Group #31805542 — subgroups

Subgroups — #31805542

Order 17792.

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

normal subgroups only

This group has 206 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
2	#22581245	C₂	1	2	yes	maximal
3	#22580702	C₂	1	1	yes	maximal, characteristic
4	#22580685	C₂	1	2	yes	maximal
5	#494×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984841	—	2	2	no	—
7	#17984840	—	2	2	no	—
8	C₄×Dic₂₇₈	C₂²	1	2	yes	—
9	#17984807	—	2	2	no	—
10	#17984775	—	2	2	no	—
11	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
12	#17984765	—	2	2	no	—
13	C₂×Q₁₆×C₁₃₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	D₄	1	2	yes	—
19	C₂×C₁₃₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
21	Dic₁₃₉⋊C₄	—	2	4	no	—
22	Dic₁₃₉⋊C₄	—	2	2	no	—
23	C₄×Dic₁₃₉	—	2	2	no	—
24	C₄×Dic₁₃₉	—	2	2	no	—
25	C₂×Q₈×C₁₃₉	—	2	1	no	—
26	C₂×Q₈×C₁₃₉	D₄	1	2	yes	—
27	Q₁₆×C₁₃₉	—	4	4	no	—
28	C₂×C₈×C₁₃₉	—	2	1	no	—
29	C₄⋊C₄×C₁₃₉	—	2	2	no	—
30	C₄⋊C₄×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	Dic₂₇₈	—	2	2	no	—
33	Dic₂₇₈	—	8	2	no	—
34	Dic₂₇₈	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₃₉	—	2	2	no	—
36	C₂×Dic₁₃₉	—	2	2	no	—
37	C₁₃₉⋊C₈	—	4	2	no	—
38	Q₈×C₁₃₉	—	4	2	no	—
39	Q₈×C₁₃₉	—	4	2	no	—
40	C₂×C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	—	4	1	no	—
42	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₃₉	—	2	2	no	—
45	C₂²×C₁₃₉	#120	1	1	yes	characteristic
46	C₄×C₁₃₉	—	2	1	no	—
47	C₄×C₁₃₉	—	4	2	no	—
48	C₄×C₁₃₉	—	8	1	no	—
49	C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₃₉	#137	1	2	yes	—
51	Dic₁₃₉	—	2	4	no	—
52	Dic₁₃₉	—	4	2	no	—
53	C₂×C₁₃₉	#451	1	2	yes	—
54	C₂×C₁₃₉	#453	1	1	yes	characteristic
55	C₁₃₉	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	139	1	no	maximal, Sylow
57	Q₈²	—	139	1	no	—
58	#494	D₁₃₉	1	1	yes	characteristic
59	#477	—	139	2	no	—
60	#333	—	139	1	no	—
61	Q₈⋊C₈	—	139	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	278	2	no	—
64	C₄⋊Q₈	—	278	2	no	—
65	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
67	C₄×Q₈	—	278	2	no	—
68	C₄×Q₈	—	139	2	no	—
69	#107	—	278	2	no	—
70	C₄⋊C₈	—	139	2	no	—
71	#103	—	278	2	no	—
72	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	278	2	no	—
75	C₂×Q₈	—	139	2	no	—
76	C₂×Q₈	C₂²⋊D₁₃₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	278	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	278	4	no	—
83	C₄⋊C₄	—	278	2	no	—
84	C₄⋊C₄	—	139	2	no	—
85	C₄²	—	278	2	no	—
86	C₄²	—	278	2	no	—
87	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	278	2	no	—
91	Q₈	—	1112	2	no	—
92	Q₈	—	139	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	278	2	no	—
96	C₂×C₄	—	278	2	no	—
97	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	556	2	no	—
101	C₂²	#17984810	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	278	4	no	—
106	C₄	—	556	2	no	—
107	C₄	Q₁₆×D₁₃₉	1	2	yes	—
108	C₄	#17984874	1	2	yes	—
109	C₂	#22581107	1	2	yes	—
110	C₂	#22581120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581683

C₂

yes

maximal, characteristic

#22581245

C₂

yes

maximal

#22580702

C₂

yes

maximal, characteristic

#22580685

C₂

yes

maximal

#494×C₁₃₉

C₂

yes

maximal, characteristic

#17984841

—

#17984840

—

C₄×Dic₂₇₈

C₂²

yes

—

#17984807

—

#17984775

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984765

—

C₂×Q₁₆×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

—

Q₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

Q₁₆

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

#137

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#451

yes

—

C₂×C₁₃₉

#453

yes

characteristic

C₁₃₉

#1633

yes

characteristic, complemented, Sylow

#1633

—

139

maximal, Sylow

Q₈²

—

139

—

#494

D₁₃₉

yes

characteristic

#477

—

139

—

#333

—

139

—

Q₈⋊C₈

—

139

—

C₂×Q₁₆

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

#107

—

278

—

C₄⋊C₈

—

139

—

#103

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂²⋊D₁₃₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

—

278

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

278

—

Q₈

—

1112

—

Q₈

—

139

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

D₄×D₁₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₈

—

100

C₈

—

556

—

101

C₂²

#17984810

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

278

—

106

C₄

—

556

—

107

C₄

Q₁₆×D₁₃₉

yes

—

108

C₄

#17984874

yes

—

109

C₂

#22581107

yes

—

110

C₂

#22581120

yes

characteristic

	subgroup	G/H
center	C₂²	#17984810
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#494×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#120
hypercenter	#494	D₁₃₉
hypocenter	C₁₃₉	#1633
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#17984810

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#494×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₂₇₈	Q₁₆	semidirect
55. C₁₃₉	#1633	semidirect

H ≅ G/N

type

34. Dic₂₇₈

Q₁₆

semidirect

55. C₁₃₉

#1633

semidirect

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

subgroup	index
#22581683	2
#22581245	2
#22580702	2
#22580685	2
#494×C₁₃₉	2
#1633	139

subgroup

index

#494×C₁₃₉

139

Subgroups — #31805542

Conjugacy classes of subgroups

Complements

Maximal subgroups