Group #31806055 — subgroups

Subgroups — #31806055

Order 17792.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22581356	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581305	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581304	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581299	C₂	1	1	yes	maximal, characteristic
5	#22581299	C₂	1	1	yes	maximal, characteristic
6	#22580675	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#17984842	C₂²	1	1	yes	characteristic
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984775	—	2	2	no	—
11	#17984767	—	2	2	no	—
12	#17984765	—	2	2	no	—
13	#17984763	—	2	2	no	—
14	#17984763	—	2	2	no	—
15	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₃₉	—	2	2	no	—
18	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	#127×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₈	—	2	1	no	—
22	C₂×Dic₂₇₈	—	2	1	no	—
23	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
25	C₄⋊Dic₁₃₉	—	2	1	no	—
26	C₄⋊Dic₁₃₉	—	2	1	no	—
27	C₄⋊Dic₁₃₉	—	2	1	no	—
28	C₄⋊Dic₁₃₉	—	2	1	no	—
29	C₂×Q₈×C₁₃₉	—	2	1	no	—
30	C₂×D₄×C₁₃₉	—	2	1	no	—
31	C₂×D₄×C₁₃₉	—	2	1	no	—
32	C₂×D₄×C₁₃₉	—	2	1	no	—
33	SD₁₆×C₁₃₉	—	4	4	no	—
34	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₃₉	—	2	1	no	—
36	C₄⋊C₄×C₁₃₉	—	2	1	no	—
37	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
38	Dic₂₇₈	—	4	2	no	—
39	Dic₂₇₈	—	4	2	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₂×Dic₁₃₉	—	4	1	no	—
42	C₁₃₉⋊C₈	—	2	2	no	—
43	C₁₃₉⋊C₈	—	2	2	no	—
44	C₂³×C₁₃₉	—	4	1	no	—
45	Q₈×C₁₃₉	—	4	2	no	—
46	D₄×C₁₃₉	—	4	2	no	—
47	D₄×C₁₃₉	—	4	2	no	—
48	D₄×C₁₃₉	—	4	2	no	—
49	C₂×C₄×C₁₃₉	—	4	1	no	—
50	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₈×C₁₃₉	—	2	2	no	—
54	C₂²×C₁₃₉	—	8	1	no	—
55	C₂²×C₁₃₉	—	8	1	no	—
56	C₂²×C₁₃₉	—	8	1	no	—
57	C₂²×C₁₃₉	#127	1	1	yes	characteristic
58	C₄×C₁₃₉	—	8	1	no	—
59	C₄×C₁₃₉	#135	1	2	yes	—
60	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₁₃₉	#137	1	2	yes	—
62	Dic₁₃₉	—	8	1	no	—
63	Dic₁₃₉	—	8	1	no	—
64	C₂×C₁₃₉	—	16	1	no	—
65	C₂×C₁₃₉	#495	1	1	yes	characteristic
66	C₂×C₁₃₉	#496	1	1	yes	characteristic
67	C₂×C₁₃₉	#497	1	1	yes	characteristic
68	C₁₃₉	#1719	1	1	yes	characteristic, complemented, Sylow
69	#1719	—	139	1	no	maximal, Sylow
70	#500	—	139	1	no	—
71	#492	D₁₃₉	1	1	yes	characteristic
72	#487	—	139	1	no	—
73	#487	—	139	1	no	—
74	#486	—	139	1	no	—
75	#486	—	139	1	no	—
76	#322	—	139	1	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	—	139	1	no	—
79	C₄⋊Q₈	—	139	1	no	—
80	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
81	#127	C₂×D₁₃₉	1	1	yes	characteristic
82	#107	—	278	2	no	—
83	#103	—	278	2	no	—
84	#103	—	278	2	no	—
85	#102	—	278	2	no	—
86	#102	—	278	2	no	—
87	C₄×C₈	—	139	1	no	—
88	C₄×C₈	—	139	1	no	—
89	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	278	1	no	—
92	C₂×Q₈	—	278	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	139	2	no	—
98	C₂×C₈	—	139	2	no	—
99	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	278	1	no	—
103	C₄⋊C₄	—	278	1	no	—
104	C₄⋊C₄	—	278	1	no	—
105	C₄⋊C₄	—	278	1	no	—
106	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
107	C₂³	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	556	2	no	—
110	Q₈	—	556	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	556	1	no	—
116	C₂×C₄	—	556	1	no	—
117	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	278	2	no	—
122	C₈	—	278	2	no	—
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	#17984838	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	1112	1	no	—
129	C₄	—	1112	1	no	—
130	C₄	#17984863	1	2	yes	—
131	C₄	#17984868	1	2	yes	—
132	C₄	#17984871	1	2	yes	—
133	C₂	—	16	1	no	—
134	C₂	#22581334	1	1	yes	characteristic
135	C₂	#22581340	1	1	yes	characteristic
136	C₂	#22581344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581356

C₂

yes

maximal, characteristic, complemented

#22581305

C₂

yes

maximal, characteristic, complemented

#22581304

C₂

yes

maximal, characteristic, complemented

#22581299

C₂

yes

maximal, characteristic

#22581299

C₂

yes

maximal, characteristic

#22580675

C₂

yes

maximal, characteristic, complemented

#492×C₁₃₉

C₂

yes

maximal, characteristic

#17984842

C₂²

yes

characteristic

#17984842

C₂²

yes

—

—

—

—

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

SD₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₉

#137

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1719

—

139

maximal, Sylow

#500

—

139

—

#492

D₁₃₉

yes

characteristic

#487

—

139

—

#487

—

139

—

#486

—

139

—

#486

—

139

—

#322

—

139

—

C₂×SD₁₆

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#127

C₂×D₁₃₉

yes

characteristic

#107

—

278

—

#103

—

278

—

#103

—

278

—

#102

—

278

—

#102

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

278

—

103

C₄⋊C₄

—

278

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄²

C₂²×D₁₃₉

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

556

—

110

Q₈

—

556

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

556

—

116

C₂×C₄

—

556

—

117

C₂×C₄

D₄×D₁₃₉

yes

characteristic

118

C₂×C₄

D₄×D₁₃₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

120

C₈

—

121

C₈

—

278

—

122

C₈

—

278

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#17984838

yes

characteristic

127

C₄

—

128

C₄

—

1112

—

129

C₄

—

1112

—

130

C₄

#17984863

yes

—

131

C₄

#17984868

yes

—

132

C₄

#17984871

yes

—

133

C₂

—

134

C₂

#22581334

yes

characteristic

135

C₂

#22581340

yes

characteristic

136

C₂

#22581344

yes

characteristic

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#492×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
prime radical	#127×C₁₃₉	C₂²
hypercenter	#492	D₁₃₉
hypocenter	C₁₃₉	#1719
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#17984838

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#492×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#127

prime radical

#127×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #22581356	C₂	semidirect
2. #22581305	C₂	semidirect
3. #22581304	C₂	semidirect
6. #22580675	C₂	semidirect
68. C₁₃₉	#1719	semidirect

H ≅ G/N

type

1. #22581356

C₂

semidirect

2. #22581305

C₂

semidirect

3. #22581304

C₂

semidirect

6. #22580675

C₂

semidirect

68. C₁₃₉

#1719

semidirect

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

subgroup	index
#22581356	2
#22581305	2
#22581304	2
#22581299	2
#22580675	2
#492×C₁₃₉	2
#1719	139

subgroup

index

#492×C₁₃₉

139

Subgroups — #31806055

Conjugacy classes of subgroups

Complements

Maximal subgroups