Group #34202987 — subgroups

Subgroups — #34202987

Order 20224.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23765656	C₂	1	4	yes	maximal, complemented
2	#23763018	C₂	1	1	yes	maximal, characteristic, complemented
3	#23763015	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579106	—	2	8	no	—
6	#19098968	—	2	2	no	—
7	#19098943	C₂²	1	2	yes	—
8	#19098940	—	2	2	no	—
9	#19098940	C₂²	1	2	yes	—
10	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
11	C₈²×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₁₆	—	2	4	no	—
13	C₂×D₆₃₂	—	2	4	no	—
14	C₂×D₆₃₂	—	4	4	no	—
15	#14579161	—	2	2	no	—
16	#14579158	—	2	2	no	—
17	#14579106	—	4	16	no	—
18	#14579095	—	4	4	no	—
19	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
20	C₄×C₈×C₇₉	D₄	1	2	yes	—
21	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₅₈	—	4	2	no	—
23	C₂×D₃₁₆	—	4	2	no	—
24	C₂×D₃₁₆	—	4	4	no	—
25	Dic₃₁₆	—	4	8	no	—
26	D₆₃₂	—	4	8	no	—
27	D₆₃₂	—	8	8	no	—
28	C₄⋊Dic₇₉	—	4	4	no	—
29	C₂×C₈×C₇₉	D₈	1	4	yes	—
30	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
31	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
32	C₁₆×C₇₉	D₈	1	8	yes	complemented
33	C₂²×D₇₉	—	8	2	no	—
34	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
36	C₂×Dic₇₉	—	8	2	no	—
37	D₃₁₆	—	8	4	no	—
38	D₃₁₆	—	8	8	no	—
39	Dic₁₅₈	—	8	4	no	—
40	C₈×C₇₉	SD₃₂	1	8	yes	complemented
41	C₈×C₇₉	C₂×D₈	1	4	yes	—
42	C₂²×C₇₉	#493	1	1	yes	characteristic
43	C₂×D₇₉	—	16	2	no	—
44	C₂×D₇₉	—	16	4	no	—
45	C₄×C₇₉	#493	1	2	yes	—
46	C₄×C₇₉	C₂×SD₃₂	1	4	yes	—
47	Dic₇₉	—	16	2	no	—
48	#12517	—	79	1	no	maximal, Sylow
49	C₂×C₇₉	#1712	1	1	yes	characteristic
50	C₂×C₇₉	#2248	1	2	yes	—
51	D₇₉	—	32	2	no	—
52	#2248	—	79	4	no	—
53	#1713	—	79	1	no	—
54	#1712	—	79	1	no	—
55	C₈×C₁₆	D₇₉	1	1	yes	characteristic, complemented
56	C₇₉	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	158	8	no	—
58	#500	—	158	2	no	—
59	#494	—	79	2	no	—
60	#493	—	79	2	no	—
61	#493	—	158	2	no	—
62	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
63	C₈²	C₂×D₇₉	1	1	yes	characteristic
64	C₂×Q₁₆	—	158	4	no	—
65	C₂×D₈	—	316	4	no	—
66	C₂×D₈	—	158	4	no	—
67	C₄⋊Q₈	—	158	2	no	—
68	#127	—	158	2	no	—
69	SD₃₂	—	316	16	no	—
70	C₂×C₁₆	D₃₁₆	1	4	yes	—
71	#107	—	316	4	no	—
72	C₄×C₈	D₃₁₆	1	2	yes	—
73	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
74	C₂×Q₈	—	316	2	no	—
75	C₂×D₄	—	316	2	no	—
76	C₂×D₄	—	316	4	no	—
77	Q₁₆	—	316	8	no	—
78	D₈	—	316	8	no	—
79	D₈	—	632	8	no	—
80	C₂×C₈	D₆₃₂	1	4	yes	—
81	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
82	C₄⋊C₄	—	316	4	no	—
83	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
84	C₁₆	D₆₃₂	1	8	yes	complemented
85	C₂³	—	632	2	no	—
86	Q₈	—	632	4	no	—
87	D₄	—	632	4	no	—
88	D₄	—	632	8	no	—
89	C₂×C₄	—	632	2	no	—
90	C₂×C₄	#14579158	1	1	yes	characteristic
91	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
92	C₈	#14579106	1	8	yes	complemented
93	C₈	C₂×D₆₃₂	1	4	yes	—
94	C₂²	—	1264	2	no	—
95	C₂²	—	1264	4	no	—
96	C₂²	#19098940	1	1	yes	characteristic
97	C₄	—	1264	2	no	—
98	C₄	#19098940	1	2	yes	—
99	C₄	C₂×#14579106	1	4	yes	—
100	C₂	—	2528	2	no	—
101	C₂	#23763015	1	1	yes	characteristic
102	C₂	#23765656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765656

C₂

yes

maximal, complemented

#23763018

C₂

yes

maximal, characteristic, complemented

#23763015

C₂

yes

maximal, characteristic

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#14579106

—

#19098968

—

#19098943

C₂²

yes

—

#19098940

—

#19098940

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

—

—

—

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

D₈

yes

complemented

C₂²×D₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

SD₃₂

yes

complemented

C₈×C₇₉

C₂×D₈

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#493

yes

—

C₄×C₇₉

C₂×SD₃₂

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₇₉

yes

characteristic, complemented

C₇₉

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

158

—

#500

—

158

—

#494

—

#493

—

#493

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₂×D₈

—

316

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

158

—

#127

—

158

—

SD₃₂

—

316

—

C₂×C₁₆

D₃₁₆

yes

—

#107

—

316

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

Q₁₆

—

316

—

D₈

—

316

—

D₈

—

632

—

C₂×C₈

D₆₃₂

yes

—

C₂×C₈

C₂×D₃₁₆

yes

—

C₄⋊C₄

—

316

—

C₄²

C₂×D₃₁₆

yes

characteristic

C₁₆

D₆₃₂

yes

complemented

C₂³

—

632

—

Q₈

—

632

—

D₄

—

632

—

D₄

—

632

—

C₂×C₄

—

632

—

C₂×C₄

#14579158

yes

characteristic

C₂×C₄

C₂×D₆₃₂

yes

—

C₈

#14579106

yes

complemented

C₈

C₂×D₆₃₂

yes

—

C₂²

—

1264

—

C₂²

—

1264

—

C₂²

#19098940

yes

characteristic

C₄

—

1264

—

C₄

#19098940

yes

—

C₄

C₂×#14579106

yes

—

100

C₂

—

2528

—

101

C₂

#23763015

yes

characteristic

102

C₂

#23765656

yes

—

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#493
prime radical	#23763015	C₂
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#12517
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19098940

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₈×C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂

C₈×C₁₆

D₇₉

hypocenter

C₇₉

#12517

LCS

G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #23765656	C₂	semidirect
2. #23763018	C₂	semidirect
4. C₈×C₁₆×C₇₉	C₂	semidirect
32. C₁₆×C₇₉	D₈	semidirect
40. C₈×C₇₉	SD₃₂	semidirect
55. C₈×C₁₆	D₇₉	semidirect
56. C₇₉	#12517	semidirect
84. C₁₆	D₆₃₂	semidirect
92. C₈	#14579106	semidirect

H ≅ G/N

type

1. #23765656

C₂

semidirect

2. #23763018

C₂

semidirect

4. C₈×C₁₆×C₇₉

C₂

semidirect

32. C₁₆×C₇₉

D₈

semidirect

40. C₈×C₇₉

SD₃₂

semidirect

55. C₈×C₁₆

D₇₉

semidirect

56. C₇₉

#12517

semidirect

84. C₁₆

D₆₃₂

semidirect

92. C₈

#14579106

semidirect

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

subgroup	index
#23765656	2
#23763018	2
#23763015	2
C₈×C₁₆×C₇₉	2
#12517	79

subgroup

index

C₈×C₁₆×C₇₉

Subgroups — #34202987

Conjugacy classes of subgroups

Complements

Maximal subgroups