Group #31807327 — subgroups

Subgroups — #31807327

Order 17792.

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

normal subgroups only

This group has 324 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	C₂×#17984842	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#17984765	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
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	—	2	1	no	—
11	C₄⋊(C₁₃₉⋊C₈)	—	2	2	no	—
12	#17984765	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	—	2	1	no	—
15	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
16	#95×C₁₃₉	—	2	1	no	—
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	—	2	4	no	—
19	C₂×Dic₂₇₈	—	2	4	no	—
20	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
21	C₂²×Dic₁₃₉	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	—	2	2	no	—
23	C₂×C₁₃₉⋊C₈	—	2	4	no	—
24	C₄⋊Dic₁₃₉	—	2	4	no	—
25	C₄⋊Dic₁₃₉	—	2	4	no	—
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	2	no	—
31	C₄⋊C₄×C₁₃₉	D₄	1	4	yes	—
32	C₄²×C₁₃₉	—	4	2	no	—
33	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₄²×C₁₃₉	D₄	1	2	yes	—
35	Dic₂₇₈	—	2	8	no	—
36	Dic₂₇₈	—	4	2	no	—
37	Dic₂₇₈	—	4	2	no	—
38	C₂×Dic₁₃₉	—	2	4	no	—
39	C₂×Dic₁₃₉	—	4	4	no	—
40	C₁₃₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄×C₁₃₉	—	2	4	no	—
44	C₂×C₄×C₁₃₉	—	4	2	no	—
45	C₂×C₄×C₁₃₉	—	4	2	no	—
46	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₃₉	C₂²⋊C₄	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	2	yes	—
52	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	#41	1	2	yes	—
54	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₃₉	#121	1	2	yes	—
56	C₂²×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₃₉	#137	1	2	yes	—
59	C₄×C₁₃₉	—	2	2	no	—
60	C₄×C₁₃₉	—	4	4	no	—
61	C₄×C₁₃₉	#103	1	4	yes	complemented
62	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
63	C₄×C₁₃₉	#124	1	1	yes	characteristic
64	C₄×C₁₃₉	#127	1	1	yes	characteristic
65	Dic₁₃₉	—	4	4	no	—
66	C₂×C₁₃₉	#390	1	1	yes	characteristic
67	C₂×C₁₃₉	C₂×#103	1	1	yes	characteristic
68	C₂×C₁₃₉	#419	1	1	yes	characteristic
69	C₂×C₁₃₉	#461	1	2	yes	—
70	C₂×C₁₃₉	#462	1	2	yes	—
71	C₁₃₉	#1981	1	1	yes	characteristic, complemented, Sylow
72	#1981	—	139	1	no	maximal, Sylow
73	C₂×C₄⋊Q₈	—	139	1	no	—
74	C₂×C₄⋊C₈	—	139	1	no	—
75	C₂×#103	—	139	4	no	—
76	C₄×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic
77	C₂²×Q₈	—	139	2	no	—
78	C₂²×C₈	—	139	2	no	—
79	C₄⋊Q₈	—	278	2	no	—
80	C₄⋊Q₈	—	139	4	no	—
81	C₂×C₄⋊C₄	—	278	1	no	—
82	C₂×C₄⋊C₄	—	278	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
86	C₄⋊C₈	—	278	2	no	—
87	#103	—	278	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	278	2	no	—
90	C₂×Q₈	—	278	4	no	—
91	C₂×Q₈	—	278	4	no	—
92	C₂×Q₈	—	139	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	278	2	no	—
95	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₂×C₈	—	278	4	no	—
99	C₂×C₈	—	278	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	278	4	no	—
102	C₄⋊C₄	—	278	4	no	—
103	C₄⋊C₄	D₅₅₆	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₁₃₉	1	2	yes	—
106	C₄²	D₅₅₆	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
108	Q₈	—	278	8	no	—
109	Q₈	—	556	2	no	—
110	Q₈	—	556	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	278	4	no	—
116	C₂×C₄	—	556	4	no	—
117	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
119	C₂×C₄	#14555773	1	2	yes	—
120	C₂×C₄	#14555775	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
122	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
123	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
124	C₂×C₄	#14555785	1	2	yes	—
125	C₈	—	556	2	no	—
126	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
127	C₂²	#17984816	1	2	yes	—
128	C₂²	C₂×#14555773	1	1	yes	characteristic
129	C₂²	C₂×#14555775	1	1	yes	characteristic
130	C₂²	#17984872	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	556	4	no	—
134	C₄	#17984765	1	4	yes	complemented
135	C₄	C₄×D₅₅₆	1	2	yes	—
136	C₄	#17984830	1	1	yes	characteristic
137	C₄	#17984839	1	1	yes	characteristic
138	C₂	#22580855	1	1	yes	characteristic
139	C₂	C₂×#17984765	1	1	yes	characteristic
140	C₂	#22580961	1	1	yes	characteristic
141	C₂	#22581158	1	2	yes	—
142	C₂	#22581163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#17984765

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984842

—

#17984842

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

—

C₄⋊(C₁₃₉⋊C₈)

—

#17984765

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#95×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

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

D₄

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

Q₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#121

yes

—

C₂²×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₉

C₂×Q₁₆

yes

characteristic

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#103

yes

complemented

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#462

yes

—

C₁₃₉

#1981

yes

characteristic, complemented, Sylow

#1981

—

139

maximal, Sylow

C₂×C₄⋊Q₈

—

139

—

C₂×C₄⋊C₈

—

139

—

C₂×#103

—

139

—

C₄×C₄⋊C₄

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

278

—

#103

—

278

—

#95

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

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

—

C₂×C₈

—

278

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

278

—

102

C₄⋊C₄

—

278

—

103

C₄⋊C₄

D₅₅₆

yes

—

104

C₄²

—

105

C₄²

C₄×D₁₃₉

yes

—

106

C₄²

D₅₅₆

yes

—

107

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

108

Q₈

—

278

—

109

Q₈

—

556

—

110

Q₈

—

556

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

278

—

116

C₂×C₄

—

556

—

117

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

119

C₂×C₄

#14555773

yes

—

120

C₂×C₄

#14555775

yes

—

121

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

122

C₂×C₄

C₂×D₅₅₆

yes

—

123

C₂×C₄

C₂×D₅₅₆

yes

characteristic

124

C₂×C₄

#14555785

yes

—

125

C₈

—

556

—

126

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

127

C₂²

#17984816

yes

—

128

C₂²

C₂×#14555773

yes

characteristic

129

C₂²

C₂×#14555775

yes

characteristic

130

C₂²

#17984872

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

556

—

134

C₄

#17984765

yes

complemented

135

C₄

C₄×D₅₅₆

yes

—

136

C₄

#17984830

yes

characteristic

137

C₄

#17984839

yes

characteristic

138

C₂

#22580855

yes

characteristic

139

C₂

C₂×#17984765

yes

characteristic

140

C₂

#22580961

yes

characteristic

141

C₂

#22581158

yes

—

142

C₂

#22581163

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₄×C₄⋊C₄×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#1981
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₄×C₄⋊C₄×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₃₉

hypocenter

C₁₃₉

#1981

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #17984842	C₄	semidirect
61. C₄×C₁₃₉	#103	semidirect
71. C₁₃₉	#1981	semidirect
134. C₄	#17984765	semidirect

H ≅ G/N

type

8. #17984842

C₄

semidirect

61. C₄×C₁₃₉

#103

semidirect

71. C₁₃₉

#1981

semidirect

134. C₄

#17984765

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
C₂×#17984765	2
C₄×C₄⋊C₄×C₁₃₉	2
#1981	139

subgroup

index

C₂×#17984842

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂×#17984765

C₄×C₄⋊C₄×C₁₃₉

#1981

139

Subgroups — #31807327

Conjugacy classes of subgroups

Complements

Maximal subgroups