Group #31807128 — subgroups

Subgroups — #31807128

Order 17792.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#17984753	C₂	1	1	yes	maximal, characteristic
4	#22580708	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₃₉⋊C₈)	—	2	4	no	—
10	C₄⋊(C₁₃₉⋊C₈)	—	2	2	no	—
11	#17984753	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₉	C₄	1	2	yes	—
13	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	#95×C₁₃₉	C₄	1	1	yes	characteristic
16	#95×C₁₃₉	C₄	1	1	yes	characteristic
17	C₂×C₁₃₉⋊C₈	—	2	1	no	—
18	C₂×C₁₃₉⋊C₈	—	2	1	no	—
19	C₂×C₁₃₉⋊C₈	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₂×C₁₃₉⋊C₈	—	2	1	no	—
22	C₂×C₁₃₉⋊C₈	—	2	1	no	—
23	C₂×C₁₃₉⋊C₈	—	2	2	no	—
24	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₃₉⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₃₉	—	2	4	no	—
32	C₄²×C₁₃₉	—	2	2	no	—
33	C₄²×C₁₃₉	—	2	2	no	—
34	C₄²×C₁₃₉	D₄	1	2	yes	—
35	C₄²×C₁₃₉	Q₈	1	2	yes	—
36	C₁₃₉⋊C₈	—	2	2	no	—
37	C₁₃₉⋊C₈	—	2	2	no	—
38	C₁₃₉⋊C₈	—	4	2	no	—
39	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄×C₁₃₉	—	2	2	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	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₃₉	M₁₆	1	4	yes	—
50	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
56	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₃₉	#131	1	2	yes	—
61	C₄×C₁₃₉	—	2	2	no	—
62	C₄×C₁₃₉	—	2	2	no	—
63	C₄×C₁₃₉	—	2	4	no	—
64	C₄×C₁₃₉	—	4	2	no	—
65	C₄×C₁₃₉	#121	1	1	yes	characteristic
66	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₁₃₉	#125	1	1	yes	characteristic
68	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₃₉	#389	1	1	yes	characteristic
70	C₂×C₁₃₉	#423	1	1	yes	characteristic
71	C₂×C₁₃₉	#424	1	1	yes	characteristic
72	C₂×C₁₃₉	#435	1	2	yes	—
73	C₂×C₁₃₉	#446	1	2	yes	—
74	C₁₃₉	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	139	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	139	2	no	—
77	C₂×C₄⋊C₈	—	139	1	no	—
78	C₂×#97	—	139	1	no	—
79	C₄×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic
80	#336	—	139	2	no	—
81	C₂²×C₈	—	139	2	no	—
82	C₂²×C₈	—	139	1	no	—
83	C₂²×C₈	—	139	1	no	—
84	C₂×C₄⋊C₄	Dic₁₃₉	1	2	yes	—
85	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
86	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
87	C₄⋊C₈	—	278	4	no	—
88	C₄⋊C₈	—	278	2	no	—
89	#97	—	278	2	no	—
90	#95	Dic₁₃₉	1	1	yes	characteristic
91	#95	Dic₁₃₉	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
97	C₂×C₈	—	278	1	no	—
98	C₂×C₈	—	278	1	no	—
99	C₂×C₈	—	278	1	no	—
100	C₂×C₈	—	278	1	no	—
101	C₂×C₈	—	278	2	no	—
102	C₂×C₈	—	278	2	no	—
103	C₂×C₈	—	278	2	no	—
104	C₂×C₈	—	139	2	no	—
105	C₂×C₈	—	139	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	D₅₅₆	1	2	yes	—
110	C₄²	Dic₂₇₈	1	2	yes	—
111	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊Dic₁₃₉	1	2	yes	—
120	C₂×C₄	C₄⋊Dic₁₃₉	1	2	yes	—
121	C₂×C₄	#14555769	1	4	yes	—
122	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
126	C₂×C₄	#14555784	1	1	yes	characteristic
127	C₂×C₄	#14555786	1	1	yes	characteristic
128	C₈	—	278	2	no	—
129	C₈	—	278	2	no	—
130	C₈	—	556	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₃₉	1	1	yes	characteristic
132	C₂²	D₄×Dic₁₃₉	1	1	yes	characteristic
133	C₂²	Q₈×Dic₁₃₉	1	1	yes	characteristic
134	C₂²	C₂×#14555769	1	2	yes	—
135	C₂²	#17984849	1	2	yes	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	2	no	—
140	C₄	#17984817	1	1	yes	characteristic
141	C₄	#17984822	1	1	yes	characteristic
142	C₄	#17984833	1	1	yes	characteristic
143	C₄	#17984841	1	1	yes	characteristic
144	C₂	#22580852	1	1	yes	characteristic
145	C₂	#22580977	1	1	yes	characteristic
146	C₂	#22580980	1	1	yes	characteristic
147	C₂	#22581025	1	2	yes	—
148	C₂	#22581084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂

yes

maximal

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#17984753

C₂

yes

maximal, characteristic

#22580708

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

—

C₄⋊(C₁₃₉⋊C₈)

—

#17984753

—

C₂×C₄⋊C₄×C₁₃₉

C₄

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#95×C₁₃₉

C₄

yes

characteristic

#95×C₁₃₉

C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

Q₈

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

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

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

M₁₆

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

C₄×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₂²×C₁₃₉

#131

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

#125

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#446

yes

—

C₁₃₉

#1940

yes

characteristic, complemented, Sylow

#1940

—

139

maximal, Sylow

C₂×C₄⋊C₈

—

139

—

C₂×C₄⋊C₈

—

139

—

C₂×#97

—

139

—

C₄×C₄⋊C₄

D₁₃₉

yes

characteristic

#336

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂×C₄⋊C₄

Dic₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

278

—

C₄⋊C₈

—

278

—

#97

—

278

—

#95

Dic₁₃₉

yes

characteristic

#95

Dic₁₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₉

yes

—

C₂²×C₄

C₂×Dic₁₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

100

C₂×C₈

—

278

—

101

C₂×C₈

—

278

—

102

C₂×C₈

—

278

—

103

C₂×C₈

—

278

—

104

C₂×C₈

—

139

—

105

C₂×C₈

—

139

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

D₅₅₆

yes

—

110

C₄²

Dic₂₇₈

yes

—

111

C₂³

C₂²×Dic₁₃₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊Dic₁₃₉

yes

—

120

C₂×C₄

C₄⋊Dic₁₃₉

yes

—

121

C₂×C₄

#14555769

yes

—

122

C₂×C₄

D₄×D₁₃₉

yes

characteristic

123

C₂×C₄

C₂×D₅₅₆

yes

characteristic

124

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

125

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

126

C₂×C₄

#14555784

yes

characteristic

127

C₂×C₄

#14555786

yes

characteristic

128

C₈

—

278

—

129

C₈

—

278

—

130

C₈

—

556

—

131

C₂²

C₂×C₄⋊Dic₁₃₉

yes

characteristic

132

C₂²

D₄×Dic₁₃₉

yes

characteristic

133

C₂²

Q₈×Dic₁₃₉

yes

characteristic

134

C₂²

C₂×#14555769

yes

—

135

C₂²

#17984849

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#17984817

yes

characteristic

141

C₄

#17984822

yes

characteristic

142

C₄

#17984833

yes

characteristic

143

C₄

#17984841

yes

characteristic

144

C₂

#22580852

yes

characteristic

145

C₂

#22580977

yes

characteristic

146

C₂

#22580980

yes

characteristic

147

C₂

#22581025

yes

—

148

C₂

#22581084

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator, socle, prime radical	C₂³×C₁₃₉	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₁₃₉	C₂
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#1940
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator, socle, prime radical

C₂³×C₁₃₉

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₁₃₉

C₂

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
74. C₁₃₉	#1940	semidirect

H ≅ G/N

type

74. C₁₃₉

#1940

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×#17984753	2
#22580708	2
C₄×C₄⋊C₄×C₁₃₉	2
#1940	139

subgroup

index

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂×#17984753

#22580708

C₄×C₄⋊C₄×C₁₃₉

#1940

139

Subgroups — #31807128

Conjugacy classes of subgroups

Complements

Maximal subgroups