	1	yes	characteristic
127	C₄	#17984890	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#22581534	1	1	yes	characteristic
130	C₂	C₂×#17984875	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581510

C₂

yes

maximal

#22581239

C₂

yes

maximal, complemented

#22581211

C₂

yes

maximal

#22581164

C₂

yes

maximal, complemented

#423×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

C₂²

yes

—

#17984842

C₂²

yes

—

#17984842

C₂²

yes

—

#17984834

—

#17984821

—

#17984821

C₂²

yes

—

C₄×Dic₂₇₈

—

C₄×Dic₂₇₈

C₂²

yes

—

#17984801

—

#17984774

C₂²

yes

—

#17984767

C₂²

yes

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

Dic₅₅₆

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#137

yes

—

C₄×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#537

yes

characteristic

C₂×C₁₃₉

C₂×#137

yes

—

C₁₃₉

#3255

yes

characteristic, complemented, Sylow

#3255

—

139

maximal, Sylow

#533

—

139

—

#475

—

139

—

#470

—

139

—

#462

—

139

—

#423

D₁₃₉

yes

characteristic

C₂×Q₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

278

—

C₂²⋊Q₈

—

278

—

C₂²⋊Q₈

—

139

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

#117

—

278

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#106

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

#103

—

139

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

Q₁₆

—

556

—

M₁₆

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

100

C₄²

—

278

—

101

C₄²

D₅₅₆

yes

characteristic

102

C₄²

D₅₅₆

yes

characteristic

103

C₄²

C₂²×D₁₃₉

yes

—

104

C₂³

C₂×D₅₅₆

yes

characteristic

105

Q₈

—

278

—

106

Q₈

—

278

—

107

Q₈

—

556

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

278

—

113

C₂×C₄

C₂×D₅₅₆

yes

characteristic

114

C₂×C₄

C₂×D₅₅₆

yes

—

115

C₂×C₄

C₂×D₅₅₆

yes

—

116

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₅₅₆

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

556

—

124

C₄

—

556

—

125

C₄

#17984875

yes

—

126

C₄

#17984887

yes

characteristic

127

C₄

#17984890

yes

characteristic

128

C₂

—

129

C₂

#22581534

yes

characteristic

130

C₂

C₂×#17984875

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#423×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#423	D₁₃₉
hypocenter	C₁₃₉	#3255
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#423×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #22581239	C₂	semidirect
4. #22581164	C₂	semidirect
65. C₁₃₉	#3255	semidirect

H ≅ G/N

type

2. #22581239

C₂

semidirect

4. #22581164

C₂

semidirect

65. C₁₃₉

#3255

semidirect

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

subgroup	index
#22581510	2
#22581239	2
#22581211	2
#22581164	2
#423×C₁₃₉	2
#3255	139

subgroup

index

#423×C₁₃₉

139

Subgroups — #31817798

Conjugacy classes of subgroups

Complements

Maximal subgroups