Group #31807364 — subgroups

Subgroups — #31807364

Order 17792.

300 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 137 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×#17984842	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22580715	C₂	1	4	yes	maximal
4	#384×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	—	2	1	no	—
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	#17984842	—	2	2	no	—
8	#17984842	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
11	C₄×C₁₃₉⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₇₈	—	2	4	no	—
17	C₂×Dic₂₇₈	—	4	2	no	—
18	C₂×Dic₂₇₈	—	4	2	no	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	Q₈	1	4	yes	complemented
23	C₄⋊Dic₁₃₉	—	2	4	no	—
24	C₄⋊Dic₁₃₉	—	2	2	no	—
25	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₃₉	—	2	2	no	—
27	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₃₉	—	2	4	no	—
31	C₄⋊C₄×C₁₃₉	—	4	2	no	—
32	C₄⋊C₄×C₁₃₉	—	4	2	no	—
33	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₄²×C₁₃₉	D₄	1	2	yes	—
35	Dic₂₇₈	—	4	8	no	—
36	C₂×Dic₁₃₉	—	2	4	no	—
37	C₂×Dic₁₃₉	—	4	2	no	—
38	C₂×Dic₁₃₉	—	4	2	no	—
39	C₁₃₉⋊C₈	—	2	4	no	—
40	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	—	4	2	no	—
42	C₂×C₄×C₁₃₉	—	4	2	no	—
43	C₂×C₄×C₁₃₉	—	4	2	no	—
44	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	D₈	1	2	yes	—
47	C₂×C₄×C₁₃₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₃₉	SD₁₆	1	2	yes	—
49	C₂×C₄×C₁₃₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₃₉	#41	1	2	yes	—
54	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₃₉	#124	1	1	yes	characteristic
56	C₂²×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₃₉	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₃₉	—	8	2	no	—
62	C₄×C₁₃₉	#102	1	4	yes	—
63	C₄×C₁₃₉	#103	1	4	yes	—
64	C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
65	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₃₉	—	4	4	no	—
67	C₂×C₁₃₉	#391	1	1	yes	characteristic
68	C₂×C₁₃₉	C₂×#102	1	1	yes	characteristic
69	C₂×C₁₃₉	C₂×#103	1	1	yes	characteristic
70	C₂×C₁₃₉	#486	1	1	yes	characteristic
71	C₂×C₁₃₉	#487	1	1	yes	characteristic
72	C₂×C₁₃₉	#498	1	1	yes	characteristic
73	C₂×C₁₃₉	#500	1	1	yes	characteristic
74	C₁₃₉	#1990	1	1	yes	characteristic, complemented, Sylow
75	#1990	—	139	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	—	139	1	no	—
77	C₂×C₄×C₈	—	139	1	no	—
78	#384	D₁₃₉	1	1	yes	characteristic
79	#340	—	139	4	no	—
80	C₂²×Q₈	—	278	1	no	—
81	C₂²×C₈	—	139	2	no	—
82	C₄⋊Q₈	—	278	2	no	—
83	C₄⋊Q₈	—	139	4	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	278	1	no	—
87	C₂×C₄⋊C₄	—	139	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
89	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
90	C₄×C₈	—	278	2	no	—
91	C₂×Q₈	—	278	4	no	—
92	C₂×Q₈	—	556	2	no	—
93	C₂×Q₈	—	556	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	278	2	no	—
96	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
99	C₂×C₈	—	278	2	no	—
100	C₂×C₈	—	278	2	no	—
101	C₂×C₈	—	139	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	278	4	no	—
106	C₄⋊C₄	—	278	2	no	—
107	C₄⋊C₄	—	139	4	no	—
108	C₄²	C₄×D₁₃₉	1	2	yes	—
109	C₄²	D₅₅₆	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
111	Q₈	—	556	8	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	278	4	no	—
116	C₂×C₄	—	556	2	no	—
117	C₂×C₄	—	556	2	no	—
118	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
119	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
120	C₂×C₄	#14555770	1	2	yes	—
121	C₂×C₄	#14555772	1	2	yes	—
122	C₂×C₄	#14555773	1	2	yes	—
123	C₂×C₄	#14555775	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
125	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
127	C₂×C₄	#14555784	1	2	yes	—
128	C₈	—	278	4	no	—
129	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
130	C₂²	#17984829	1	1	yes	characteristic
131	C₂²	#17984840	1	1	yes	characteristic
132	C₂²	C₂×#14555770	1	1	yes	characteristic
133	C₂²	C₂×#14555772	1	1	yes	characteristic
134	C₂²	C₂×#14555773	1	1	yes	characteristic
135	C₂²	C₂×#14555775	1	1	yes	characteristic
136	C₄	—	8	2	no	—
137	C₄	—	556	4	no	—
138	C₄	#17984762	1	4	yes	—
139	C₄	#17984765	1	4	yes	—
140	C₄	#17984807	1	2	yes	—
141	C₄	#17984822	1	2	yes	—
142	C₂	#22580858	1	1	yes	characteristic
143	C₂	C₂×#17984762	1	1	yes	characteristic
144	C₂	C₂×#17984765	1	1	yes	characteristic
145	C₂	#22581299	1	1	yes	characteristic
146	C₂	#22581303	1	1	yes	characteristic
147	C₂	#22581349	1	1	yes	characteristic
148	C₂	#22581356	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984842

C₂

yes

maximal, characteristic

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#22580715

C₂

yes

maximal

#384×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984842

—

#17984842

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₄×C₁₃₉⋊C₈

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

D₈

yes

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

Q₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#124

yes

characteristic

C₂²×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×D₈

yes

characteristic

C₂²×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

#102

yes

—

C₄×C₁₃₉

#103

yes

—

C₄×C₁₃₉

C₄×Q₈

yes

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1990

—

139

maximal, Sylow

C₂×C₄⋊Q₈

—

139

—

C₂×C₄×C₈

—

139

—

#384

D₁₃₉

yes

characteristic

#340

—

139

—

C₂²×Q₈

—

278

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄×C₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

556

—

C₂×Q₈

—

556

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

100

C₂×C₈

—

278

—

101

C₂×C₈

—

139

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

278

—

107

C₄⋊C₄

—

139

—

108

C₄²

C₄×D₁₃₉

yes

—

109

C₄²

D₅₅₆

yes

—

110

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

111

Q₈

—

556

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

278

—

116

C₂×C₄

—

556

—

117

C₂×C₄

—

556

—

118

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

119

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

120

C₂×C₄

#14555770

yes

—

121

C₂×C₄

#14555772

yes

—

122

C₂×C₄

#14555773

yes

—

123

C₂×C₄

#14555775

yes

—

124

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

125

C₂×C₄

C₂×D₅₅₆

yes

characteristic

126

C₂×C₄

Q₈×D₁₃₉

yes

—

127

C₂×C₄

#14555784

yes

—

128

C₈

—

278

—

129

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

130

C₂²

#17984829

yes

characteristic

131

C₂²

#17984840

yes

characteristic

132

C₂²

C₂×#14555770

yes

characteristic

133

C₂²

C₂×#14555772

yes

characteristic

134

C₂²

C₂×#14555773

yes

characteristic

135

C₂²

C₂×#14555775

yes

characteristic

136

C₄

—

137

C₄

—

556

—

138

C₄

#17984762

yes

—

139

C₄

#17984765

yes

—

140

C₄

#17984807

yes

—

141

C₄

#17984822

yes

—

142

C₂

#22580858

yes

characteristic

143

C₂

C₂×#17984762

yes

characteristic

144

C₂

C₂×#17984765

yes

characteristic

145

C₂

#22581299

yes

characteristic

146

C₂

#22581303

yes

characteristic

147

C₂

#22581349

yes

characteristic

148

C₂

#22581356

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#384×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#384	D₁₃₉
hypocenter	C₁₃₉	#1990
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#384×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #17984842	C₄	semidirect
22. C₂×C₁₃₉⋊C₈	Q₈	semidirect
74. C₁₃₉	#1990	semidirect

H ≅ G/N

type

8. #17984842

C₄

semidirect

22. C₂×C₁₃₉⋊C₈

Q₈

semidirect

74. C₁₃₉

#1990

semidirect

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

subgroup	index
C₂×#17984842	2
C₂×C₄×C₁₃₉⋊C₈	2
#22580715	2
#384×C₁₃₉	2
#1990	139

subgroup

index

C₂×#17984842

C₂×C₄×C₁₃₉⋊C₈

#22580715

#384×C₁₃₉

#1990

139

Subgroups — #31807364

Conjugacy classes of subgroups

Complements

Maximal subgroups