Group #31807551 — subgroups

Subgroups — #31807551

Order 17792.

294 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 83 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×#17984834	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984764	C₂	1	2	yes	maximal
3	#22580855	C₂	1	1	yes	maximal, characteristic
4	#22580716	C₂	1	2	yes	maximal, complemented
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
7	#17984834	—	2	4	no	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
10	C₂×Dic₁₃₉⋊C₄	—	2	2	no	—
11	C₂×(C₂×D₁₃₉)⋊C₄	—	2	2	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	2	no	—
21	C₂²×Dic₁₃₉	—	2	1	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₄⋊Dic₁₃₉	D₄	1	4	yes	—
25	Dic₁₃₉⋊C₄	—	2	8	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	—	2	2	no	—
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	—	2	4	no	—
41	C₂×Dic₁₃₉	—	4	2	no	—
42	C₂×Dic₁₃₉	—	4	2	no	—
43	C₂×Dic₁₃₉	—	4	1	no	—
44	C₂×Dic₁₃₉	—	4	2	no	—
45	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	—	2	2	no	—
48	C₂×C₄×C₁₃₉	—	2	2	no	—
49	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	#41	1	2	yes	—
53	C₈×C₁₃₉	—	4	2	no	—
54	C₂²×C₁₃₉	#120	1	1	yes	characteristic
55	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₃₉	#124	1	1	yes	characteristic
57	C₂²×C₁₃₉	#135	1	2	yes	—
58	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₁₃₉	—	4	2	no	—
60	C₂×D₁₃₉	—	4	4	no	—
61	C₂×D₁₃₉	—	8	2	no	—
62	C₂×D₁₃₉	—	8	1	no	—
63	C₂×D₁₃₉	—	8	1	no	—
64	C₄×C₁₃₉	—	4	2	no	—
65	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₃₉	#124	1	2	yes	—
67	Dic₁₃₉	—	4	4	no	—
68	Dic₁₃₉	—	8	1	no	—
69	C₂×C₁₃₉	#393	1	1	yes	characteristic
70	C₂×C₁₃₉	#449	1	2	yes	—
71	C₂×C₁₃₉	#478	1	2	yes	—
72	C₂×C₁₃₉	#488	1	1	yes	characteristic
73	C₂×C₁₃₉	#490	1	1	yes	characteristic
74	D₁₃₉	—	8	2	no	—
75	C₁₃₉	#2025	1	1	yes	characteristic, complemented, Sylow
76	#2025	—	139	1	no	maximal, Sylow
77	C₂×#125	—	139	1	no	—
78	C₂×#102	—	139	2	no	—
79	#390	—	139	1	no	—
80	#340	—	139	2	no	—
81	#336	D₁₃₉	1	1	yes	characteristic, complemented
82	C₂²×D₄	—	139	1	no	—
83	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
84	#125	—	278	4	no	—
85	C₂×C₄⋊C₄	—	278	2	no	—
86	C₂×C₄⋊C₄	—	139	1	no	—
87	C₂×C₄⋊C₄	—	139	2	no	—
88	C₂×C₂²⋊C₄	—	278	2	no	—
89	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
90	#102	—	278	4	no	—
91	C₂⁴	—	278	1	no	—
92	C₂×D₄	—	278	2	no	—
93	C₂×D₄	—	278	1	no	—
94	C₂×D₄	—	278	2	no	—
95	C₂×D₄	—	139	2	no	—
96	C₂²×C₄	—	278	2	no	—
97	C₂²×C₄	—	278	1	no	—
98	C₂²×C₄	D₅₅₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	278	8	no	—
104	C₄⋊C₄	—	278	2	no	—
105	C₄⋊C₄	—	278	2	no	—
106	C₄⋊C₄	—	139	4	no	—
107	C₂²⋊C₄	—	556	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	278	2	no	—
110	C₂³	—	556	1	no	—
111	C₂³	—	556	1	no	—
112	C₂³	—	556	2	no	—
113	C₂³	—	556	2	no	—
114	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
115	D₄	—	278	4	no	—
116	D₄	—	556	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	278	4	no	—
121	C₂×C₄	—	556	2	no	—
122	C₂×C₄	—	556	2	no	—
123	C₂×C₄	—	556	1	no	—
124	C₂×C₄	—	556	2	no	—
125	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
126	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
127	C₂×C₄	#14555784	1	1	yes	characteristic
128	C₂×C₄	#14555785	1	2	yes	—
129	C₈	—	4	2	no	—
130	C₂²	—	556	4	no	—
131	C₂²	—	556	2	no	—
132	C₂²	—	1112	1	no	—
133	C₂²	—	1112	1	no	—
134	C₂²	—	1112	2	no	—
135	C₂²	#17984812	1	1	yes	characteristic
136	C₂²	#17984822	1	1	yes	characteristic
137	C₂²	#17984830	1	1	yes	characteristic
138	C₂²	#17984864	1	2	yes	—
139	C₂²	#17984870	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	556	4	no	—
142	C₄	—	1112	1	no	—
143	C₄	#17984820	1	2	yes	—
144	C₄	#17984828	1	2	yes	—
145	C₂	—	1112	2	no	—
146	C₂	#22580866	1	1	yes	characteristic
147	C₂	#22581100	1	2	yes	—
148	C₂	#22581254	1	2	yes	—
149	C₂	#22581308	1	1	yes	characteristic
150	C₂	#22581318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984834

C₂

yes

maximal, characteristic, complemented

C₂×#17984764

C₂

yes

maximal

#22580855

C₂

yes

maximal, characteristic

#22580716

C₂

yes

maximal, complemented

#336×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₅₆

C₂²

yes

characteristic

#17984834

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

—

C₂×(C₂×D₁₃₉)⋊C₄

—

#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

D₄

yes

—

Dic₁₃₉⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

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

#124

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#393

yes

characteristic

C₂×C₁₃₉

#449

yes

—

C₂×C₁₃₉

#478

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#2025

yes

characteristic, complemented, Sylow

#2025

—

139

maximal, Sylow

C₂×#125

—

139

—

C₂×#102

—

139

—

#390

—

139

—

#340

—

139

—

#336

D₁₃₉

yes

characteristic, complemented

C₂²×D₄

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

#125

—

278

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

278

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#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

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

278

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

139

—

107

C₂²⋊C₄

—

556

—

108

C₄²

—

109

C₂³

—

278

—

110

C₂³

—

556

—

111

C₂³

—

556

—

112

C₂³

—

556

—

113

C₂³

—

556

—

114

C₂³

C₂×D₅₅₆

yes

characteristic

115

D₄

—

278

—

116

D₄

—

556

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

278

—

121

C₂×C₄

—

556

—

122

C₂×C₄

—

556

—

123

C₂×C₄

—

556

—

124

C₂×C₄

—

556

—

125

C₂×C₄

D₄×D₁₃₉

yes

—

126

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

127

C₂×C₄

#14555784

yes

characteristic

128

C₂×C₄

#14555785

yes

—

129

C₈

—

130

C₂²

—

556

—

131

C₂²

—

556

—

132

C₂²

—

1112

—

133

C₂²

—

1112

—

134

C₂²

—

1112

—

135

C₂²

#17984812

yes

characteristic

136

C₂²

#17984822

yes

characteristic

137

C₂²

#17984830

yes

characteristic

138

C₂²

#17984864

yes

—

139

C₂²

#17984870

yes

—

140

C₄

—

141

C₄

—

556

—

142

C₄

—

1112

—

143

C₄

#17984820

yes

—

144

C₄

#17984828

yes

—

145

C₂

—

1112

—

146

C₂

#22580866

yes

characteristic

147

C₂

#22581100

yes

—

148

C₂

#22581254

yes

—

149

C₂

#22581308

yes

characteristic

150

C₂

#22581318

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	#2025
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×#17984834	C₂	semidirect
4. #22580716	C₂	semidirect
5. #336×C₁₃₉	C₂	semidirect
75. C₁₃₉	#2025	semidirect
81. #336	D₁₃₉	semidirect

H ≅ G/N

type

1. C₂×#17984834

C₂

semidirect

4. #22580716

C₂

semidirect

5. #336×C₁₃₉

C₂

semidirect

75. C₁₃₉

#2025

semidirect

81. #336

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984834	2
C₂×#17984764	2
#22580855	2
#22580716	2
#336×C₁₃₉	2
#2025	139

subgroup

index

#336×C₁₃₉

139

Subgroups — #31807551

Conjugacy classes of subgroups

Complements

Maximal subgroups