Group #34202974 — subgroups

Subgroups — #34202974

Order 20224.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23765625	C₂	1	4	yes	maximal, complemented
2	#23763015	C₂	1	1	yes	maximal, characteristic
3	#23761317	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098940	—	2	2	no	—
6	#19098940	C₂²	1	2	yes	—
7	#19098363	—	2	8	no	—
8	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₇₉	C₂²	1	2	yes	—
10	#487×C₇₉	—	2	2	no	—
11	C₈²×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×D₆₃₂	—	2	4	no	—
13	C₂×D₆₃₂	—	4	4	no	—
14	#14579158	—	2	2	no	—
15	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
16	C₄⋊Q₈×C₇₉	—	2	2	no	—
17	#107×C₇₉	—	2	4	no	—
18	#103×C₇₉	—	4	4	no	—
19	C₄×C₈×C₇₉	D₄	1	2	yes	—
20	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
21	C₂×D₃₁₆	—	4	2	no	—
22	C₂×D₃₁₆	—	4	4	no	—
23	D₆₃₂	—	4	8	no	—
24	D₆₃₂	—	8	8	no	—
25	C₇₉⋊C₁₆	—	2	4	no	—
26	C₂×Q₈×C₇₉	—	4	2	no	—
27	C₂×C₈×C₇₉	D₈	1	4	yes	—
28	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
29	C₄⋊C₄×C₇₉	—	4	2	no	—
30	C₄⋊C₄×C₇₉	—	4	2	no	—
31	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₇₉	—	8	2	no	—
33	Q₈×C₇₉	—	8	4	no	—
34	C₂×C₄×C₇₉	—	8	2	no	—
35	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
37	D₃₁₆	—	8	4	no	—
38	D₃₁₆	—	8	8	no	—
39	C₈×C₇₉	—	2	4	no	—
40	C₈×C₇₉	#135	1	4	yes	—
41	C₂²×C₇₉	#493	1	1	yes	characteristic
42	C₂×D₇₉	—	16	2	no	—
43	C₂×D₇₉	—	16	4	no	—
44	C₄×C₇₉	—	16	2	no	—
45	C₄×C₇₉	#495	1	2	yes	—
46	C₄×C₇₉	#509	1	4	yes	—
47	#12513	—	79	1	no	maximal, Sylow
48	C₂×C₇₉	#1709	1	1	yes	characteristic
49	C₂×C₇₉	#2250	1	2	yes	—
50	D₇₉	—	32	2	no	—
51	#2240	—	79	4	no	—
52	#1712	—	79	1	no	—
53	#1710	D₇₉	1	1	yes	characteristic, complemented
54	#1311	—	79	1	no	—
55	C₇₉	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	C₂×D₇₉	1	2	yes	—
57	#493	—	79	2	no	—
58	#493	—	158	2	no	—
59	#487	—	2	2	no	—
60	#357	—	158	8	no	—
61	C₄×C₁₆	—	79	2	no	—
62	C₈²	C₂×D₇₉	1	1	yes	characteristic
63	C₂×D₈	—	316	4	no	—
64	C₂×D₈	—	158	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	158	2	no	—
67	C₂×C₁₆	—	79	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
71	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	316	2	no	—
74	C₂×D₄	—	316	4	no	—
75	D₈	—	316	8	no	—
76	D₈	—	632	8	no	—
77	C₂×C₈	#11929609	1	4	yes	—
78	C₂×C₈	D₄×D₇₉	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
82	C₁₆	—	158	4	no	—
83	C₂³	—	632	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	632	4	no	—
86	D₄	—	632	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14579157	1	1	yes	characteristic
89	C₂×C₄	C₂×#11929609	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14579184	1	4	yes	—
92	C₂²	—	1264	2	no	—
93	C₂²	—	1264	4	no	—
94	C₂²	#19098939	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#19098948	1	2	yes	—
97	C₄	#19098998	1	4	yes	—
98	C₂	—	2528	2	no	—
99	C₂	#23763007	1	1	yes	characteristic
100	C₂	#23765667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765625

C₂

yes

maximal, complemented

#23763015

C₂

yes

maximal, characteristic

#23761317

C₂

yes

maximal, characteristic, complemented

C₇₉×#1710

C₂

yes

maximal, characteristic, complemented

#19098940

—

#19098940

C₂²

yes

—

#19098363

—

C₄×C₇₉⋊C₁₆

C₂²

yes

—

#500×C₇₉

C₂²

yes

—

#487×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579158

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

C₄⋊Q₈×C₇₉

—

#107×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

D₆₃₂

—

D₆₃₂

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₈×C₇₉

#135

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#495

yes

—

C₄×C₇₉

#509

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#2240

—

#1712

—

#1710

D₇₉

yes

characteristic, complemented

#1311

—

C₇₉

#12513

yes

characteristic, complemented, Sylow

#500

C₂×D₇₉

yes

—

#493

—

#493

—

158

—

#487

—

#357

—

158

—

C₄×C₁₆

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×D₈

—

316

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

#127

—

158

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

D₈

—

316

—

D₈

—

632

—

C₂×C₈

#11929609

yes

—

C₂×C₈

D₄×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

C₁₆

—

158

—

C₂³

—

632

—

Q₈

—

D₄

—

632

—

D₄

—

632

—

C₂×C₄

—

C₂×C₄

#14579157

yes

characteristic

C₂×C₄

C₂×#11929609

yes

—

C₈

—

C₈

#14579184

yes

—

C₂²

—

1264

—

C₂²

—

1264

—

C₂²

#19098939

yes

characteristic

C₄

—

C₄

#19098948

yes

—

C₄

#19098998

yes

—

C₂

—

2528

—

C₂

#23763007

yes

characteristic

100

C₂

#23765667

yes

—

	subgroup	G/H
center	C₂²	#19098939
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1710	C₂
socle	C₂²×C₇₉	#493
prime radical	#23763015	C₂
hypercenter	#1710	D₇₉
hypocenter	C₇₉	#12513
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19098939

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1710

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #23765625	C₂	semidirect
3. #23761317	C₂	semidirect
4. C₇₉×#1710	C₂	semidirect
53. #1710	D₇₉	semidirect
55. C₇₉	#12513	semidirect

H ≅ G/N

type

1. #23765625

C₂

semidirect

3. #23761317

C₂

semidirect

4. C₇₉×#1710

C₂

semidirect

53. #1710

D₇₉

semidirect

55. C₇₉

#12513

semidirect

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

subgroup	index
#23765625	2
#23763015	2
#23761317	2
C₇₉×#1710	2
#12513	79

subgroup

index

C₇₉×#1710

Subgroups — #34202974

Conjugacy classes of subgroups

Complements

Maximal subgroups