Group #31816917 — subgroups

Subgroups — #31816917

Order 17792.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22581509	C₂	1	2	yes	maximal
2	#22581357	C₂	1	2	yes	maximal, complemented
3	#22581355	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581352	C₂	1	2	yes	maximal, complemented
5	#22581350	C₂	1	2	yes	maximal, complemented
6	#22580990	C₂	1	2	yes	maximal
7	#22580988	C₂	1	2	yes	maximal
8	C₄×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
12	#17984841	C₂²	1	4	yes	—
13	#17984834	C₂²	1	2	yes	—
14	#17984820	—	2	4	no	—
15	#17984807	—	2	4	no	—
16	#17984800	C₂²	1	4	yes	—
17	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
18	#17984776	C₂²	1	2	yes	—
19	#17984775	C₂²	1	4	yes	—
20	#17984774	C₂²	1	2	yes	—
21	#17984773	C₂²	1	4	yes	—
22	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	#125×C₁₃₉	C₂²	1	2	yes	—
26	#107×C₁₃₉	C₂²	1	2	yes	—
27	#106×C₁₃₉	C₂²	1	2	yes	—
28	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₇₈	—	2	4	no	—
30	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
31	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
32	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
34	C₈×D₁₃₉	Q₈	1	4	yes	complemented
35	C₈×D₁₃₉	Q₈	1	4	yes	complemented
36	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
37	Dic₁₃₉⋊C₄	—	2	4	no	—
38	Dic₁₃₉⋊C₄	—	2	4	no	—
39	(C₂×D₁₃₉)⋊C₄	—	2	4	no	—
40	C₄×Dic₁₃₉	—	2	4	no	—
41	C₄×Dic₁₃₉	D₄	1	2	yes	—
42	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₃₉	—	2	4	no	—
46	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
47	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
48	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
49	Dic₂₇₈	—	4	8	no	—
50	C₂×Dic₁₃₉	—	2	4	no	—
51	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
53	C₄×D₁₃₉	—	2	4	no	—
54	C₄×D₁₃₉	C₂×Q₈	1	2	yes	—
55	C₄×D₁₃₉	C₂×Q₈	1	2	yes	—
56	C₁₃₉⋊C₈	C₂×Q₈	1	2	yes	—
57	C₁₃₉⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₃₉	—	2	4	no	—
59	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
62	C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₁₃₉	—	2	2	no	—
65	C₂×D₁₃₉	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₃₉	—	2	2	no	—
67	C₄×C₁₃₉	—	4	4	no	—
68	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₃₉	—	4	4	no	—
71	Dic₁₃₉	C₄⋊Q₈	1	2	yes	—
72	Dic₁₃₉	#135	1	4	yes	—
73	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₃₉	C₂×#135	1	2	yes	—
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	2	no	—
80	#499	—	139	2	no	—
81	#499	—	139	1	no	—
82	#499	D₁₃₉	1	1	yes	characteristic, complemented
83	#498	—	139	2	no	—
84	#427	—	139	2	no	—
85	#427	—	139	2	no	—
86	C₂×C₄×C₈	—	139	1	no	—
87	C₂²×C₈	—	139	1	no	—
88	C₂²×C₈	—	139	1	no	—
89	C₄⋊Q₈	—	139	4	no	—
90	#125	—	139	2	no	—
91	#125	C₂×D₁₃₉	1	2	yes	—
92	C₂²⋊Q₈	—	278	4	no	—
93	C₄×Q₈	—	278	4	no	—
94	#117	—	139	4	no	—
95	C₂×C₄²	—	139	1	no	—
96	#107	—	139	2	no	—
97	#107	—	139	4	no	—
98	#107	C₂×D₁₃₉	1	2	yes	—
99	#106	—	139	2	no	—
100	#106	—	139	4	no	—
101	#106	C₂×D₁₃₉	1	2	yes	—
102	C₄×C₈	—	139	1	no	—
103	C₄×C₈	—	139	1	no	—
104	C₄×C₈	—	139	1	no	—
105	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
106	C₂×Q₈	—	278	4	no	—
107	C₂²×C₄	—	139	2	no	—
108	C₂²×C₄	—	139	1	no	—
109	C₂×C₈	—	139	1	no	—
110	C₂×C₈	—	139	1	no	—
111	C₂×C₈	—	139	4	no	—
112	C₂×C₈	—	139	4	no	—
113	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	278	4	no	—
117	C₄⋊C₄	—	278	4	no	—
118	C₄⋊C₄	—	139	4	no	—
119	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
120	C₂²⋊C₄	—	278	4	no	—
121	C₄²	—	278	4	no	—
122	C₄²	—	139	2	no	—
123	C₄²	—	139	1	no	—
124	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
125	C₂³	—	139	1	no	—
126	Q₈	—	556	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	278	4	no	—
129	C₂×C₄	—	278	4	no	—
130	C₂×C₄	—	139	1	no	—
131	C₂×C₄	—	139	2	no	—
132	C₂×C₄	—	139	2	no	—
133	C₂×C₄	—	139	2	no	—
134	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
135	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
136	C₈	—	139	2	no	—
137	C₈	—	139	2	no	—
138	C₈	Q₈×D₁₃₉	1	2	yes	—
139	C₈	Q₈×D₁₃₉	1	2	yes	—
140	C₂²	—	278	2	no	—
141	C₂²	—	139	2	no	—
142	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	556	4	no	—
146	C₄	—	139	4	no	—
147	C₄	—	139	2	no	—
148	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
150	C₂	—	278	2	no	—
151	C₂	C₄⋊Q₈×D₁₃₉	1	1	yes	characteristic
152	C₂	#135×D₁₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581509

C₂

yes

maximal

#22581357

C₂

yes

maximal, complemented

#22581355

C₂

yes

maximal, characteristic, complemented

#22581352

C₂

yes

maximal, complemented

#22581350

C₂

yes

maximal, complemented

#22580990

C₂

yes

maximal

#22580988

C₂

yes

maximal

C₄×C₈×D₁₃₉

C₂

yes

maximal, characteristic

#499×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₉

C₂²

yes

characteristic

C₂×C₈×D₁₃₉

C₂²

yes

characteristic

#17984841

C₂²

yes

—

#17984834

C₂²

yes

—

#17984820

—

#17984807

—

#17984800

C₂²

yes

—

C₄²×D₁₃₉

C₂²

yes

characteristic

#17984776

C₂²

yes

—

#17984775

C₂²

yes

—

#17984774

C₂²

yes

—

#17984773

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

characteristic

#125×C₁₃₉

C₂²

yes

—

#107×C₁₃₉

C₂²

yes

—

#106×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₈×D₁₃₉

Q₈

yes

complemented

C₈×D₁₃₉

Q₈

yes

complemented

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

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

—

C₄×D₁₃₉

—

C₄×D₁₃₉

C₂×Q₈

yes

—

C₄×D₁₃₉

C₂×Q₈

yes

—

C₁₃₉⋊C₈

C₂×Q₈

yes

—

C₁₃₉⋊C₈

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

C₄⋊Q₈

yes

—

Dic₁₃₉

#135

yes

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#135

yes

—

D₁₃₉

—

C₁₃₉

#3160

yes

characteristic, complemented, Sylow

#3160

—

139

maximal, Sylow

#533

—

139

—

#500

—

139

—

#499

—

139

—

#499

—

139

—

#499

D₁₃₉

yes

characteristic, complemented

#498

—

139

—

#427

—

139

—

#427

—

139

—

C₂×C₄×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

139

—

#125

C₂×D₁₃₉

yes

—

C₂²⋊Q₈

—

278

—

C₄×Q₈

—

278

—

#117

—

139

—

C₂×C₄²

—

139

—

#107

—

139

—

#107

—

139

—

#107

C₂×D₁₃₉

yes

—

#106

—

139

—

100

#106

—

139

—

101

#106

C₂×D₁₃₉

yes

—

102

C₄×C₈

—

139

—

103

C₄×C₈

—

139

—

104

C₄×C₈

—

139

—

105

C₄×C₈

C₂×D₁₃₉

yes

characteristic

106

C₂×Q₈

—

278

—

107

C₂²×C₄

—

139

—

108

C₂²×C₄

—

139

—

109

C₂×C₈

—

139

—

110

C₂×C₈

—

139

—

111

C₂×C₈

—

139

—

112

C₂×C₈

—

139

—

113

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

114

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

278

—

117

C₄⋊C₄

—

278

—

118

C₄⋊C₄

—

139

—

119

C₄⋊C₄

C₂²×D₁₃₉

yes

—

120

C₂²⋊C₄

—

278

—

121

C₄²

—

278

—

122

C₄²

—

139

—

123

C₄²

—

139

—

124

C₄²

C₂²×D₁₃₉

yes

characteristic

125

C₂³

—

139

—

126

Q₈

—

556

—

127

C₂×C₄

—

128

C₂×C₄

—

278

—

129

C₂×C₄

—

278

—

130

C₂×C₄

—

139

—

131

C₂×C₄

—

139

—

132

C₂×C₄

—

139

—

133

C₂×C₄

—

139

—

134

C₂×C₄

D₄×D₁₃₉

yes

—

135

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

136

C₈

—

139

—

137

C₈

—

139

—

138

C₈

Q₈×D₁₃₉

yes

—

139

C₈

Q₈×D₁₃₉

yes

—

140

C₂²

—

278

—

141

C₂²

—

139

—

142

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

556

—

146

C₄

—

139

—

147

C₄

—

139

—

148

C₄

C₂×Q₈×D₁₃₉

yes

characteristic

149

C₄

C₂×Q₈×D₁₃₉

yes

characteristic

150

C₂

—

278

—

151

C₂

C₄⋊Q₈×D₁₃₉

yes

characteristic

152

C₂

#135×D₁₃₉

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#499×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#499	D₁₃₉
hypocenter	C₁₃₉	#3160
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#499×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 ◁ #499

N	H ≅ G/N	type
2. #22581357	C₂	semidirect
3. #22581355	C₂	semidirect
4. #22581352	C₂	semidirect
5. #22581350	C₂	semidirect
9. #499×C₁₃₉	C₂	semidirect
34. C₈×D₁₃₉	Q₈	semidirect
35. C₈×D₁₃₉	Q₈	semidirect
76. C₁₃₉	#3160	semidirect
82. #499	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581357

C₂

semidirect

3. #22581355

C₂

semidirect

4. #22581352

C₂

semidirect

5. #22581350

C₂

semidirect

9. #499×C₁₃₉

C₂

semidirect

34. C₈×D₁₃₉

Q₈

semidirect

35. C₈×D₁₃₉

Q₈

semidirect

76. C₁₃₉

#3160

semidirect

82. #499

D₁₃₉

semidirect

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

subgroup	index
#22581509	2
#22581357	2
#22581355	2
#22581352	2
#22581350	2
#22580990	2
#22580988	2
C₄×C₈×D₁₃₉	2
#499×C₁₃₉	2
#3160	139

subgroup

index

C₄×C₈×D₁₃₉

#499×C₁₃₉

#3160

139

Subgroups — #31816917

Conjugacy classes of subgroups

Complements

Maximal subgroups