Group #34182046 — subgroups

Subgroups — #34182046

Order 20224.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#23763560	C₂	1	2	yes	maximal
2	#23763528	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579095	—	2	4	no	—
5	#19098433	C₂²	1	4	yes	—
6	#19098326	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₈²×C₇₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₉	—	2	8	no	—
11	C₂×C₄⋊Dic₇₉	—	2	4	no	—
12	#14579095	—	4	8	no	—
13	C₂²×C₈×C₇₉	D₄	1	4	yes	—
14	C₂²×C₈×C₇₉	D₄	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₉	Q₈	1	4	yes	—
19	C₄×C₈×C₇₉	Q₈	1	2	yes	—
20	C₂²×Dic₇₉	—	4	4	no	—
21	C₄⋊Dic₇₉	—	4	8	no	—
22	C₄⋊Dic₇₉	—	4	16	no	—
23	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
25	C₂×C₈×C₇₉	—	2	8	no	—
26	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
27	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₇₉	D₈	1	4	yes	—
29	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
30	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
33	C₂³×C₇₉	#127	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
40	C₂×C₄×C₇₉	#135	1	8	yes	—
41	C₂×Dic₇₉	—	8	4	no	—
42	C₂×Dic₇₉	—	8	8	no	—
43	C₈×C₇₉	—	2	8	no	—
44	C₈×C₇₉	#107	1	8	yes	—
45	C₂²×C₇₉	#384	1	1	yes	characteristic
46	C₂²×C₇₉	#493	1	1	yes	characteristic
47	C₂²×C₇₉	#494	1	1	yes	characteristic
48	C₂²×C₇₉	#495	1	4	yes	—
49	C₄×C₇₉	C₂×#107	1	2	yes	—
50	C₄×C₇₉	#427	1	4	yes	—
51	C₄×C₇₉	#499	1	4	yes	—
52	C₄×C₇₉	#500	1	2	yes	—
53	Dic₇₉	—	16	4	no	—
54	#8816	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	#1709	1	2	yes	—
56	C₂×C₇₉	#1710	1	2	yes	—
57	C₂×C₇₉	#1845	1	1	yes	characteristic
58	C₂×C₇₉	#1846	1	2	yes	—
59	#1845	—	79	2	no	—
60	#1834	—	79	4	no	—
61	C₂×C₈²	D₇₉	1	1	yes	characteristic
62	C₇₉	#8816	1	1	yes	characteristic, complemented, Sylow
63	C₂×#107	—	158	4	no	—
64	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
66	#384	—	79	4	no	—
67	#340	—	158	8	no	—
68	C₈²	Dic₇₉	1	4	yes	complemented
69	C₂²×C₈	D₃₁₆	1	4	yes	—
70	C₂²×C₈	D₃₁₆	1	2	yes	—
71	C₂×C₄⋊C₄	—	158	8	no	—
72	C₂×C₄⋊C₄	—	158	4	no	—
73	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
74	#107	—	316	8	no	—
75	C₄×C₈	Dic₁₅₈	1	4	yes	—
76	C₄×C₈	Dic₁₅₈	1	2	yes	—
77	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
78	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
79	C₂²×C₄	—	316	4	no	—
80	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
84	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
85	C₂×C₈	D₆₃₂	1	4	yes	—
86	C₂×C₈	Dic₃₁₆	1	4	yes	—
87	C₄⋊C₄	—	316	8	no	—
88	C₄⋊C₄	—	316	16	no	—
89	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
90	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
91	C₄²	C₂×Dic₁₅₈	1	2	yes	—
92	C₂³	#14579158	1	1	yes	characteristic
93	C₂×C₄	—	632	4	no	—
94	C₂×C₄	—	632	8	no	—
95	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
97	C₂×C₄	#14579161	1	2	yes	—
98	C₂×C₄	#14579161	1	1	yes	characteristic
99	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
100	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
101	C₂×C₄	#14579183	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#14579095	1	8	yes	—
104	C₂²	#19098433	1	1	yes	characteristic
105	C₂²	#19098940	1	1	yes	characteristic
106	C₂²	#19098943	1	1	yes	characteristic
107	C₂²	#19098947	1	4	yes	—
108	C₄	—	1264	4	no	—
109	C₄	C₂×#14579095	1	2	yes	—
110	C₄	#19098601	1	4	yes	—
111	C₄	#19098965	1	4	yes	—
112	C₄	#19098968	1	2	yes	—
113	C₂	#23763006	1	2	yes	—
114	C₂	#23763010	1	2	yes	—
115	C₂	#23763560	1	1	yes	characteristic
116	C₂	#23763564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763560

C₂

yes

maximal

#23763528

C₂

yes

maximal

C₂×C₈²×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579095

—

#19098433

C₂²

yes

—

#19098326

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579095

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#107

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

C₂×#107

yes

—

C₄×C₇₉

#427

yes

—

C₄×C₇₉

#499

yes

—

C₄×C₇₉

#500

yes

—

Dic₇₉

—

#8816

—

maximal, Sylow

C₂×C₇₉

#1709

yes

—

C₂×C₇₉

#1710

yes

—

C₂×C₇₉

#1845

yes

characteristic

C₂×C₇₉

#1846

yes

—

#1845

—

#1834

—

C₂×C₈²

D₇₉

yes

characteristic

C₇₉

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#384

—

#340

—

158

—

C₈²

Dic₇₉

yes

complemented

C₂²×C₈

D₃₁₆

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#107

—

316

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×D₃₁₆

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

D₆₃₂

yes

—

C₂×C₈

Dic₃₁₆

yes

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

C₂²×Dic₇₉

yes

characteristic

C₄²

C₂×Dic₁₅₈

yes

characteristic

C₄²

C₂×Dic₁₅₈

yes

—

C₂³

#14579158

yes

characteristic

C₂×C₄

—

632

—

C₂×C₄

—

632

—

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

C₂×C₄

#14579161

yes

—

C₂×C₄

#14579161

yes

characteristic

C₂×C₄

C₂×D₆₃₂

yes

—

100

C₂×C₄

C₂×Dic₃₁₆

yes

—

101

C₂×C₄

#14579183

yes

—

102

C₈

—

103

C₈

#14579095

yes

—

104

C₂²

#19098433

yes

characteristic

105

C₂²

#19098940

yes

characteristic

106

C₂²

#19098943

yes

characteristic

107

C₂²

#19098947

yes

—

108

C₄

—

1264

—

109

C₄

C₂×#14579095

yes

—

110

C₄

#19098601

yes

—

111

C₄

#19098965

yes

—

112

C₄

#19098968

yes

—

113

C₂

#23763006

yes

—

114

C₂

#23763010

yes

—

115

C₂

#23763560

yes

characteristic

116

C₂

#23763564

yes

—

	subgroup	G/H
center	C₂³	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#8816
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×C₈²×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₂×C₈²

D₇₉

hypocenter

C₇₉

#8816

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₇₉	C₄	semidirect
62. C₇₉	#8816	semidirect
68. C₈²	Dic₇₉	semidirect

H ≅ G/N

type

9. C₈²×C₇₉

C₄

semidirect

62. C₇₉

#8816

semidirect

68. C₈²

Dic₇₉

semidirect

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

subgroup	index
#23763560	2
#23763528	2
C₂×C₈²×C₇₉	2
#8816	79

subgroup

index

#23763560

#23763528

C₂×C₈²×C₇₉

#8816

Subgroups — #34182046

Conjugacy classes of subgroups

Complements

Maximal subgroups