Group #31806000 — subgroups

Subgroups — #31806000

Order 17792.

180 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 65 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
2	#22581349	C₂	1	1	yes	maximal, characteristic
3	#22581349	C₂	1	1	yes	maximal, characteristic
4	#22581332	C₂	1	1	yes	maximal, characteristic
5	#22581304	C₂	1	1	yes	maximal, characteristic
6	C₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#487×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555775	—	2	2	no	—
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984842	C₂²	1	1	yes	characteristic
11	#17984775	—	2	2	no	—
12	#17984773	—	2	2	no	—
13	#17984773	—	2	2	no	—
14	#17984767	—	2	2	no	—
15	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	#103×C₁₃₉	—	2	2	no	—
20	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₈	—	2	1	no	—
22	C₂×Dic₂₇₈	—	2	1	no	—
23	#14555775	—	4	4	no	—
24	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₃₉	—	2	1	no	—
27	C₄⋊Dic₁₃₉	—	2	1	no	—
28	C₄⋊Dic₁₃₉	—	2	1	no	—
29	C₄⋊Dic₁₃₉	—	2	1	no	—
30	C₂×Q₈×C₁₃₉	—	2	1	no	—
31	C₂×Q₈×C₁₃₉	—	2	1	no	—
32	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₃₉	—	2	1	no	—
34	C₄⋊C₄×C₁₃₉	—	2	1	no	—
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	SD₁₆	1	4	yes	—
43	C₁₃₉⋊C₈	Q₁₆	1	4	yes	complemented
44	Q₈×C₁₃₉	—	4	2	no	—
45	Q₈×C₁₃₉	—	4	2	no	—
46	C₂×C₄×C₁₃₉	—	4	1	no	—
47	C₂×C₄×C₁₃₉	—	4	1	no	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₃₉	—	2	2	no	—
52	C₂²×C₁₃₉	#127	1	1	yes	characteristic
53	C₄×C₁₃₉	—	8	1	no	—
54	C₄×C₁₃₉	—	8	1	no	—
55	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₃₉	#135	1	2	yes	—
58	Dic₁₃₉	—	8	1	no	—
59	Dic₁₃₉	—	8	1	no	—
60	C₂×C₁₃₉	#492	1	1	yes	characteristic
61	C₂×C₁₃₉	#494	1	1	yes	characteristic
62	C₂×C₁₃₉	#495	1	1	yes	characteristic
63	C₁₃₉	#1707	1	1	yes	characteristic, complemented, Sylow
64	#1707	—	139	1	no	maximal, Sylow
65	#500	—	139	1	no	—
66	#498	—	139	1	no	—
67	#498	—	139	1	no	—
68	#494	—	139	1	no	—
69	#487	—	139	1	no	—
70	#487	D₁₃₉	1	1	yes	characteristic
71	C₈²	—	139	1	no	—
72	C₂×Q₁₆	—	278	2	no	—
73	C₄⋊Q₈	—	139	1	no	—
74	C₄⋊Q₈	—	139	1	no	—
75	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
77	#107	—	278	2	no	—
78	#106	—	278	2	no	—
79	#106	—	278	2	no	—
80	#103	—	2	2	no	—
81	#103	—	278	2	no	—
82	C₄×C₈	—	139	1	no	—
83	C₄×C₈	—	139	1	no	—
84	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	278	1	no	—
88	C₂×Q₈	—	278	1	no	—
89	Q₁₆	—	556	4	no	—
90	C₂×C₈	—	139	2	no	—
91	C₂×C₈	—	139	2	no	—
92	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	278	1	no	—
98	C₄⋊C₄	—	278	1	no	—
99	C₄⋊C₄	—	278	1	no	—
100	C₄⋊C₄	—	278	1	no	—
101	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	556	2	no	—
105	Q₈	—	556	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	556	1	no	—
109	C₂×C₄	—	556	1	no	—
110	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	139	4	no	—
115	C₈	—	139	4	no	—
116	C₂²	#17984838	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1112	1	no	—
120	C₄	—	1112	1	no	—
121	C₄	SD₁₆×D₁₃₉	1	2	yes	—
122	C₄	Q₁₆×D₁₃₉	1	2	yes	—
123	C₄	#17984862	1	2	yes	—
124	C₂	#22581324	1	1	yes	characteristic
125	C₂	#22581331	1	1	yes	characteristic
126	C₂	#22581335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581356

C₂

yes

maximal, characteristic

#22581349

C₂

yes

maximal, characteristic

#22581349

C₂

yes

maximal, characteristic

#22581332

C₂

yes

maximal, characteristic

#22581304

C₂

yes

maximal, characteristic

C₈×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#487×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555775

—

#17984842

C₂²

yes

characteristic

#17984842

C₂²

yes

—

—

—

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#103×C₁₃₉

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

#14555775

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

SD₁₆

yes

—

C₁₃₉⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×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

#127

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

#135

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1707

—

139

maximal, Sylow

#500

—

139

—

#498

—

139

—

#498

—

139

—

#494

—

139

—

#487

—

139

—

#487

D₁₃₉

yes

characteristic

C₈²

—

139

—

C₂×Q₁₆

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#107

—

278

—

#106

—

278

—

#106

—

278

—

#103

—

#103

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

Q₁₆

—

556

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

100

C₄⋊C₄

—

278

—

101

C₄²

C₂²×D₁₃₉

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

556

—

105

Q₈

—

556

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

556

—

109

C₂×C₄

—

556

—

110

C₂×C₄

D₄×D₁₃₉

yes

characteristic

111

C₂×C₄

D₄×D₁₃₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

113

C₈

—

114

C₈

—

139

—

115

C₈

—

139

—

116

C₂²

#17984838

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1112

—

120

C₄

—

1112

—

121

C₄

SD₁₆×D₁₃₉

yes

—

122

C₄

Q₁₆×D₁₃₉

yes

—

123

C₄

#17984862

yes

—

124

C₂

#22581324

yes

characteristic

125

C₂

#22581331

yes

characteristic

126

C₂

#22581335

yes

characteristic

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#487×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#127
hypercenter	#487	D₁₃₉
hypocenter	C₁₃₉	#1707
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#17984838

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#487×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
43. C₁₃₉⋊C₈	Q₁₆	semidirect
63. C₁₃₉	#1707	semidirect

H ≅ G/N

type

43. C₁₃₉⋊C₈

Q₁₆

semidirect

63. C₁₃₉

#1707

semidirect

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

subgroup	index
#22581356	2
#22581349	2
#22581332	2
#22581304	2
C₈×C₁₃₉⋊C₈	2
#487×C₁₃₉	2
#1707	139

subgroup

index

C₈×C₁₃₉⋊C₈

#487×C₁₃₉

#1707

139

Subgroups — #31806000

Conjugacy classes of subgroups

Complements

Maximal subgroups