Group #31816988 — subgroups

Subgroups — #31816988

Order 17792.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#22581303	C₂	1	2	yes	maximal, complemented
4	#22581191	C₂	1	4	yes	maximal
5	Q₁₆×Dic₁₃₉	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#494×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
9	#17984842	C₂²	1	2	yes	—
10	#17984833	—	2	2	no	—
11	#17984822	—	2	2	no	—
12	#17984818	C₂²	1	4	yes	—
13	Q₈×Dic₁₃₉	C₂²	1	4	yes	—
14	#17984807	—	2	2	no	—
15	#17984800	—	2	2	no	—
16	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
17	#17984776	C₂²	1	4	yes	—
18	#17984766	C₂²	1	8	yes	—
19	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₉	C₂²	1	2	yes	—
21	C₂×Q₁₆×C₁₃₉	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₁₃₉	C₂²	1	2	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	—	2	4	no	—
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	C₂³	1	4	yes	—
40	Q₁₆×C₁₃₉	—	2	8	no	—
41	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₃₉	—	2	2	no	—
43	C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
46	Dic₂₇₈	—	4	4	no	—
47	C₂×Dic₁₃₉	—	2	4	no	—
48	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₄×D₁₃₉	—	2	2	no	—
52	C₄×D₁₃₉	—	2	2	no	—
53	C₄×D₁₃₉	Q₁₆	1	4	yes	complemented
54	C₁₃₉⋊C₈	—	2	2	no	—
55	Q₈×C₁₃₉	—	2	8	no	—
56	C₂×C₄×C₁₃₉	—	2	4	no	—
57	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₃₉	#41	1	4	yes	—
61	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₃₉	—	2	1	no	—
63	C₂×D₁₃₉	—	2	1	no	—
64	C₂×D₁₃₉	C₂×Q₁₆	1	2	yes	—
65	C₄×C₁₃₉	—	4	4	no	—
66	C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
67	C₄×C₁₃₉	#135	1	2	yes	—
68	C₄×C₁₃₉	C₂×#41	1	2	yes	—
69	Dic₁₃₉	—	2	1	no	—
70	Dic₁₃₉	—	2	1	no	—
71	Dic₁₃₉	—	4	4	no	—
72	Dic₁₃₉	C₂×Q₁₆	1	2	yes	—
73	C₂×C₁₃₉	#532	1	1	yes	characteristic
74	C₂×C₁₃₉	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
76	D₁₃₉	—	2	2	no	—
77	C₁₃₉	#3170	1	1	yes	characteristic, complemented, Sylow
78	#3170	—	139	1	no	maximal, Sylow
79	#533	—	139	2	no	—
80	#500	—	139	1	no	—
81	#494	D₁₃₉	1	1	yes	characteristic, complemented
82	#487	—	139	2	no	—
83	#467	—	139	4	no	—
84	C₄×Q₁₆	—	139	4	no	—
85	C₂×C₄×C₈	—	139	1	no	—
86	C₂×Q₁₆	C₂×D₁₃₉	1	4	yes	—
87	C₂²×C₈	—	139	2	no	—
88	C₄⋊Q₈	—	139	2	no	—
89	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
90	#125	—	278	2	no	—
91	C₂²⋊Q₈	—	278	2	no	—
92	C₂²⋊Q₈	—	139	4	no	—
93	C₄×Q₈	—	278	2	no	—
94	C₄×Q₈	—	139	4	no	—
95	#117	—	278	2	no	—
96	C₂×C₄²	—	139	1	no	—
97	#107	—	139	4	no	—
98	#103	—	139	8	no	—
99	C₄×C₈	—	139	2	no	—
100	C₄×C₈	—	139	1	no	—
101	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
102	C₂×Q₈	—	278	2	no	—
103	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
104	C₂²×C₄	—	139	1	no	—
105	C₂²×C₄	—	139	1	no	—
106	C₂²×C₄	—	139	1	no	—
107	Q₁₆	—	2	8	no	—
108	C₂×C₈	—	278	4	no	—
109	C₂×C₈	—	139	2	no	—
110	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	278	4	no	—
114	C₄⋊C₄	—	278	4	no	—
115	C₄⋊C₄	—	278	2	no	—
116	C₄⋊C₄	—	139	4	no	—
117	C₂²⋊C₄	—	278	4	no	—
118	C₄²	—	278	4	no	—
119	C₄²	—	139	1	no	—
120	C₄²	—	139	1	no	—
121	C₄²	—	139	1	no	—
122	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
123	C₂³	—	139	1	no	—
124	Q₈	—	2	8	no	—
125	Q₈	—	556	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	278	2	no	—
128	C₂×C₄	—	278	2	no	—
129	C₂×C₄	—	278	4	no	—
130	C₂×C₄	—	139	1	no	—
131	C₂×C₄	—	139	1	no	—
132	C₂×C₄	—	139	1	no	—
133	C₂×C₄	—	139	4	no	—
134	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
137	C₈	—	278	2	no	—
138	C₈	#14555784	1	4	yes	—
139	C₂²	—	278	1	no	—
140	C₂²	—	278	1	no	—
141	C₂²	—	139	2	no	—
142	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	278	1	no	—
145	C₄	—	278	1	no	—
146	C₄	—	556	4	no	—
147	C₄	—	139	2	no	—
148	C₄	Q₁₆×D₁₃₉	1	2	yes	—
149	C₄	#17984865	1	2	yes	—
150	C₄	C₂×#14555784	1	2	yes	—
151	C₂	—	278	2	no	—
152	C₂	#22581501	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₁₃₉	1	1	yes	characteristic
154	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

#22581303

C₂

yes

maximal, complemented

#22581191

C₂

yes

maximal

Q₁₆×Dic₁₃₉

C₂

yes

maximal, complemented

C₄×C₈×D₁₃₉

C₂

yes

maximal, characteristic

#494×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₉

C₂²

yes

—

#17984842

C₂²

yes

—

#17984833

—

#17984822

—

#17984818

C₂²

yes

—

Q₈×Dic₁₃₉

C₂²

yes

—

#17984807

—

#17984800

—

C₄²×D₁₃₉

C₂²

yes

characteristic

#17984776

C₂²

yes

—

#17984766

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

—

C₂×Q₁₆×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×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

—

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

C₂³

yes

—

Q₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

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

Q₁₆

yes

complemented

C₁₃₉⋊C₈

—

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

#41

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

C₂×#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₂×Q₁₆

yes

—

C₂×C₁₃₉

#532

yes

characteristic

C₂×C₁₃₉

C₂²×Q₁₆

yes

characteristic

C₂×C₁₃₉

C₂×#135

yes

characteristic

D₁₃₉

—

C₁₃₉

#3170

yes

characteristic, complemented, Sylow

#3170

—

139

maximal, Sylow

#533

—

139

—

#500

—

139

—

#494

D₁₃₉

yes

characteristic, complemented

#487

—

139

—

#467

—

139

—

C₄×Q₁₆

—

139

—

C₂×C₄×C₈

—

139

—

C₂×Q₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

278

—

C₂²⋊Q₈

—

278

—

C₂²⋊Q₈

—

139

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

#117

—

278

—

C₂×C₄²

—

139

—

#107

—

139

—

#103

—

139

—

C₄×C₈

—

139

—

100

C₄×C₈

—

139

—

101

C₄×C₈

C₂×D₁₃₉

yes

characteristic

102

C₂×Q₈

—

278

—

103

C₂×Q₈

C₂²×D₁₃₉

yes

—

104

C₂²×C₄

—

139

—

105

C₂²×C₄

—

139

—

106

C₂²×C₄

—

139

—

107

Q₁₆

—

108

C₂×C₈

—

278

—

109

C₂×C₈

—

139

—

110

C₂×C₈

C₂²×D₁₃₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

278

—

114

C₄⋊C₄

—

278

—

115

C₄⋊C₄

—

278

—

116

C₄⋊C₄

—

139

—

117

C₂²⋊C₄

—

278

—

118

C₄²

—

278

—

119

C₄²

—

139

—

120

C₄²

—

139

—

121

C₄²

—

139

—

122

C₄²

C₂²×D₁₃₉

yes

characteristic

123

C₂³

—

139

—

124

Q₈

—

125

Q₈

—

556

—

126

C₂×C₄

—

127

C₂×C₄

—

278

—

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

characteristic

135

C₂×C₄

D₄×D₁₃₉

yes

characteristic

136

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

137

C₈

—

278

—

138

C₈

#14555784

yes

—

139

C₂²

—

278

—

140

C₂²

—

278

—

141

C₂²

—

139

—

142

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

143

C₄

—

144

C₄

—

278

—

145

C₄

—

278

—

146

C₄

—

556

—

147

C₄

—

139

—

148

C₄

Q₁₆×D₁₃₉

yes

—

149

C₄

#17984865

yes

—

150

C₄

C₂×#14555784

yes

—

151

C₂

—

278

—

152

C₂

#22581501

yes

characteristic

153

C₂

C₂×Q₁₆×D₁₃₉

yes

characteristic

154

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	#494×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#494	D₁₃₉
hypocenter	C₁₃₉	#3170
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#494×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 ◁ #494

N	H ≅ G/N	type
2. #22581358	C₂	semidirect
3. #22581303	C₂	semidirect
5. Q₁₆×Dic₁₃₉	C₂	semidirect
7. #494×C₁₃₉	C₂	semidirect
53. C₄×D₁₃₉	Q₁₆	semidirect
77. C₁₃₉	#3170	semidirect
81. #494	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581358

C₂

semidirect

3. #22581303

C₂

semidirect

5. Q₁₆×Dic₁₃₉

C₂

semidirect

7. #494×C₁₃₉

C₂

semidirect

53. C₄×D₁₃₉

Q₁₆

semidirect

77. C₁₃₉

#3170

semidirect

81. #494

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
#22581303	2
#22581191	2
Q₁₆×Dic₁₃₉	2
C₄×C₈×D₁₃₉	2
#494×C₁₃₉	2
#3170	139

subgroup

index

Q₁₆×Dic₁₃₉

C₄×C₈×D₁₃₉

#494×C₁₃₉

#3170

139

Subgroups — #31816988

Conjugacy classes of subgroups

Complements

Maximal subgroups