Group #23776553 — subgroups

Subgroups — #23776553

Order 10112.

350 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 129 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	#19099380	C₂	1	2	yes	maximal, complemented
2	#19099006	C₂	1	6	yes	maximal, complemented
3	#19098995	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₆₃₂	C₂	1	3	yes	maximal, complemented
5	#504×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579208	—	2	2	no	—
7	#14579194	—	2	6	no	—
8	#14579183	C₂²	1	6	yes	—
9	C₂×Dic₃₁₆	C₂²	1	6	yes	—
10	Dic₆₃₂	C₂²	1	3	yes	—
11	Dic₆₃₂	C₂²	1	6	yes	—
12	#14579106	C₂²	1	6	yes	—
13	D₁₂₆₄	C₂²	1	1	yes	characteristic
14	#131×C₇₉	C₂²	1	1	yes	characteristic
15	M₃₂×C₇₉	C₂²	1	3	yes	—
16	C₂×C₁₆×C₇₉	C₂²	1	3	yes	—
17	#11929624	—	2	6	no	—
18	#11929623	—	4	6	no	—
19	C₂×Dic₁₅₈	—	2	6	no	—
20	Q₈×D₇₉	—	4	2	no	—
21	Dic₃₁₆	—	2	6	no	—
22	Dic₃₁₆	C₂³	1	6	yes	—
23	#11929613	—	2	6	no	—
24	D₆₃₂	C₂³	1	2	yes	—
25	#41×C₇₉	D₄	1	1	yes	characteristic
26	M₁₆×C₇₉	D₄	1	3	yes	—
27	C₂×C₈×C₇₉	C₂³	1	3	yes	—
28	C₁₆×C₇₉	C₂³	1	1	yes	characteristic
29	C₁₆×C₇₉	C₂³	1	3	yes	—
30	Q₈×C₇₉	D₈	1	1	yes	characteristic
31	D₄×C₇₉	D₈	1	3	yes	—
32	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
33	C₂²⋊D₇₉	—	4	6	no	—
34	C₂×Dic₇₉	—	4	6	no	—
35	D₃₁₆	—	2	2	no	—
36	C₄×D₇₉	—	4	6	no	—
37	Dic₁₅₈	—	2	6	no	—
38	Dic₁₅₈	—	4	6	no	—
39	C₈×C₇₉	C₂×D₄	1	3	yes	—
40	C₈×C₇₉	C₂⁴	1	1	yes	characteristic
41	C₂²×C₇₉	C₂×D₈	1	3	yes	—
42	C₂×D₇₉	—	4	2	no	—
43	C₄×C₇₉	C₂×D₈	1	3	yes	—
44	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
45	Dic₇₉	—	4	6	no	—
46	C₂×C₇₉	—	2	3	no	—
47	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
48	D₇₉	—	8	2	no	—
49	#3417	—	79	1	no	maximal, Sylow
50	C₇₉	#3417	1	1	yes	characteristic, complemented, Sylow
51	#578	—	79	2	no	—
52	#510	—	79	6	no	—
53	#508	—	79	3	no	—
54	C₂×Q₃₂	—	79	3	no	—
55	#504	D₇₉	1	1	yes	characteristic, complemented
56	2₋¹⁺⁴	—	158	2	no	—
57	#137	—	158	6	no	—
58	#135	—	79	6	no	—
59	C₂×Q₁₆	—	79	6	no	—
60	#131	C₂×D₇₉	1	1	yes	characteristic
61	Q₃₂	—	79	3	no	—
62	Q₃₂	—	79	6	no	—
63	SD₃₂	—	79	6	no	—
64	D₁₆	—	79	1	no	—
65	M₃₂	C₂×D₇₉	1	3	yes	—
66	C₂×C₁₆	C₂×D₇₉	1	3	yes	—
67	#41	—	316	6	no	—
68	#41	—	158	6	no	—
69	#41	D₃₁₆	1	1	yes	characteristic
70	C₂×Q₈	—	316	2	no	—
71	C₂×Q₈	—	158	6	no	—
72	Q₁₆	—	79	6	no	—
73	Q₁₆	—	158	6	no	—
74	SD₁₆	—	158	6	no	—
75	D₈	—	79	2	no	—
76	M₁₆	D₃₁₆	1	3	yes	—
77	C₂×C₈	C₂²×D₇₉	1	3	yes	—
78	C₁₆	C₂²×D₇₉	1	1	yes	characteristic
79	C₁₆	C₂²×D₇₉	1	3	yes	—
80	Q₈	—	316	6	no	—
81	Q₈	—	158	6	no	—
82	Q₈	D₆₃₂	1	1	yes	characteristic
83	D₄	—	316	6	no	—
84	D₄	—	158	2	no	—
85	D₄	D₆₃₂	1	3	yes	—
86	C₂×C₄	—	316	6	no	—
87	C₂×C₄	—	316	6	no	—
88	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
89	C₈	C₂×D₃₁₆	1	3	yes	—
90	C₈	C₂³×D₇₉	1	1	yes	characteristic
91	C₂²	—	316	2	no	—
92	C₂²	C₂×D₆₃₂	1	3	yes	—
93	C₄	—	316	6	no	—
94	C₄	C₂×D₆₃₂	1	3	yes	—
95	C₄	C₂²×D₃₁₆	1	1	yes	characteristic
96	C₂	—	2	3	no	—
97	C₂	—	632	2	no	—
98	C₂	C₂²×D₆₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099380

C₂

yes

maximal, complemented

#19099006

C₂

yes

maximal, complemented

#19098995

C₂

yes

maximal, complemented

C₂×Dic₆₃₂

C₂

yes

maximal, complemented

#504×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579208

—

#14579194

—

#14579183

C₂²

yes

—

C₂×Dic₃₁₆

C₂²

yes

—

Dic₆₃₂

C₂²

yes

—

Dic₆₃₂

C₂²

yes

—

#14579106

C₂²

yes

—

D₁₂₆₄

C₂²

yes

characteristic

#131×C₇₉

C₂²

yes

characteristic

M₃₂×C₇₉

C₂²

yes

—

C₂×C₁₆×C₇₉

C₂²

yes

—

#11929624

—

#11929623

—

C₂×Dic₁₅₈

—

Q₈×D₇₉

—

Dic₃₁₆

—

Dic₃₁₆

C₂³

yes

—

#11929613

—

D₆₃₂

C₂³

yes

—

#41×C₇₉

D₄

yes

characteristic

M₁₆×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₁₆×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

C₂³

yes

—

Q₈×C₇₉

D₈

yes

characteristic

D₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂×D₄

yes

—

C₈×C₇₉

C₂⁴

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

—

C₂×D₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂²×D₄

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂²×D₈

yes

characteristic

D₇₉

—

#3417

—

maximal, Sylow

C₇₉

#3417

yes

characteristic, complemented, Sylow

#578

—

#510

—

#508

—

C₂×Q₃₂

—

#504

D₇₉

yes

characteristic, complemented

2₋¹⁺⁴

—

158

—

#137

—

158

—

#135

—

C₂×Q₁₆

—

#131

C₂×D₇₉

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

C₂×D₇₉

yes

—

C₂×C₁₆

C₂×D₇₉

yes

—

#41

—

316

—

#41

—

158

—

#41

D₃₁₆

yes

characteristic

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

Q₁₆

—

Q₁₆

—

158

—

SD₁₆

—

158

—

D₈

—

M₁₆

D₃₁₆

yes

—

C₂×C₈

C₂²×D₇₉

yes

—

C₁₆

C₂²×D₇₉

yes

characteristic

C₁₆

C₂²×D₇₉

yes

—

Q₈

—

316

—

Q₈

—

158

—

Q₈

D₆₃₂

yes

characteristic

D₄

—

316

—

D₄

—

158

—

D₄

D₆₃₂

yes

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₈

C₂×D₃₁₆

yes

—

C₈

C₂³×D₇₉

yes

characteristic

C₂²

—

316

—

C₂²

C₂×D₆₃₂

yes

—

C₄

—

316

—

C₄

C₂×D₆₃₂

yes

—

C₄

C₂²×D₃₁₆

yes

characteristic

C₂

—

C₂

—

632

—

C₂

C₂²×D₆₃₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₃₂
commutator, cosocle	C₈×C₇₉	C₂⁴
Frattini	C₈	C₂³×D₇₉
Fitting	#504×C₇₉	C₂
socle	C₂×C₇₉	C₂²×D₈
hypercenter	#504	D₇₉
hypocenter	C₇₉	#3417
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

subgroup

G/H

center

C₂

C₂²×D₆₃₂

commutator, cosocle

C₈×C₇₉

C₂⁴

Frattini

C₈

C₂³×D₇₉

Fitting

#504×C₇₉

C₂

socle

C₂×C₇₉

C₂²×D₈

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

N	H ≅ G/N	type
1. #19099380	C₂	semidirect
2. #19099006	C₂	semidirect
3. #19098995	C₂	semidirect
4. C₂×Dic₆₃₂	C₂	semidirect
5. #504×C₇₉	C₂	semidirect
50. C₇₉	#3417	semidirect
55. #504	D₇₉	semidirect

H ≅ G/N

type

1. #19099380

C₂

semidirect

2. #19099006

C₂

semidirect

3. #19098995

C₂

semidirect

4. C₂×Dic₆₃₂

C₂

semidirect

5. #504×C₇₉

C₂

semidirect

50. C₇₉

#3417

semidirect

55. #504

D₇₉

semidirect

Subgroups — #23776553

Conjugacy classes of subgroups

Complements