Group #31807526 — subgroups

Subgroups — #31807526

Order 17792.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#17984875	C₂	1	2	yes	maximal
2	C₂²×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984760	C₂	1	2	yes	maximal, complemented
4	#22580919	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555785	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
8	#17984875	—	2	4	no	—
9	#17984875	—	2	4	no	—
10	C₂×Dic₅₅₆	—	2	4	no	—
11	C₂×Dic₅₅₆	—	2	2	no	—
12	C₂×Dic₅₅₆	—	2	2	no	—
13	C₂×#14555774	—	2	2	no	—
14	C₂×#14555769	C₂²	1	1	yes	characteristic
15	#17984760	—	2	4	no	—
16	(C₂×D₁₃₉)⋊C₈	—	2	2	no	—
17	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	#14555785	—	2	2	no	—
20	#14555785	—	2	2	no	—
21	C₂×Dic₂₇₈	—	2	2	no	—
22	C₂×Dic₂₇₈	—	2	2	no	—
23	C₂×Dic₂₇₈	—	2	2	no	—
24	C₂×Dic₂₇₈	—	2	2	no	—
25	C₂×Dic₂₇₈	—	2	2	no	—
26	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
27	C₂×Dic₂₇₈	D₄	1	4	yes	—
28	C₂×D₅₅₆	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
30	C₂²×Dic₁₃₉	—	2	2	no	—
31	C₂×C₄×D₁₃₉	—	2	1	no	—
32	Dic₅₅₆	—	2	8	no	—
33	Dic₅₅₆	—	2	8	no	—
34	#14555774	—	2	8	no	—
35	#14555769	—	2	2	no	—
36	C₂×C₁₃₉⋊C₈	—	2	1	no	—
37	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₃₉	—	2	2	no	—
39	M₁₆×C₁₃₉	D₄	1	4	yes	—
40	C₂×C₈×C₁₃₉	—	2	2	no	—
41	C₂×C₈×C₁₃₉	—	2	2	no	—
42	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
43	C₂²×D₁₃₉	—	2	1	no	—
44	Dic₂₇₈	—	2	4	no	—
45	Dic₂₇₈	—	2	4	no	—
46	Dic₂₇₈	—	2	2	no	—
47	Dic₂₇₈	—	4	2	no	—
48	Dic₂₇₈	—	4	2	no	—
49	D₅₅₆	—	2	2	no	—
50	C₂²⋊D₁₃₉	—	4	2	no	—
51	C₂×Dic₁₃₉	—	2	1	no	—
52	C₂×Dic₁₃₉	—	2	4	no	—
53	C₂×Dic₁₃₉	—	4	2	no	—
54	C₂×Dic₁₃₉	—	4	2	no	—
55	C₄×D₁₃₉	—	4	1	no	—
56	C₄×D₁₃₉	—	4	1	no	—
57	C₁₃₉⋊C₈	—	4	1	no	—
58	C₂³×C₁₃₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
63	C₈×C₁₃₉	—	2	2	no	—
64	C₈×C₁₃₉	—	2	4	no	—
65	C₂²×C₁₃₉	—	2	2	no	—
66	C₂²×C₁₃₉	#121	1	2	yes	—
67	C₂²×C₁₃₉	#121	1	1	yes	characteristic
68	C₂×D₁₃₉	—	4	1	no	—
69	C₂×D₁₃₉	—	4	2	no	—
70	C₄×C₁₃₉	#120	1	1	yes	characteristic
71	C₄×C₁₃₉	#120	1	2	yes	—
72	C₄×C₁₃₉	#127	1	1	yes	characteristic
73	Dic₁₃₉	—	4	1	no	—
74	Dic₁₃₉	—	4	4	no	—
75	C₂×C₁₃₉	—	2	2	no	—
76	C₂×C₁₃₉	#392	1	1	yes	characteristic
77	C₂×C₁₃₉	#473	1	2	yes	—
78	D₁₃₉	—	8	1	no	—
79	C₁₃₉	#2020	1	1	yes	characteristic, complemented, Sylow
80	#2020	—	139	1	no	maximal, Sylow
81	C₂×#137	—	139	2	no	—
82	C₂²×Q₁₆	—	139	1	no	—
83	C₂×#101	—	139	2	no	—
84	#408	—	139	1	no	—
85	#341	D₁₃₉	1	1	yes	characteristic, complemented
86	C₂×#41	—	139	1	no	—
87	C₂²×Q₈	—	139	2	no	—
88	#137	—	278	4	no	—
89	#137	—	278	4	no	—
90	C₂×Q₁₆	—	278	4	no	—
91	C₂×Q₁₆	—	278	2	no	—
92	C₂×Q₁₆	—	278	2	no	—
93	C₂×SD₁₆	—	278	2	no	—
94	C₂×M₁₆	—	139	1	no	—
95	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
96	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
97	#101	—	278	4	no	—
98	C₂²⋊C₈	—	278	2	no	—
99	#41	—	278	2	no	—
100	#41	—	278	2	no	—
101	C₂×Q₈	—	278	2	no	—
102	C₂×Q₈	—	278	2	no	—
103	C₂×Q₈	—	278	2	no	—
104	C₂×Q₈	—	278	2	no	—
105	C₂×Q₈	—	278	2	no	—
106	C₂×Q₈	—	139	1	no	—
107	C₂×Q₈	—	139	4	no	—
108	C₂×D₄	—	278	1	no	—
109	C₂×D₄	—	139	1	no	—
110	C₂²×C₄	—	278	2	no	—
111	C₂²×C₄	—	278	1	no	—
112	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
113	Q₁₆	—	278	8	no	—
114	Q₁₆	—	278	8	no	—
115	SD₁₆	—	278	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	278	2	no	—
118	M₁₆	D₅₅₆	1	4	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	278	1	no	—
122	C₂×C₈	D₅₅₆	1	2	yes	—
123	C₂³	—	278	1	no	—
124	C₂³	#14555786	1	1	yes	characteristic
125	Q₈	—	278	4	no	—
126	Q₈	—	278	4	no	—
127	Q₈	—	278	2	no	—
128	Q₈	—	556	2	no	—
129	Q₈	—	556	2	no	—
130	D₄	—	278	2	no	—
131	D₄	—	556	2	no	—
132	C₂×C₄	—	278	4	no	—
133	C₂×C₄	—	278	1	no	—
134	C₂×C₄	—	556	2	no	—
135	C₂×C₄	—	556	2	no	—
136	C₂×C₄	—	556	1	no	—
137	C₂×C₄	—	556	1	no	—
138	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
139	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
141	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	—	556	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	556	1	no	—
147	C₂²	—	556	2	no	—
148	C₂²	#17984817	1	2	yes	—
149	C₂²	#17984817	1	1	yes	characteristic
150	C₄	—	556	1	no	—
151	C₄	—	556	4	no	—
152	C₄	#17984812	1	1	yes	characteristic
153	C₄	#17984812	1	2	yes	—
154	C₄	#17984839	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	1112	1	no	—
157	C₂	#22580863	1	1	yes	characteristic
158	C₂	#22581229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984875

C₂

yes

maximal

C₂²×Dic₅₅₆

C₂

yes

maximal, characteristic, complemented

C₂×#17984760

C₂

yes

maximal, complemented

#22580919

C₂

yes

maximal, characteristic

#341×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555785

C₂²

yes

characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

#17984875

—

#17984875

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×#14555774

—

C₂×#14555769

C₂²

yes

characteristic

#17984760

—

(C₂×D₁₃₉)⋊C₈

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#14555785

—

#14555785

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

characteristic

C₂×Dic₂₇₈

D₄

yes

—

C₂×D₅₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

—

Dic₅₅₆

—

Dic₅₅₆

—

#14555774

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#121

yes

—

C₂²×C₁₃₉

#121

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

#120

yes

—

C₄×C₁₃₉

#127

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#392

yes

characteristic

C₂×C₁₃₉

#473

yes

—

D₁₃₉

—

C₁₃₉

#2020

yes

characteristic, complemented, Sylow

#2020

—

139

maximal, Sylow

C₂×#137

—

139

—

C₂²×Q₁₆

—

139

—

C₂×#101

—

139

—

#408

—

139

—

#341

D₁₃₉

yes

characteristic, complemented

C₂×#41

—

139

—

C₂²×Q₈

—

139

—

#137

—

278

—

#137

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×SD₁₆

—

278

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#101

—

278

—

C₂²⋊C₈

—

278

—

#41

—

278

—

100

#41

—

278

—

101

C₂×Q₈

—

278

—

102

C₂×Q₈

—

278

—

103

C₂×Q₈

—

278

—

104

C₂×Q₈

—

278

—

105

C₂×Q₈

—

278

—

106

C₂×Q₈

—

139

—

107

C₂×Q₈

—

139

—

108

C₂×D₄

—

278

—

109

C₂×D₄

—

139

—

110

C₂²×C₄

—

278

—

111

C₂²×C₄

—

278

—

112

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

113

Q₁₆

—

278

—

114

Q₁₆

—

278

—

115

SD₁₆

—

278

—

116

M₁₆

—

117

M₁₆

—

278

—

118

M₁₆

D₅₅₆

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

278

—

122

C₂×C₈

D₅₅₆

yes

—

123

C₂³

—

278

—

124

C₂³

#14555786

yes

characteristic

125

Q₈

—

278

—

126

Q₈

—

278

—

127

Q₈

—

278

—

128

Q₈

—

556

—

129

Q₈

—

556

—

130

D₄

—

278

—

131

D₄

—

556

—

132

C₂×C₄

—

278

—

133

C₂×C₄

—

278

—

134

C₂×C₄

—

556

—

135

C₂×C₄

—

556

—

136

C₂×C₄

—

556

—

137

C₂×C₄

—

556

—

138

C₂×C₄

D₄×D₁₃₉

yes

—

139

C₂×C₄

D₄×D₁₃₉

yes

characteristic

140

C₂×C₄

C₂×D₅₅₆

yes

—

141

C₂×C₄

C₂×D₅₅₆

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

556

—

145

C₂²

—

146

C₂²

—

556

—

147

C₂²

—

556

—

148

C₂²

#17984817

yes

—

149

C₂²

#17984817

yes

characteristic

150

C₄

—

556

—

151

C₄

—

556

—

152

C₄

#17984812

yes

characteristic

153

C₄

#17984812

yes

—

154

C₄

#17984839

yes

characteristic

155

C₂

—

156

C₂

—

1112

—

157

C₂

#22580863

yes

characteristic

158

C₂

#22581229

yes

—

	subgroup	G/H
center	C₂²	#17984817
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#341×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#121
prime radical	C₂×#14555785	C₂²
hypercenter	#341	D₁₃₉
hypocenter	C₁₃₉	#2020
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#17984817

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#341×C₁₃₉

C₂

C₂²×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂²×Dic₅₅₆	C₂	semidirect
3. C₂×#17984760	C₂	semidirect
5. #341×C₁₃₉	C₂	semidirect
79. C₁₃₉	#2020	semidirect
85. #341	D₁₃₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₅₅₆

C₂

semidirect

3. C₂×#17984760

C₂

semidirect

5. #341×C₁₃₉

C₂

semidirect

79. C₁₃₉

#2020

semidirect

85. #341

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984875	2
C₂²×Dic₅₅₆	2
C₂×#17984760	2
#22580919	2
#341×C₁₃₉	2
#2020	139

subgroup

index

C₂²×Dic₅₅₆

#341×C₁₃₉

139

Subgroups — #31807526

Conjugacy classes of subgroups

Complements

Maximal subgroups