Group #31816915 — subgroups

Subgroups — #31816915

Order 17792.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#22581508	C₂	1	2	yes	maximal
2	#22581358	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581354	C₂	1	4	yes	maximal, complemented
4	#22581349	C₂	1	2	yes	maximal, complemented
5	#22580990	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
9	#17984842	C₂²	1	2	yes	—
10	#17984833	C₂²	1	4	yes	—
11	#17984822	—	2	2	no	—
12	#17984818	—	2	2	no	—
13	Q₈×Dic₁₃₉	—	2	2	no	—
14	#17984807	—	2	2	no	—
15	#17984800	C₂²	1	4	yes	—
16	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
17	#17984776	C₂²	1	4	yes	—
18	#17984773	C₂²	1	8	yes	—
19	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
22	#107×C₁₃₉	C₂²	1	4	yes	—
23	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₇₈	—	2	2	no	—
25	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
27	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
29	C₈×D₁₃₉	Q₈	1	8	yes	complemented
30	C₄⋊Dic₁₃₉	—	2	4	no	—
31	C₄⋊Dic₁₃₉	—	2	2	no	—
32	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
33	Dic₁₃₉⋊C₄	—	2	4	no	—
34	(C₂×D₁₃₉)⋊C₄	—	2	4	no	—
35	C₄×Dic₁₃₉	—	2	4	no	—
36	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
37	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
38	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₃₉	—	2	2	no	—
40	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
41	C₄⋊C₄×C₁₃₉	—	2	2	no	—
42	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
43	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
45	Dic₂₇₈	—	4	4	no	—
46	C₂×Dic₁₃₉	—	2	4	no	—
47	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₃₉	—	2	2	no	—
51	C₄×D₁₃₉	—	2	2	no	—
52	C₄×D₁₃₉	C₂×Q₈	1	4	yes	—
53	C₁₃₉⋊C₈	C₂×Q₈	1	4	yes	—
54	Q₈×C₁₃₉	—	4	4	no	—
55	C₂×C₄×C₁₃₉	—	2	4	no	—
56	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
59	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
60	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₁₃₉	—	2	1	no	—
62	C₂×D₁₃₉	—	2	1	no	—
63	C₂×D₁₃₉	C₄⋊Q₈	1	2	yes	—
64	C₄×C₁₃₉	—	4	4	no	—
65	C₄×C₁₃₉	#135	1	2	yes	—
66	C₄×C₁₃₉	#135	1	2	yes	—
67	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
68	Dic₁₃₉	—	2	1	no	—
69	Dic₁₃₉	—	2	1	no	—
70	Dic₁₃₉	—	4	4	no	—
71	Dic₁₃₉	C₄⋊Q₈	1	2	yes	—
72	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
74	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
75	D₁₃₉	—	2	2	no	—
76	C₁₃₉	#3160	1	1	yes	characteristic, complemented, Sylow
77	#3160	—	139	1	no	maximal, Sylow
78	#533	—	139	2	no	—
79	#500	—	139	1	no	—
80	#500	D₁₃₉	1	1	yes	characteristic, complemented
81	#499	—	139	4	no	—
82	#498	—	139	2	no	—
83	#427	—	139	4	no	—
84	C₂×C₄×C₈	—	139	1	no	—
85	C₂²×C₈	—	139	2	no	—
86	C₄⋊Q₈	—	139	2	no	—
87	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
88	#125	—	139	4	no	—
89	C₂²⋊Q₈	—	278	2	no	—
90	C₂²⋊Q₈	—	278	2	no	—
91	C₄×Q₈	—	278	2	no	—
92	C₄×Q₈	—	278	2	no	—
93	#117	—	139	4	no	—
94	C₂×C₄²	—	139	1	no	—
95	#107	—	139	4	no	—
96	#107	C₂×D₁₃₉	1	4	yes	—
97	#106	—	139	8	no	—
98	C₄×C₈	—	139	2	no	—
99	C₄×C₈	—	139	1	no	—
100	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	278	2	no	—
103	C₂²×C₄	—	139	1	no	—
104	C₂²×C₄	—	139	1	no	—
105	C₂²×C₄	—	139	1	no	—
106	C₂×C₈	—	139	2	no	—
107	C₂×C₈	—	139	8	no	—
108	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	278	4	no	—
111	C₄⋊C₄	—	278	4	no	—
112	C₄⋊C₄	—	278	2	no	—
113	C₄⋊C₄	—	139	4	no	—
114	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
115	C₂²⋊C₄	—	278	4	no	—
116	C₄²	—	278	4	no	—
117	C₄²	—	139	1	no	—
118	C₄²	—	139	1	no	—
119	C₄²	—	139	1	no	—
120	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
121	C₂³	—	139	1	no	—
122	Q₈	—	4	4	no	—
123	Q₈	—	556	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	278	2	no	—
126	C₂×C₄	—	278	2	no	—
127	C₂×C₄	—	278	4	no	—
128	C₂×C₄	—	139	1	no	—
129	C₂×C₄	—	139	1	no	—
130	C₂×C₄	—	139	1	no	—
131	C₂×C₄	—	139	4	no	—
132	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
133	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
135	C₈	—	139	4	no	—
136	C₈	Q₈×D₁₃₉	1	4	yes	—
137	C₂²	—	278	1	no	—
138	C₂²	—	278	1	no	—
139	C₂²	—	139	2	no	—
140	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	278	1	no	—
143	C₄	—	278	1	no	—
144	C₄	—	556	4	no	—
145	C₄	—	139	2	no	—
146	C₄	#17984861	1	2	yes	—
147	C₄	#17984865	1	2	yes	—
148	C₄	C₂×Q₈×D₁₃₉	1	2	yes	—
149	C₂	—	278	2	no	—
150	C₂	C₄⋊Q₈×D₁₃₉	1	1	yes	characteristic
151	C₂	C₂×#17984861	1	1	yes	characteristic
152	C₂	C₂×#17984865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581508

C₂

yes

maximal

#22581358

C₂

yes

maximal, characteristic, complemented

#22581354

C₂

yes

maximal, complemented

#22581349

C₂

yes

maximal, complemented

#22580990

C₂

yes

maximal

C₄×C₈×D₁₃₉

C₂

yes

maximal, characteristic

#500×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₉

C₂²

yes

—

#17984842

C₂²

yes

—

#17984833

C₂²

yes

—

#17984822

—

#17984818

—

Q₈×Dic₁₃₉

—

#17984807

—

#17984800

C₂²

yes

—

C₄²×D₁₃₉

C₂²

yes

characteristic

#17984776

C₂²

yes

—

#17984773

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

#107×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×C₄×D₁₃₉

D₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

characteristic

C₂×C₄×D₁₃₉

C₂³

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₈×D₁₃₉

Q₈

yes

complemented

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

characteristic

C₄×Dic₁₃₉

C₂³

yes

characteristic

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂×D₄

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

characteristic

C₂×Dic₁₃₉

C₂×D₄

yes

characteristic

C₂×Dic₁₃₉

C₂×D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

C₂×Q₈

yes

—

C₁₃₉⋊C₈

C₂×Q₈

yes

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₄⋊Q₈

yes

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#3160

yes

characteristic, complemented, Sylow

#3160

—

139

maximal, Sylow

#533

—

139

—

#500

—

139

—

#500

D₁₃₉

yes

characteristic, complemented

#499

—

139

—

#498

—

139

—

#427

—

139

—

C₂×C₄×C₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

139

—

C₂²⋊Q₈

—

278

—

C₂²⋊Q₈

—

278

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

#117

—

139

—

C₂×C₄²

—

139

—

#107

—

139

—

#107

C₂×D₁₃₉

yes

—

#106

—

139

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

100

C₄×C₈

C₂×D₁₃₉

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

278

—

103

C₂²×C₄

—

139

—

104

C₂²×C₄

—

139

—

105

C₂²×C₄

—

139

—

106

C₂×C₈

—

139

—

107

C₂×C₈

—

139

—

108

C₂×C₈

C₂²×D₁₃₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

278

—

111

C₄⋊C₄

—

278

—

112

C₄⋊C₄

—

278

—

113

C₄⋊C₄

—

139

—

114

C₄⋊C₄

C₂²×D₁₃₉

yes

—

115

C₂²⋊C₄

—

278

—

116

C₄²

—

278

—

117

C₄²

—

139

—

118

C₄²

—

139

—

119

C₄²

—

139

—

120

C₄²

C₂²×D₁₃₉

yes

characteristic

121

C₂³

—

139

—

122

Q₈

—

123

Q₈

—

556

—

124

C₂×C₄

—

125

C₂×C₄

—

278

—

126

C₂×C₄

—

278

—

127

C₂×C₄

—

278

—

128

C₂×C₄

—

139

—

129

C₂×C₄

—

139

—

130

C₂×C₄

—

139

—

131

C₂×C₄

—

139

—

132

C₂×C₄

D₄×D₁₃₉

yes

characteristic

133

C₂×C₄

D₄×D₁₃₉

yes

characteristic

134

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

135

C₈

—

139

—

136

C₈

Q₈×D₁₃₉

yes

—

137

C₂²

—

278

—

138

C₂²

—

278

—

139

C₂²

—

139

—

140

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

141

C₄

—

142

C₄

—

278

—

143

C₄

—

278

—

144

C₄

—

556

—

145

C₄

—

139

—

146

C₄

#17984861

yes

—

147

C₄

#17984865

yes

—

148

C₄

C₂×Q₈×D₁₃₉

yes

—

149

C₂

—

278

—

150

C₂

C₄⋊Q₈×D₁₃₉

yes

characteristic

151

C₂

C₂×#17984861

yes

characteristic

152

C₂

C₂×#17984865

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#500×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#500	D₁₃₉
hypocenter	C₁₃₉	#3160
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#500×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂²×D₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #22581358	C₂	semidirect
3. #22581354	C₂	semidirect
4. #22581349	C₂	semidirect
7. #500×C₁₃₉	C₂	semidirect
29. C₈×D₁₃₉	Q₈	semidirect
76. C₁₃₉	#3160	semidirect
80. #500	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581358

C₂

semidirect

3. #22581354

C₂

semidirect

4. #22581349

C₂

semidirect

7. #500×C₁₃₉

C₂

semidirect

29. C₈×D₁₃₉

Q₈

semidirect

76. C₁₃₉

#3160

semidirect

80. #500

D₁₃₉

semidirect

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

subgroup	index
#22581508	2
#22581358	2
#22581354	2
#22581349	2
#22580990	2
C₄×C₈×D₁₃₉	2
#500×C₁₃₉	2
#3160	139

subgroup

index

C₄×C₈×D₁₃₉

#500×C₁₃₉

#3160

139

Subgroups — #31816915

Conjugacy classes of subgroups

Complements

Maximal subgroups