Group #31805483 — subgroups

Subgroups — #31805483

Order 17792.

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

normal subgroups only

This group has 228 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	#22581683	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581327	C₂	1	1	yes	maximal, characteristic
3	#22581154	C₂	1	2	yes	maximal
4	#22580683	C₂	1	2	yes	maximal, complemented
5	#333×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555774	—	2	2	no	—
7	C₂×#14555772	—	2	2	no	—
8	#17984841	—	2	2	no	—
9	#17984840	—	2	2	no	—
10	#17984839	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₇₈	C₂²	1	2	yes	—
12	#17984807	—	2	2	no	—
13	#17984764	—	2	2	no	—
14	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	D₄	1	2	yes	—
19	C₂×D₅₅₆	—	2	1	no	—
20	C₂×D₅₅₆	—	2	2	no	—
21	#14555774	—	4	4	no	—
22	#14555772	—	4	4	no	—
23	C₂×C₁₃₉⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
25	Dic₁₃₉⋊C₄	—	2	4	no	—
26	Dic₁₃₉⋊C₄	—	2	2	no	—
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₄×Dic₁₃₉	—	2	2	no	—
29	C₂×Q₈×C₁₃₉	D₄	1	2	yes	—
30	C₂×C₈×C₁₃₉	—	2	2	no	—
31	C₄⋊C₄×C₁₃₉	—	2	2	no	—
32	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₃₉	—	4	1	no	—
34	Dic₂₇₈	—	2	2	no	—
35	Dic₂₇₈	—	8	2	no	—
36	Dic₂₇₈	SD₁₆	1	4	yes	complemented
37	D₅₅₆	—	4	2	no	—
38	D₅₅₆	—	4	2	no	—
39	D₅₅₆	—	4	2	no	—
40	C₂×Dic₁₃₉	—	2	2	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₁₃₉⋊C₈	—	2	2	no	—
43	Q₈×C₁₃₉	—	4	2	no	—
44	C₂×C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₃₉	—	4	2	no	—
48	C₂²×C₁₃₉	#120	1	1	yes	characteristic
49	C₂×D₁₃₉	—	8	2	no	—
50	C₂×D₁₃₉	—	8	1	no	—
51	C₄×C₁₃₉	—	2	1	no	—
52	C₄×C₁₃₉	—	4	2	no	—
53	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₃₉	#137	1	2	yes	—
55	Dic₁₃₉	—	2	4	no	—
56	Dic₁₃₉	—	4	2	no	—
57	C₂×C₁₃₉	#448	1	2	yes	—
58	C₂×C₁₃₉	#453	1	1	yes	characteristic
59	D₁₃₉	—	16	1	no	—
60	C₁₃₉	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	139	1	no	maximal, Sylow
62	Q₈²	—	139	1	no	—
63	#492	—	139	1	no	—
64	#460	—	139	2	no	—
65	#333	D₁₃₉	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	139	2	no	—
67	C₂×SD₁₆	—	278	2	no	—
68	C₂×SD₁₆	—	278	2	no	—
69	C₄⋊Q₈	—	278	2	no	—
70	C₄⋊Q₈	—	278	2	no	—
71	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
72	#127	—	139	1	no	—
73	C₄×Q₈	—	278	2	no	—
74	C₄×Q₈	—	139	2	no	—
75	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
76	#102	—	278	2	no	—
77	C₄×C₈	—	139	1	no	—
78	C₂×Q₈	—	278	2	no	—
79	C₂×Q₈	—	139	2	no	—
80	C₂×Q₈	D₅₅₆	1	2	yes	—
81	C₂×D₄	—	278	2	no	—
82	C₂×D₄	—	278	1	no	—
83	SD₁₆	—	556	4	no	—
84	SD₁₆	—	556	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	278	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	278	4	no	—
89	C₄⋊C₄	—	278	2	no	—
90	C₄⋊C₄	—	139	2	no	—
91	C₄²	—	278	2	no	—
92	C₄²	—	278	2	no	—
93	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
94	C₂³	—	556	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	278	2	no	—
97	Q₈	—	1112	2	no	—
98	Q₈	—	139	4	no	—
99	D₄	—	556	2	no	—
100	D₄	—	556	2	no	—
101	D₄	—	556	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	278	2	no	—
104	C₂×C₄	—	278	2	no	—
105	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
106	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	278	2	no	—
109	C₂²	—	1112	2	no	—
110	C₂²	—	1112	1	no	—
111	C₂²	#17984812	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	278	4	no	—
115	C₄	—	556	2	no	—
116	C₄	SD₁₆×D₁₃₉	1	2	yes	—
117	C₄	#17984874	1	2	yes	—
118	C₂	—	2224	1	no	—
119	C₂	#22581093	1	2	yes	—
120	C₂	#22581123	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581683

C₂

yes

maximal, characteristic, complemented

#22581327

C₂

yes

maximal, characteristic

#22581154

C₂

yes

maximal

#22580683

C₂

yes

maximal, complemented

#333×C₁₃₉

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₂₇₈

C₂²

yes

—

#17984807

—

#17984764

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

#14555774

—

#14555772

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

SD₁₆

yes

complemented

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

#137

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#448

yes

—

C₂×C₁₃₉

#453

yes

characteristic

D₁₃₉

—

C₁₃₉

#1623

yes

characteristic, complemented, Sylow

#1623

—

139

maximal, Sylow

Q₈²

—

139

—

#492

—

139

—

#460

—

139

—

#333

D₁₃₉

yes

characteristic, complemented

Q₈⋊C₈

—

139

—

C₂×SD₁₆

—

278

—

C₂×SD₁₆

—

278

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#127

—

139

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

#102

—

278

—

C₄×C₈

—

139

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

D₅₅₆

yes

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

SD₁₆

—

556

—

SD₁₆

—

556

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

—

278

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

556

—

Q₈

—

Q₈

—

278

—

Q₈

—

1112

—

Q₈

—

139

—

D₄

—

556

—

100

D₄

—

556

—

101

D₄

—

556

—

102

C₂×C₄

—

103

C₂×C₄

—

278

—

104

C₂×C₄

—

278

—

105

C₂×C₄

D₄×D₁₃₉

yes

—

106

C₂×C₄

C₂×D₅₅₆

yes

characteristic

107

C₈

—

108

C₈

—

278

—

109

C₂²

—

1112

—

110

C₂²

—

1112

—

111

C₂²

#17984812

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

278

—

115

C₄

—

556

—

116

C₄

SD₁₆×D₁₃₉

yes

—

117

C₄

#17984874

yes

—

118

C₂

—

2224

—

119

C₂

#22581093

yes

—

120

C₂

#22581123

yes

characteristic

	subgroup	G/H
center	C₂²	#17984812
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#333×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	#17984839	C₂²
hypercenter	#333	D₁₃₉
hypocenter	C₁₃₉	#1623
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#17984812

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#333×C₁₃₉

C₂

C₂²×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #22581683	C₂	semidirect
4. #22580683	C₂	semidirect
5. #333×C₁₃₉	C₂	semidirect
36. Dic₂₇₈	SD₁₆	semidirect
60. C₁₃₉	#1623	semidirect
65. #333	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581683

C₂

semidirect

4. #22580683

C₂

semidirect

5. #333×C₁₃₉

C₂

semidirect

36. Dic₂₇₈

SD₁₆

semidirect

60. C₁₃₉

#1623

semidirect

65. #333

D₁₃₉

semidirect

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

subgroup	index
#22581683	2
#22581327	2
#22581154	2
#22580683	2
#333×C₁₃₉	2
#1623	139

subgroup

index

#333×C₁₃₉

139

Subgroups — #31805483

Conjugacy classes of subgroups

Complements

Maximal subgroups