Group #31807566 — subgroups

Subgroups — #31807566

Order 17792.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22581584	C₂	1	1	yes	maximal, characteristic
2	#22580963	C₂	1	2	yes	maximal
3	#22580858	C₂	1	1	yes	maximal, characteristic
4	#22580714	C₂	1	2	yes	maximal
5	#343×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
7	#17984841	—	2	2	no	—
8	#17984833	—	2	2	no	—
9	#17984821	—	2	2	no	—
10	#17984801	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
12	#17984767	—	2	2	no	—
13	#17984767	—	2	2	no	—
14	C₁₃₉⋊#95	—	2	2	no	—
15	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	—	2	2	no	—
19	C₂×Dic₂₇₈	—	2	1	no	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	Q₈	1	2	yes	complemented
22	C₂²×Dic₁₃₉	—	2	1	no	—
23	C₂²×Dic₁₃₉	—	2	1	no	—
24	C₄⋊Dic₁₃₉	—	2	2	no	—
25	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
26	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
27	Dic₁₃₉⋊C₄	—	2	4	no	—
28	C₂²⋊Dic₁₃₉	—	2	2	no	—
29	C₄×Dic₁₃₉	—	2	2	no	—
30	C₄×Dic₁₃₉	—	2	2	no	—
31	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₃₉	—	2	2	no	—
34	M₁₆×C₁₃₉	—	2	2	no	—
35	C₂×C₈×C₁₃₉	—	2	2	no	—
36	C₄⋊C₄×C₁₃₉	—	2	2	no	—
37	Dic₂₇₈	—	2	4	no	—
38	Dic₂₇₈	—	4	2	no	—
39	Dic₂₇₈	—	8	2	no	—
40	C₂×Dic₁₃₉	—	2	2	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₂×Dic₁₃₉	—	2	2	no	—
43	C₂×Dic₁₃₉	—	2	2	no	—
44	C₂×Dic₁₃₉	—	4	1	no	—
45	C₂×Dic₁₃₉	—	4	1	no	—
46	C₂×Dic₁₃₉	—	4	1	no	—
47	C₂×Dic₁₃₉	—	4	1	no	—
48	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	—	2	2	no	—
50	C₂×C₄×C₁₃₉	—	4	2	no	—
51	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	#41	1	2	yes	—
55	C₈×C₁₃₉	—	4	2	no	—
56	C₂²×C₁₃₉	—	2	1	no	—
57	C₂²×C₁₃₉	—	2	1	no	—
58	C₂²×C₁₃₉	#120	1	1	yes	characteristic
59	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₃₉	#124	1	1	yes	characteristic
61	C₄×C₁₃₉	—	4	2	no	—
62	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₁₃₉	#124	1	2	yes	—
64	Dic₁₃₉	—	4	2	no	—
65	Dic₁₃₉	—	4	2	no	—
66	Dic₁₃₉	—	4	2	no	—
67	Dic₁₃₉	—	4	2	no	—
68	C₂×C₁₃₉	—	2	1	no	—
69	C₂×C₁₃₉	—	2	1	no	—
70	C₂×C₁₃₉	#393	1	1	yes	characteristic
71	C₂×C₁₃₉	#455	1	1	yes	characteristic
72	C₂×C₁₃₉	#456	1	1	yes	characteristic
73	C₁₃₉	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	139	1	no	maximal, Sylow
75	#544	—	139	1	no	—
76	#419	—	139	2	no	—
77	#391	—	139	1	no	—
78	#343	D₁₃₉	1	1	yes	characteristic
79	#339	—	139	2	no	—
80	C₂²×Q₈	—	139	1	no	—
81	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
82	C₄⋊Q₈	—	278	2	no	—
83	#125	—	278	2	no	—
84	C₂²⋊Q₈	—	278	2	no	—
85	#117	—	139	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄²	—	139	1	no	—
88	#103	—	278	2	no	—
89	#103	—	278	2	no	—
90	#95	—	278	2	no	—
91	C₂×Q₈	—	278	2	no	—
92	C₂×Q₈	—	278	1	no	—
93	C₂×Q₈	—	278	2	no	—
94	C₂×Q₈	—	278	2	no	—
95	C₂×Q₈	—	139	2	no	—
96	C₂²×C₄	—	278	1	no	—
97	C₂²×C₄	—	278	1	no	—
98	C₂²×C₄	D₅₅₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	278	4	no	—
105	C₄⋊C₄	—	278	2	no	—
106	C₄⋊C₄	—	139	2	no	—
107	C₄⋊C₄	—	139	2	no	—
108	C₂²⋊C₄	—	278	2	no	—
109	C₄²	—	278	2	no	—
110	C₄²	—	278	2	no	—
111	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
112	Q₈	—	278	4	no	—
113	Q₈	—	556	2	no	—
114	Q₈	—	1112	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	278	2	no	—
118	C₂×C₄	—	278	2	no	—
119	C₂×C₄	—	278	2	no	—
120	C₂×C₄	—	278	2	no	—
121	C₂×C₄	—	556	1	no	—
122	C₂×C₄	—	556	1	no	—
123	C₂×C₄	—	556	1	no	—
124	C₂×C₄	—	556	1	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	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#17984812	1	1	yes	characteristic
133	C₂²	#17984822	1	1	yes	characteristic
134	C₂²	#17984830	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	556	2	no	—
137	C₄	—	556	2	no	—
138	C₄	—	556	2	no	—
139	C₄	—	556	2	no	—
140	C₄	#17984820	1	2	yes	—
141	C₄	#17984828	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#22580866	1	1	yes	characteristic
145	C₂	#22581133	1	1	yes	characteristic
146	C₂	#22581138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581584

C₂

yes

maximal, characteristic

#22580963

C₂

yes

maximal

#22580858

C₂

yes

maximal, characteristic

#22580714

C₂

yes

maximal

#343×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984767

—

#17984767

—

C₁₃₉⋊#95

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

Q₈

yes

complemented

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

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×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#124

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

#124

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2028

—

139

maximal, Sylow

#544

—

139

—

#419

—

139

—

#391

—

139

—

#343

D₁₃₉

yes

characteristic

#339

—

139

—

C₂²×Q₈

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

278

—

#125

—

278

—

C₂²⋊Q₈

—

278

—

#117

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

—

139

—

#103

—

278

—

#103

—

278

—

#95

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

139

—

107

C₄⋊C₄

—

139

—

108

C₂²⋊C₄

—

278

—

109

C₄²

—

278

—

110

C₄²

—

278

—

111

C₂³

C₂×D₅₅₆

yes

characteristic

112

Q₈

—

278

—

113

Q₈

—

556

—

114

Q₈

—

1112

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

278

—

118

C₂×C₄

—

278

—

119

C₂×C₄

—

278

—

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

—

131

C₂²

—

132

C₂²

#17984812

yes

characteristic

133

C₂²

#17984822

yes

characteristic

134

C₂²

#17984830

yes

characteristic

135

C₄

—

136

C₄

—

556

—

137

C₄

—

556

—

138

C₄

—

556

—

139

C₄

—

556

—

140

C₄

#17984820

yes

—

141

C₄

#17984828

yes

—

142

C₂

—

143

C₂

—

144

C₂

#22580866

yes

characteristic

145

C₂

#22581133

yes

characteristic

146

C₂

#22581138

yes

characteristic

	subgroup	G/H
center	C₂²	#17984812
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#343×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#343	D₁₃₉
hypocenter	C₁₃₉	#2028
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#17984812

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#343×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#120

prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
21. C₂×Dic₂₇₈	Q₈	semidirect
73. C₁₃₉	#2028	semidirect

H ≅ G/N

type

21. C₂×Dic₂₇₈

Q₈

semidirect

73. C₁₃₉

#2028

semidirect

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

subgroup	index
#22581584	2
#22580963	2
#22580858	2
#22580714	2
#343×C₁₃₉	2
#2028	139

subgroup

index

#343×C₁₃₉

139

Subgroups — #31807566

Conjugacy classes of subgroups

Complements

Maximal subgroups