Group #31805797 — subgroups

Subgroups — #31805797

Order 17792.

156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22581356	C₂	1	1	yes	maximal, characteristic
2	#22581164	C₂	1	1	yes	maximal, characteristic
3	#22581160	C₂	1	1	yes	maximal, characteristic
4	#22581077	C₂	1	1	yes	maximal, characteristic
5	#22580705	C₂	1	1	yes	maximal, characteristic
6	#22580698	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₅₆	—	2	1	no	—
9	C₂×#14555775	—	2	1	no	—
10	#17984842	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
12	#17984775	—	2	1	no	—
13	#17984775	—	2	1	no	—
14	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₁₃₉⋊C₈	—	2	1	no	—
16	#17984767	—	2	1	no	—
17	#17984766	—	2	1	no	—
18	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₃₉	—	2	1	no	—
20	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₇₈	—	2	1	no	—
24	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
25	Dic₅₅₆	—	4	2	no	—
26	#14555775	—	4	2	no	—
27	C₂×C₁₃₉⋊C₈	—	2	1	no	—
28	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₃₉	—	2	1	no	—
31	C₄⋊Dic₁₃₉	—	2	1	no	—
32	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
33	Dic₁₃₉⋊C₄	—	2	1	no	—
34	C₄×Dic₁₃₉	—	2	1	no	—
35	C₂×Q₈×C₁₃₉	—	2	1	no	—
36	C₂×C₈×C₁₃₉	—	2	1	no	—
37	C₂×C₈×C₁₃₉	—	2	1	no	—
38	C₄⋊C₄×C₁₃₉	—	2	1	no	—
39	C₄⋊C₄×C₁₃₉	—	2	1	no	—
40	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
41	Dic₂₇₈	—	2	1	no	—
42	Dic₂₇₈	—	4	1	no	—
43	Dic₂₇₈	—	4	1	no	—
44	Dic₂₇₈	D₈	1	2	yes	—
45	C₂×Dic₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	2	1	no	—
47	C₂×Dic₁₃₉	—	4	1	no	—
48	C₁₃₉⋊C₈	—	2	1	no	—
49	C₁₃₉⋊C₈	—	4	1	no	—
50	C₁₃₉⋊C₈	Q₁₆	1	2	yes	complemented
51	Q₈×C₁₃₉	—	4	1	no	—
52	Q₈×C₁₃₉	—	4	1	no	—
53	C₂×C₄×C₁₃₉	—	4	1	no	—
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
57	C₈×C₁₃₉	—	4	1	no	—
58	C₈×C₁₃₉	—	4	1	no	—
59	C₂²×C₁₃₉	#121	1	1	yes	characteristic
60	C₄×C₁₃₉	—	2	1	no	—
61	C₄×C₁₃₉	—	8	1	no	—
62	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
63	C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₃₉	#135	1	1	yes	characteristic
65	C₄×C₁₃₉	#137	1	1	yes	characteristic
66	Dic₁₃₉	—	2	2	no	—
67	Dic₁₃₉	—	4	1	no	—
68	Dic₁₃₉	—	8	1	no	—
69	C₂×C₁₃₉	#462	1	1	yes	characteristic
70	C₂×C₁₃₉	#466	1	1	yes	characteristic
71	C₂×C₁₃₉	#473	1	1	yes	characteristic
72	C₁₃₉	#1674	1	1	yes	characteristic, complemented, Sylow
73	#1674	—	139	1	no	maximal, Sylow
74	#500	—	139	1	no	—
75	#462	—	139	1	no	—
76	#462	—	139	1	no	—
77	C₈×Q₈	—	139	1	no	—
78	#335	—	139	1	no	—
79	#332	—	139	1	no	—
80	#332	D₁₃₉	1	1	yes	characteristic
81	C₂×Q₁₆	—	278	1	no	—
82	C₂×Q₁₆	—	278	1	no	—
83	C₄⋊Q₈	—	139	1	no	—
84	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
85	C₄×Q₈	—	139	1	no	—
86	#107	—	278	1	no	—
87	#107	—	278	1	no	—
88	C₄⋊C₈	—	278	1	no	—
89	C₄⋊C₈	—	139	1	no	—
90	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
92	#103	—	278	1	no	—
93	#103	—	278	1	no	—
94	C₄×C₈	—	278	1	no	—
95	C₄×C₈	—	139	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	278	1	no	—
98	C₂×Q₈	—	139	1	no	—
99	Q₁₆	—	556	2	no	—
100	Q₁₆	—	556	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	278	1	no	—
104	C₂×C₈	—	139	1	no	—
105	C₂×C₈	—	139	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	278	1	no	—
109	C₄⋊C₄	—	278	1	no	—
110	C₄⋊C₄	—	278	1	no	—
111	C₄⋊C₄	—	139	1	no	—
112	C₄²	—	278	1	no	—
113	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	278	1	no	—
117	Q₈	—	556	1	no	—
118	Q₈	—	556	1	no	—
119	Q₈	—	139	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	278	1	no	—
122	C₂×C₄	—	278	1	no	—
123	C₂×C₄	—	556	1	no	—
124	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
126	C₂×C₄	#14555785	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	278	1	no	—
130	C₈	—	556	1	no	—
131	C₈	—	139	2	no	—
132	C₂²	#17984815	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	278	2	no	—
136	C₄	—	556	1	no	—
137	C₄	—	1112	1	no	—
138	C₄	D₈×D₁₃₉	1	1	yes	characteristic
139	C₄	Q₁₆×D₁₃₉	1	1	yes	characteristic
140	C₄	#17984863	1	1	yes	characteristic
141	C₄	#17984871	1	1	yes	characteristic
142	C₂	#22581162	1	1	yes	characteristic
143	C₂	#22581188	1	1	yes	characteristic
144	C₂	#22581227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581356

C₂

yes

maximal, characteristic

#22581164

C₂

yes

maximal, characteristic

#22581160

C₂

yes

maximal, characteristic

#22581077

C₂

yes

maximal, characteristic

#22580705

C₂

yes

maximal, characteristic

#22580698

C₂

yes

maximal, characteristic

#332×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

—

C₂×#14555775

—

#17984842

C₂²

yes

characteristic

C₄×Dic₂₇₈

C₂²

yes

characteristic

#17984775

—

#17984775

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

#17984767

—

#17984766

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

characteristic

Dic₅₅₆

—

#14555775

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

D₈

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₈

yes

characteristic

C₄×C₁₃₉

C₂×Q₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1674

—

139

maximal, Sylow

#500

—

139

—

#462

—

139

—

#462

—

139

—

C₈×Q₈

—

139

—

#335

—

139

—

#332

—

139

—

#332

D₁₃₉

yes

characteristic

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

C₄×Q₈

—

139

—

#107

—

278

—

#107

—

278

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

#103

—

278

—

C₄×C₈

—

278

—

C₄×C₈

—

139

—

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

Q₁₆

—

556

—

100

Q₁₆

—

556

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

278

—

104

C₂×C₈

—

139

—

105

C₂×C₈

—

139

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

278

—

109

C₄⋊C₄

—

278

—

110

C₄⋊C₄

—

278

—

111

C₄⋊C₄

—

139

—

112

C₄²

—

278

—

113

C₄²

C₂²×D₁₃₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

278

—

117

Q₈

—

556

—

118

Q₈

—

556

—

119

Q₈

—

139

—

120

C₂×C₄

—

121

C₂×C₄

—

278

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

556

—

124

C₂×C₄

D₄×D₁₃₉

yes

characteristic

125

C₂×C₄

D₄×D₁₃₉

yes

characteristic

126

C₂×C₄

#14555785

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

278

—

130

C₈

—

556

—

131

C₈

—

139

—

132

C₂²

#17984815

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

278

—

136

C₄

—

556

—

137

C₄

—

1112

—

138

C₄

D₈×D₁₃₉

yes

characteristic

139

C₄

Q₁₆×D₁₃₉

yes

characteristic

140

C₄

#17984863

yes

characteristic

141

C₄

#17984871

yes

characteristic

142

C₂

#22581162

yes

characteristic

143

C₂

#22581188

yes

characteristic

144

C₂

#22581227

yes

characteristic

	subgroup	G/H
center	C₂²	#17984815
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#332×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#121
hypercenter	#332	D₁₃₉
hypocenter	C₁₃₉	#1674
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#17984815

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#332×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
50. C₁₃₉⋊C₈	Q₁₆	semidirect
72. C₁₃₉	#1674	semidirect

H ≅ G/N

type

50. C₁₃₉⋊C₈

Q₁₆

semidirect

72. C₁₃₉

#1674

semidirect

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

subgroup	index
#22581356	2
#22581164	2
#22581160	2
#22581077	2
#22580705	2
#22580698	2
#332×C₁₃₉	2
#1674	139

subgroup

index

#332×C₁₃₉

139

Subgroups — #31805797

Conjugacy classes of subgroups

Complements

Maximal subgroups