Group #31805774 — subgroups

Subgroups — #31805774

Order 17792.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22581333	C₂	1	1	yes	maximal, characteristic
2	#22581238	C₂	1	2	yes	maximal
3	#22580706	C₂	1	1	yes	maximal, characteristic
4	#22580700	C₂	1	2	yes	maximal
5	C₈×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₆	—	2	1	no	—
7	C₂×Dic₅₅₆	—	2	1	no	—
8	#17984842	C₂²	1	2	yes	—
9	#17984773	—	2	2	no	—
10	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
11	#17984766	—	2	2	no	—
12	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₉	—	2	1	no	—
14	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₉	—	2	1	no	—
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	—	2	2	no	—
18	Dic₅₅₆	—	4	2	no	—
19	Dic₅₅₆	—	4	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₉	—	2	1	no	—
25	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₉	—	2	1	no	—
28	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
29	C₄²×C₁₃₉	—	2	1	no	—
30	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
31	Dic₂₇₈	—	4	2	no	—
32	Dic₂₇₈	—	4	2	no	—
33	C₂×Dic₁₃₉	—	4	2	no	—
34	C₁₃₉⋊C₈	—	4	2	no	—
35	Q₈×C₁₃₉	—	2	1	no	—
36	Q₈×C₁₃₉	Q₁₆	1	2	yes	complemented
37	C₂×C₄×C₁₃₉	—	2	1	no	—
38	C₂×C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
42	C₈×C₁₃₉	—	2	1	no	—
43	C₈×C₁₃₉	—	4	1	no	—
44	C₈×C₁₃₉	SD₁₆	1	2	yes	—
45	C₂²×C₁₃₉	#121	1	1	yes	characteristic
46	C₄×C₁₃₉	—	2	2	no	—
47	C₄×C₁₃₉	—	2	1	no	—
48	C₄×C₁₃₉	—	4	1	no	—
49	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₁₃₉	#135	1	1	yes	characteristic
52	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₁₃₉	—	8	2	no	—
54	C₂×C₁₃₉	#460	1	1	yes	characteristic
55	C₂×C₁₃₉	#467	1	1	yes	characteristic
56	C₂×C₁₃₉	#473	1	1	yes	characteristic
57	C₁₃₉	#1670	1	1	yes	characteristic, complemented, Sylow
58	#1670	—	139	1	no	maximal, Sylow
59	#494	—	139	1	no	—
60	#475	—	139	2	no	—
61	C₈×Q₈	D₁₃₉	1	1	yes	characteristic
62	#335	—	139	1	no	—
63	#332	—	139	2	no	—
64	C₂×Q₁₆	—	278	1	no	—
65	C₂×Q₁₆	—	278	1	no	—
66	C₄⋊Q₈	—	139	2	no	—
67	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
68	#106	—	278	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	139	2	no	—
71	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
72	#103	—	278	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
75	C₂×Q₈	—	278	2	no	—
76	C₂×Q₈	D₅₅₆	1	1	yes	characteristic
77	Q₁₆	—	556	2	no	—
78	Q₁₆	—	556	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	278	2	no	—
81	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	278	2	no	—
85	C₄⋊C₄	—	278	2	no	—
86	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	556	2	no	—
91	Q₈	—	556	2	no	—
92	Q₈	Dic₅₅₆	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	556	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
97	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
98	C₂×C₄	#14555785	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	556	2	no	—
102	C₈	#14555772	1	2	yes	—
103	C₂²	#17984816	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	1112	2	no	—
108	C₄	C₂×#14555772	1	1	yes	characteristic
109	C₄	C₂×Dic₅₅₆	1	1	yes	characteristic
110	C₄	#17984864	1	1	yes	characteristic
111	C₄	#17984867	1	1	yes	characteristic
112	C₂	#22581153	1	1	yes	characteristic
113	C₂	#22581194	1	1	yes	characteristic
114	C₂	#22581228	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581333

C₂

yes

maximal, characteristic

#22581238

C₂

yes

maximal

#22580706

C₂

yes

maximal, characteristic

#22580700

C₂

yes

maximal

C₈×Q₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

#17984842

C₂²

yes

—

#17984773

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984766

—

C₄×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

—

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

Dic₅₅₆

—

Dic₅₅₆

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

Q₁₆

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

SD₁₆

yes

—

C₂²×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₃₉

C₂×Q₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1670

—

139

maximal, Sylow

#494

—

139

—

#475

—

139

—

C₈×Q₈

D₁₃₉

yes

characteristic

#335

—

139

—

#332

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₄⋊Q₈

—

139

—

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

#106

—

278

—

C₄⋊C₈

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

D₅₅₆

yes

characteristic

Q₁₆

—

556

—

Q₁₆

—

556

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

D₅₅₆

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₃₉

yes

characteristic

Q₈

—

Q₈

—

556

—

Q₈

—

556

—

Q₈

Dic₅₅₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

556

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

#14555785

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

556

—

102

C₈

#14555772

yes

—

103

C₂²

#17984816

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

1112

—

108

C₄

C₂×#14555772

yes

characteristic

109

C₄

C₂×Dic₅₅₆

yes

characteristic

110

C₄

#17984864

yes

characteristic

111

C₄

#17984867

yes

characteristic

112

C₂

#22581153

yes

characteristic

113

C₂

#22581194

yes

characteristic

114

C₂

#22581228

yes

characteristic

	subgroup	G/H
center	C₂²	#17984816
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	C₈×Q₈×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#121
hypercenter	C₈×Q₈	D₁₃₉
hypocenter	C₁₃₉	#1670
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#17984816

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

C₈×Q₈×C₁₃₉

C₂

socle, prime radical

C₂²×C₁₃₉

#121

hypercenter

C₈×Q₈

D₁₃₉

hypocenter

C₁₃₉

#1670

LCS

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
36. Q₈×C₁₃₉	Q₁₆	semidirect
57. C₁₃₉	#1670	semidirect
92. Q₈	Dic₅₅₆	semidirect

H ≅ G/N

type

36. Q₈×C₁₃₉

Q₁₆

semidirect

57. C₁₃₉

#1670

semidirect

92. Q₈

Dic₅₅₆

semidirect

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

subgroup	index
#22581333	2
#22581238	2
#22580706	2
#22580700	2
C₈×Q₈×C₁₃₉	2
#1670	139

subgroup

index

C₈×Q₈×C₁₃₉

139

Subgroups — #31805774

Conjugacy classes of subgroups

Complements

Maximal subgroups