Group #34174990 — subgroups

Subgroups — #34174990

Order 20224.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#23761464	C₂	1	2	yes	maximal, complemented
2	C₇₉×#1709	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
4	#494×C₇₉	C₄	1	1	yes	characteristic
5	#493×C₇₉	C₄	1	1	yes	characteristic
6	#486×C₇₉	—	2	2	no	—
7	C₈²×C₇₉	C₂²	1	1	yes	characteristic
8	C₂×C₇₉⋊C₁₆	—	4	2	no	—
9	C₂×Q₁₆×C₇₉	—	4	1	no	—
10	C₂×D₈×C₇₉	—	4	1	no	—
11	C₄⋊Q₈×C₇₉	—	2	1	no	—
12	#127×C₇₉	—	2	1	no	—
13	#102×C₇₉	—	4	2	no	—
14	#102×C₇₉	—	4	2	no	—
15	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₄×C₈×C₇₉	D₄	1	2	yes	—
17	C₇₉⋊C₁₆	—	8	2	no	—
18	C₂×Q₈×C₇₉	—	4	1	no	—
19	C₂×D₄×C₇₉	—	4	1	no	—
20	C₂×D₄×C₇₉	—	4	2	no	—
21	Q₁₆×C₇₉	—	8	2	no	—
22	D₈×C₇₉	—	8	2	no	—
23	C₂×C₈×C₇₉	—	2	1	no	—
24	C₂×C₈×C₇₉	D₈	1	2	yes	—
25	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
26	C₄⋊C₄×C₇₉	—	4	2	no	—
27	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂³×C₇₉	—	8	1	no	—
29	Q₈×C₇₉	—	8	2	no	—
30	D₄×C₇₉	—	8	2	no	—
31	D₄×C₇₉	—	8	2	no	—
32	D₄×C₇₉	—	8	2	no	—
33	C₂×C₄×C₇₉	—	8	1	no	—
34	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
35	C₂×C₄×C₇₉	#102	1	2	yes	—
36	C₈×C₇₉	—	2	2	no	—
37	C₈×C₇₉	—	2	2	no	—
38	C₈×C₇₉	—	2	2	no	—
39	C₂²×C₇₉	—	16	1	no	—
40	C₂²×C₇₉	—	16	2	no	—
41	C₂²×C₇₉	#331	1	1	yes	characteristic
42	C₄×C₇₉	—	2	1	no	—
43	C₄×C₇₉	—	16	1	no	—
44	C₄×C₇₉	#359	1	2	yes	—
45	C₄×C₇₉	#360	1	2	yes	—
46	#7092	—	79	1	no	maximal, Sylow
47	C₂×C₇₉	—	32	1	no	—
48	C₂×C₇₉	#1357	1	1	yes	characteristic
49	C₂×C₇₉	#1360	1	2	yes	—
50	#1709	D₇₉	1	1	yes	characteristic
51	#1367	—	79	2	no	—
52	C₇₉	#7092	1	1	yes	characteristic, complemented, Sylow
53	#494	Dic₇₉	1	1	yes	characteristic
54	#493	Dic₇₉	1	1	yes	characteristic
55	#486	—	2	2	no	—
56	C₄⋊C₁₆	—	158	2	no	—
57	C₈²	C₂×D₇₉	1	1	yes	characteristic
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₄⋊Q₈	—	2	1	no	—
61	#127	—	2	1	no	—
62	C₂×C₁₆	—	316	2	no	—
63	#102	—	4	2	no	—
64	#102	—	4	2	no	—
65	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	Q₁₆	—	8	2	no	—
71	D₈	—	8	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	#11929609	1	2	yes	—
74	C₂×C₈	#11929612	1	2	yes	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
77	C₁₆	—	632	2	no	—
78	C₂³	—	8	1	no	—
79	Q₈	—	8	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	2	no	—
82	D₄	—	8	2	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	#14579078	1	1	yes	characteristic
85	C₂×C₄	#14579082	1	2	yes	—
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	16	1	no	—
90	C₂²	—	16	2	no	—
91	C₂²	#19098307	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	16	1	no	—
94	C₄	#19098370	1	2	yes	—
95	C₄	#19098373	1	2	yes	—
96	C₂	—	32	1	no	—
97	C₂	#23761437	1	1	yes	characteristic
98	C₂	#23761443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23761464

C₂

yes

maximal, complemented

C₇₉×#1709

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₁₆)

—

#494×C₇₉

C₄

yes

characteristic

#493×C₇₉

C₄

yes

characteristic

#486×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×Q₁₆×C₇₉

—

C₂×D₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

#127×C₇₉

—

#102×C₇₉

—

#102×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#100

yes

characteristic

C₂×C₄×C₇₉

#102

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#331

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#359

yes

—

C₄×C₇₉

#360

yes

—

#7092

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1357

yes

characteristic

C₂×C₇₉

#1360

yes

—

#1709

D₇₉

yes

characteristic

#1367

—

C₇₉

#7092

yes

characteristic, complemented, Sylow

#494

Dic₇₉

yes

characteristic

#493

Dic₇₉

yes

characteristic

#486

—

C₄⋊C₁₆

—

158

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

316

—

#102

—

#102

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

#11929609

yes

—

C₂×C₈

#11929612

yes

—

C₄⋊C₄

—

C₄²

C₂²⋊Dic₇₉

yes

characteristic

C₁₆

—

632

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#14579078

yes

characteristic

C₂×C₄

#14579082

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#19098307

yes

characteristic

C₄

—

C₄

—

C₄

#19098370

yes

—

C₄

#19098373

yes

—

C₂

—

C₂

#23761437

yes

characteristic

C₂

#23761443

yes

—

	subgroup	G/H
center	C₂²	#19098307
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1709	C₂
socle	C₂²×C₇₉	#331
prime radical	#493×C₇₉	C₄
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1709	D₇₉
hypocenter	C₇₉	#7092
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#19098307

commutator

C₄×C₈×C₇₉

C₂×C₄

Frattini

C₈²

C₂×D₇₉

Fitting

C₇₉×#1709

C₂

socle

C₂²×C₇₉

#331

prime radical

#493×C₇₉

C₄

cosocle

C₈²×C₇₉

C₂²

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #23761464	C₂	semidirect
52. C₇₉	#7092	semidirect

H ≅ G/N

type

1. #23761464

C₂

semidirect

52. C₇₉

#7092

semidirect

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

subgroup	index
#23761464	2
C₇₉×#1709	2
#7092	79

subgroup

index

#23761464

C₇₉×#1709

#7092

Subgroups — #34174990

Conjugacy classes of subgroups

Complements

Maximal subgroups