Group #31806618 — subgroups

Subgroups — #31806618

Order 17792.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22581431	C₂	1	1	yes	maximal, characteristic
2	#22580993	C₂	1	2	yes	maximal
3	C₂×#17984761	C₂	1	1	yes	maximal, characteristic
4	#22580713	C₂	1	2	yes	maximal
5	#391×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	Q₈×Dic₁₃₉	—	2	2	no	—
8	Q₈×Dic₁₃₉	—	2	2	no	—
9	#17984801	—	2	2	no	—
10	#17984801	C₂²	1	2	yes	—
11	#17984775	—	2	2	no	—
12	#17984773	—	2	2	no	—
13	#17984761	C₄	1	4	yes	complemented
14	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
17	#95×C₁₃₉	—	2	2	no	—
18	#14555769	—	2	2	no	—
19	#14555769	C₂×C₄	1	4	yes	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
24	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
25	C₂²⋊Dic₁₃₉	—	2	2	no	—
26	C₄×Dic₁₃₉	—	2	4	no	—
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₂×Q₈×C₁₃₉	—	2	2	no	—
29	C₂×Q₈×C₁₃₉	—	2	2	no	—
30	C₂×Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₃₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₃₉	—	2	1	no	—
33	C₂²×C₄×C₁₃₉	—	2	1	no	—
34	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₃₉	—	2	2	no	—
37	C₄²×C₁₃₉	—	2	2	no	—
38	C₂×Dic₁₃₉	—	2	2	no	—
39	C₂×Dic₁₃₉	—	2	2	no	—
40	C₁₃₉⋊C₈	—	2	4	no	—
41	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₃₉	—	2	4	no	—
43	Q₈×C₁₃₉	—	2	4	no	—
44	C₂×C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	—	4	1	no	—
48	C₂×C₄×C₁₃₉	—	4	1	no	—
49	C₂×C₄×C₁₃₉	—	4	2	no	—
50	C₂×C₄×C₁₃₉	—	4	1	no	—
51	C₂×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	#41	1	2	yes	—
56	C₂²×C₁₃₉	—	2	1	no	—
57	C₂²×C₁₃₉	—	2	1	no	—
58	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₃₉	#120	1	1	yes	characteristic
60	C₂²×C₁₃₉	#123	1	1	yes	characteristic
61	C₄×C₁₃₉	—	2	4	no	—
62	C₄×C₁₃₉	—	4	2	no	—
63	C₄×C₁₃₉	—	8	1	no	—
64	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
65	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₃₉	—	4	2	no	—
67	Dic₁₃₉	—	4	2	no	—
68	C₂×C₁₃₉	—	2	1	no	—
69	C₂×C₁₃₉	—	2	1	no	—
70	C₂×C₁₃₉	#385	1	1	yes	characteristic
71	C₂×C₁₃₉	#454	1	1	yes	characteristic
72	C₂×C₁₃₉	#456	1	1	yes	characteristic
73	C₁₃₉	#1857	1	1	yes	characteristic, complemented, Sylow
74	#1857	—	139	1	no	maximal, Sylow
75	#519	—	139	1	no	—
76	#428	—	139	2	no	—
77	C₂×#101	—	139	1	no	—
78	#391	D₁₃₉	1	1	yes	characteristic
79	#339	—	139	2	no	—
80	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
81	C₂×M₁₆	—	139	2	no	—
82	C₄×Q₈	—	278	2	no	—
83	C₄×Q₈	—	278	2	no	—
84	#117	—	278	2	no	—
85	#117	—	139	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
88	#107	—	278	2	no	—
89	#106	—	278	2	no	—
90	#101	—	139	4	no	—
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₄×D₁₃₉	1	2	yes	—
95	C₂×Q₈	Dic₂₇₈	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
100	M₁₆	—	278	2	no	—
101	M₁₆	—	139	4	no	—
102	C₂×C₈	—	278	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	278	2	no	—
105	C₄⋊C₄	—	278	2	no	—
106	C₄⋊C₄	—	139	2	no	—
107	C₄⋊C₄	—	139	2	no	—
108	C₂²⋊C₄	—	278	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	278	4	no	—
111	C₄²	—	278	2	no	—
112	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	278	2	no	—
123	C₂×C₄	—	278	2	no	—
124	C₂×C₄	Dic₁₃₉⋊C₄	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
128	C₂×C₄	#14555784	1	2	yes	—
129	C₈	—	278	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
133	C₂²	#17984810	1	1	yes	characteristic
134	C₂²	#17984823	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	556	2	no	—
139	C₄	—	556	2	no	—
140	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
141	C₄	#17984819	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#22580825	1	1	yes	characteristic
145	C₂	#22581125	1	1	yes	characteristic
146	C₂	#22581135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581431

C₂

yes

maximal, characteristic

#22580993

C₂

yes

maximal

C₂×#17984761

C₂

yes

maximal, characteristic

#22580713

C₂

yes

maximal

#391×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

Q₈×Dic₁₃₉

—

Q₈×Dic₁₃₉

—

#17984801

—

#17984801

C₂²

yes

—

#17984775

—

#17984773

—

#17984761

C₄

yes

complemented

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#95×C₁₃₉

—

#14555769

—

#14555769

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×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×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1857

—

139

maximal, Sylow

#519

—

139

—

#428

—

139

—

C₂×#101

—

139

—

#391

D₁₃₉

yes

characteristic

#339

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

#117

—

278

—

#117

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#107

—

278

—

#106

—

278

—

#101

—

139

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₁₃₉

yes

—

C₂×Q₈

Dic₂₇₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

100

M₁₆

—

278

—

101

M₁₆

—

139

—

102

C₂×C₈

—

278

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

139

—

107

C₄⋊C₄

—

139

—

108

C₂²⋊C₄

—

278

—

109

C₄²

—

110

C₄²

—

278

—

111

C₄²

—

278

—

112

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

278

—

124

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

125

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

126

C₂×C₄

D₄×D₁₃₉

yes

—

127

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

128

C₂×C₄

#14555784

yes

—

129

C₈

—

278

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

133

C₂²

#17984810

yes

characteristic

134

C₂²

#17984823

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

556

—

139

C₄

—

556

—

140

C₄

(C₂²⋊D₁₃₉)⋊C₄

yes

—

141

C₄

#17984819

yes

—

142

C₂

—

143

C₂

—

144

C₂

#22580825

yes

characteristic

145

C₂

#22581125

yes

characteristic

146

C₂

#22581135

yes

characteristic

	subgroup	G/H
center	C₂²	#17984810
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#391×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#391	D₁₃₉
hypocenter	C₁₃₉	#1857
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#17984810

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#391×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#120

prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
13. #17984761	C₄	semidirect
73. C₁₃₉	#1857	semidirect

H ≅ G/N

type

13. #17984761

C₄

semidirect

73. C₁₃₉

#1857

semidirect

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

subgroup	index
#22581431	2
#22580993	2
C₂×#17984761	2
#22580713	2
#391×C₁₃₉	2
#1857	139

subgroup

index

#391×C₁₃₉

139

Subgroups — #31806618

Conjugacy classes of subgroups

Complements

Maximal subgroups