Group #34176127 — subgroups

Subgroups — #34176127

Order 20224.

186 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 27 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	#23761563	C₂	1	2	yes	maximal, complemented
2	C₇₉×#1712	C₂	1	1	yes	maximal, characteristic
3	#19098356	—	2	2	no	—
4	#493×C₇₉	—	2	2	no	—
5	#493×C₇₉	C₄	1	1	yes	characteristic, complemented
6	#493×C₇₉	C₂²	1	1	yes	characteristic
7	C₈²×C₇₉	C₄	1	1	yes	characteristic, complemented
8	#14579076	—	4	2	no	—
9	C₂×D₈×C₇₉	—	4	1	no	—
10	C₂×D₈×C₇₉	—	4	1	no	—
11	C₂×D₈×C₇₉	—	4	4	no	—
12	#127×C₇₉	—	2	1	no	—
13	#127×C₇₉	D₄	1	2	yes	—
14	C₄×C₈×C₇₉	—	2	1	no	—
15	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊Dic₇₉	—	8	2	no	—
17	C₂×D₄×C₇₉	—	2	2	no	—
18	C₂×D₄×C₇₉	—	4	2	no	—
19	C₂×D₄×C₇₉	—	4	2	no	—
20	C₂×D₄×C₇₉	—	4	1	no	—
21	D₈×C₇₉	—	8	2	no	—
22	D₈×C₇₉	—	8	2	no	—
23	D₈×C₇₉	—	8	8	no	—
24	C₂×C₈×C₇₉	—	2	2	no	—
25	C₂×C₈×C₇₉	—	2	1	no	—
26	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂³×C₇₉	—	4	2	no	—
28	C₂³×C₇₉	—	8	1	no	—
29	D₄×C₇₉	—	8	2	no	—
30	D₄×C₇₉	—	8	4	no	—
31	D₄×C₇₉	—	8	4	no	—
32	D₄×C₇₉	—	8	2	no	—
33	C₂×C₄×C₇₉	—	2	1	no	—
34	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
35	C₂×Dic₇₉	—	16	2	no	—
36	C₈×C₇₉	—	2	4	no	—
37	C₈×C₇₉	—	2	2	no	—
38	C₂²×C₇₉	—	8	2	no	—
39	C₂²×C₇₉	—	16	2	no	—
40	C₂²×C₇₉	—	16	2	no	—
41	C₂²×C₇₉	—	16	1	no	—
42	C₂²×C₇₉	#353	1	1	yes	characteristic
43	C₄×C₇₉	—	2	2	no	—
44	C₄×C₇₉	—	2	1	no	—
45	Dic₇₉	—	32	2	no	—
46	#7529	—	79	1	no	maximal, Sylow
47	C₂×C₇₉	—	2	1	no	—
48	C₂×C₇₉	—	16	2	no	—
49	C₂×C₇₉	—	32	1	no	—
50	C₂×C₇₉	#1408	1	1	yes	characteristic
51	#1712	D₇₉	1	1	yes	characteristic
52	#1408	—	79	2	no	—
53	C₇₉	#7529	1	1	yes	characteristic, complemented, Sylow
54	#493	—	2	2	no	—
55	#493	Dic₇₉	1	1	yes	characteristic, complemented
56	#493	C₂×D₇₉	1	1	yes	characteristic
57	#353	—	158	2	no	—
58	C₈²	Dic₇₉	1	1	yes	characteristic, complemented
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	4	1	no	—
61	C₂×D₈	—	4	4	no	—
62	#127	—	2	1	no	—
63	#127	C₂²⋊D₇₉	1	2	yes	—
64	#99	—	316	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	1	no	—
71	D₈	—	8	2	no	—
72	D₈	—	8	2	no	—
73	D₈	—	8	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂²⋊C₄	—	632	2	no	—
77	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
78	C₂³	—	4	2	no	—
79	C₂³	—	8	1	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	8	4	no	—
83	D₄	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	1264	2	no	—
86	C₂×C₄	#14579076	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	16	2	no	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	1	no	—
93	C₂²	#19098356	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2528	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	16	2	no	—
99	C₂	—	32	1	no	—
100	C₂	#23761563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761563

C₂

yes

maximal, complemented

C₇₉×#1712

C₂

yes

maximal, characteristic

#19098356

—

#493×C₇₉

—

#493×C₇₉

C₄

yes

characteristic, complemented

#493×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₄

yes

characteristic, complemented

#14579076

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#127×C₇₉

—

#127×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²⋊Dic₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#99

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#353

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

#7529

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1408

yes

characteristic

#1712

D₇₉

yes

characteristic

#1408

—

C₇₉

#7529

yes

characteristic, complemented, Sylow

#493

—

#493

Dic₇₉

yes

characteristic, complemented

#493

C₂×D₇₉

yes

characteristic

#353

—

158

—

C₈²

Dic₇₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

C₂²⋊D₇₉

yes

—

#99

—

316

—

C₄×C₈

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

632

—

C₄²

C₂²⋊Dic₇₉

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

1264

—

C₂×C₄

#14579076

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#19098356

yes

characteristic

C₄

—

C₄

—

C₄

—

2528

—

C₂

—

C₂

—

C₂

—

100

C₂

#23761563

yes

characteristic

	subgroup	G/H
center	C₂	#23761563
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	#493	C₂×D₇₉
Fitting, prime radical	C₇₉×#1712	C₂
socle	C₂×C₇₉	#1408
cosocle	#493×C₇₉	C₂²
hypercenter	#1712	D₇₉
hypocenter	C₇₉	#7529
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

subgroup

G/H

center

C₂

#23761563

commutator

C₄×C₈×C₇₉

C₂×C₄

Frattini

#493

C₂×D₇₉

Fitting, prime radical

C₇₉×#1712

C₂

socle

C₂×C₇₉

#1408

cosocle

#493×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

N	H ≅ G/N	type
1. #23761563	C₂	semidirect
5. #493×C₇₉	C₄	semidirect
7. C₈²×C₇₉	C₄	semidirect
53. C₇₉	#7529	semidirect
55. #493	Dic₇₉	semidirect
58. C₈²	Dic₇₉	semidirect

H ≅ G/N

type

1. #23761563

C₂

semidirect

5. #493×C₇₉

C₄

semidirect

7. C₈²×C₇₉

C₄

semidirect

53. C₇₉

#7529

semidirect

55. #493

Dic₇₉

semidirect

58. C₈²

Dic₇₉

semidirect

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

subgroup	index
#23761563	2
C₇₉×#1712	2
#7529	79

subgroup

index

#23761563

C₇₉×#1712

#7529

Subgroups — #34176127

Conjugacy classes of subgroups

Complements

Maximal subgroups