Group #31807936 — subgroups

Subgroups — #31807936

Order 17792.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#17984774	C₂	1	2	yes	maximal, complemented
2	C₂×#17984764	C₂	1	2	yes	maximal
3	#22580855	C₂	1	1	yes	maximal, characteristic
4	#22580823	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
11	#17984774	—	2	4	no	—
12	#17984764	—	2	4	no	—
13	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₃₉	—	2	1	no	—
16	C₂×D₅₅₆	—	2	2	no	—
17	C₂×D₅₅₆	—	2	2	no	—
18	C₂×D₅₅₆	—	2	1	no	—
19	C₂×D₅₅₆	Q₈	1	2	yes	—
20	C₂²×Dic₁₃₉	—	2	1	no	—
21	C₂²×Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₄⋊Dic₁₃₉	—	2	2	no	—
25	C₄⋊Dic₁₃₉	—	4	4	no	—
26	(C₂×D₁₃₉)⋊C₄	—	4	4	no	—
27	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
28	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	—	2	2	no	—
31	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
32	C₄²×C₁₃₉	—	2	2	no	—
33	C₂²×D₁₃₉	—	2	2	no	—
34	C₂²×D₁₃₉	—	4	1	no	—
35	C₂²×D₁₃₉	—	4	1	no	—
36	C₂²×D₁₃₉	—	4	2	no	—
37	C₂²×D₁₃₉	—	4	2	no	—
38	D₅₅₆	—	2	4	no	—
39	D₅₅₆	—	4	2	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₂×Dic₁₃₉	—	4	2	no	—
42	C₂×Dic₁₃₉	—	4	2	no	—
43	C₂×Dic₁₃₉	—	4	4	no	—
44	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	SD₁₆	1	4	yes	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	#41	1	2	yes	—
51	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
52	C₈×C₁₃₉	—	2	4	no	—
53	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₃₉	#123	1	1	yes	characteristic
55	C₂²×C₁₃₉	#127	1	1	yes	characteristic
56	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×D₁₃₉	—	4	2	no	—
59	C₂×D₁₃₉	—	4	4	no	—
60	C₂×D₁₃₉	—	8	2	no	—
61	C₂×D₁₃₉	—	8	1	no	—
62	C₂×D₁₃₉	—	8	1	no	—
63	C₄×C₁₃₉	—	2	4	no	—
64	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₃₉	#123	1	2	yes	—
66	Dic₁₃₉	—	8	1	no	—
67	Dic₁₃₉	—	8	2	no	—
68	C₂×C₁₃₉	#399	1	1	yes	characteristic
69	C₂×C₁₃₉	#476	1	2	yes	—
70	C₂×C₁₃₉	#481	1	2	yes	—
71	C₂×C₁₃₉	#492	1	1	yes	characteristic
72	C₂×C₁₃₉	#496	1	1	yes	characteristic
73	D₁₃₉	—	8	2	no	—
74	C₁₃₉	#2094	1	1	yes	characteristic, complemented, Sylow
75	#2094	—	139	1	no	maximal, Sylow
76	C₂×#106	—	139	2	no	—
77	C₂×#102	—	139	2	no	—
78	#390	—	139	1	no	—
79	#384	—	139	1	no	—
80	#336	D₁₃₉	1	1	yes	characteristic, complemented
81	C₂²×D₄	—	139	1	no	—
82	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
83	C₂×C₄⋊C₄	—	278	2	no	—
84	C₂×C₄⋊C₄	—	139	2	no	—
85	C₂×C₄⋊C₄	—	139	1	no	—
86	C₂×C₂²⋊C₄	—	278	2	no	—
87	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
88	#106	—	278	4	no	—
89	#102	—	278	4	no	—
90	C₂⁴	—	278	1	no	—
91	C₂×D₄	—	278	2	no	—
92	C₂×D₄	—	278	2	no	—
93	C₂×D₄	—	278	1	no	—
94	C₂×D₄	—	139	2	no	—
95	C₂²×C₄	—	278	1	no	—
96	C₂²×C₄	—	278	2	no	—
97	C₂²×C₄	D₅₅₆	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₅₅₆	1	4	yes	—
102	C₄⋊C₄	—	278	2	no	—
103	C₄⋊C₄	—	278	2	no	—
104	C₄⋊C₄	—	278	2	no	—
105	C₄⋊C₄	—	556	4	no	—
106	C₂²⋊C₄	—	556	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	278	2	no	—
109	C₂³	—	556	1	no	—
110	C₂³	—	556	1	no	—
111	C₂³	—	556	2	no	—
112	C₂³	—	556	2	no	—
113	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
114	D₄	—	278	4	no	—
115	D₄	—	556	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	556	1	no	—
119	C₂×C₄	—	556	2	no	—
120	C₂×C₄	—	556	2	no	—
121	C₂×C₄	—	556	4	no	—
122	C₂×C₄	#14555774	1	4	yes	—
123	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
124	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
125	C₂×C₄	#14555784	1	2	yes	—
126	C₂×C₄	#14555784	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂²	—	556	2	no	—
129	C₂²	—	556	4	no	—
130	C₂²	—	1112	2	no	—
131	C₂²	—	1112	1	no	—
132	C₂²	—	1112	1	no	—
133	C₂²	#17984822	1	1	yes	characteristic
134	C₂²	#17984826	1	1	yes	characteristic
135	C₂²	#17984839	1	1	yes	characteristic
136	C₂²	C₂×#14555774	1	2	yes	—
137	C₂²	#17984870	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	1112	1	no	—
140	C₄	—	1112	2	no	—
141	C₄	#17984822	1	2	yes	—
142	C₄	#17984826	1	2	yes	—
143	C₂	—	1112	2	no	—
144	C₂	#22580890	1	1	yes	characteristic
145	C₂	#22581244	1	2	yes	—
146	C₂	#22581272	1	2	yes	—
147	C₂	#22581326	1	1	yes	characteristic
148	C₂	#22581342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984774

C₂

yes

maximal, complemented

C₂×#17984764

C₂

yes

maximal

#22580855

C₂

yes

maximal, characteristic

#22580823

C₂

yes

maximal, characteristic, complemented

#336×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₅₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×(C₂×D₁₃₉)⋊C₄

—

#17984774

—

#17984764

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

Q₈

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂²⋊Q₈

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

#123

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#399

yes

characteristic

C₂×C₁₃₉

#476

yes

—

C₂×C₁₃₉

#481

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#2094

yes

characteristic, complemented, Sylow

#2094

—

139

maximal, Sylow

C₂×#106

—

139

—

C₂×#102

—

139

—

#390

—

139

—

#384

—

139

—

#336

D₁₃₉

yes

characteristic, complemented

C₂²×D₄

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

278

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#106

—

278

—

#102

—

278

—

C₂⁴

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₅₅₆

yes

—

102

C₄⋊C₄

—

278

—

103

C₄⋊C₄

—

278

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

556

—

106

C₂²⋊C₄

—

556

—

107

C₄²

—

108

C₂³

—

278

—

109

C₂³

—

556

—

110

C₂³

—

556

—

111

C₂³

—

556

—

112

C₂³

—

556

—

113

C₂³

C₂×D₅₅₆

yes

characteristic

114

D₄

—

278

—

115

D₄

—

556

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

556

—

119

C₂×C₄

—

556

—

120

C₂×C₄

—

556

—

121

C₂×C₄

—

556

—

122

C₂×C₄

#14555774

yes

—

123

C₂×C₄

C₂×D₅₅₆

yes

—

124

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

125

C₂×C₄

#14555784

yes

—

126

C₂×C₄

#14555784

yes

characteristic

127

C₈

—

128

C₂²

—

556

—

129

C₂²

—

556

—

130

C₂²

—

1112

—

131

C₂²

—

1112

—

132

C₂²

—

1112

—

133

C₂²

#17984822

yes

characteristic

134

C₂²

#17984826

yes

characteristic

135

C₂²

#17984839

yes

characteristic

136

C₂²

C₂×#14555774

yes

—

137

C₂²

#17984870

yes

—

138

C₄

—

139

C₄

—

1112

—

140

C₄

—

1112

—

141

C₄

#17984822

yes

—

142

C₄

#17984826

yes

—

143

C₂

—

1112

—

144

C₂

#22580890

yes

characteristic

145

C₂

#22581244

yes

—

146

C₂

#22581272

yes

—

147

C₂

#22581326

yes

characteristic

148

C₂

#22581342

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#336×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂×D₄
prime radical	C₂²×D₅₅₆	C₂²
hypercenter	#336	D₁₃₉
hypocenter	C₁₃₉	#2094
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#336×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂×D₄

prime radical

C₂²×D₅₅₆

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#17984774	C₂	semidirect
4. #22580823	C₂	semidirect
5. #336×C₁₃₉	C₂	semidirect
74. C₁₃₉	#2094	semidirect
80. #336	D₁₃₉	semidirect

H ≅ G/N

type

1. C₂×#17984774

C₂

semidirect

4. #22580823

C₂

semidirect

5. #336×C₁₃₉

C₂

semidirect

74. C₁₃₉

#2094

semidirect

80. #336

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984774	2
C₂×#17984764	2
#22580855	2
#22580823	2
#336×C₁₃₉	2
#2094	139

subgroup

index

#336×C₁₃₉

139

Subgroups — #31807936

Conjugacy classes of subgroups

Complements

Maximal subgroups