Group #23765568 — subgroups

Subgroups — #23765568

Order 10112.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19098681	C₂	1	1	yes	maximal, characteristic
2	#19098385	C₂	1	2	yes	maximal, complemented
3	#19098384	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098372	C₂	1	2	yes	maximal
5	#498×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579183	C₂²	1	1	yes	characteristic
7	#14579167	—	2	1	no	—
8	#14579101	C₂²	1	2	yes	—
9	#14579097	C₂²	1	1	yes	characteristic
10	#14579089	—	2	1	no	—
11	C₄⋊Q₈×C₇₉	—	2	1	no	—
12	#106×C₇₉	—	2	1	no	—
13	#106×C₇₉	C₂²	1	2	yes	—
14	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#11929624	—	2	1	no	—
16	Dic₃₁₆	Q₈	1	1	yes	characteristic, complemented
17	#11929613	—	2	1	no	—
18	D₆₃₂	Q₈	1	1	yes	characteristic, complemented
19	#11929608	—	2	1	no	—
20	#11929607	—	4	1	no	—
21	C₈×D₇₉	—	4	1	no	—
22	C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×Q₈×C₇₉	—	4	1	no	—
24	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₄⋊C₄×C₇₉	—	4	1	no	—
28	C₄²×C₇₉	D₄	1	1	yes	characteristic
29	Q₈×C₇₉	—	8	2	no	—
30	C₂×C₄×C₇₉	—	2	1	no	—
31	C₂×C₄×C₇₉	—	4	2	no	—
32	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂²⋊D₇₉	—	4	1	no	—
34	D₃₁₆	—	2	1	no	—
35	C₄×D₇₉	—	4	1	no	—
36	Dic₁₅₈	—	2	1	no	—
37	C₈×C₇₉	SD₁₆	1	1	yes	characteristic
38	C₈×C₇₉	SD₁₆	1	1	yes	characteristic
39	C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₇₉	#41	1	1	yes	characteristic
41	C₇₉⋊C₈	—	4	1	no	—
42	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₇₉	—	4	1	no	—
44	C₄×C₇₉	—	2	2	no	—
45	C₄×C₇₉	—	8	2	no	—
46	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₇₉	—	4	1	no	—
49	C₂×C₇₉	—	2	1	no	—
50	C₂×C₇₉	#476	1	1	yes	characteristic
51	D₇₉	—	8	1	no	—
52	#2230	—	79	1	no	maximal, Sylow
53	C₇₉	#2230	1	1	yes	characteristic, complemented, Sylow
54	#498	D₇₉	1	1	yes	characteristic, complemented
55	#443	—	79	1	no	—
56	#365	—	79	2	no	—
57	#364	—	79	1	no	—
58	#360	—	79	2	no	—
59	#135	—	79	1	no	—
60	#131	—	158	1	no	—
61	C₄⋊Q₈	—	2	1	no	—
62	M₃₂	—	79	2	no	—
63	#108	—	79	1	no	—
64	#106	—	2	1	no	—
65	#106	C₂×D₇₉	1	2	yes	—
66	#104	—	158	1	no	—
67	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
68	#41	—	158	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	Q₁₆	—	79	1	no	—
71	SD₁₆	—	158	1	no	—
72	D₈	—	79	1	no	—
73	M₁₆	—	316	1	no	—
74	M₁₆	—	158	1	no	—
75	C₂×C₈	—	316	1	no	—
76	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
81	C₁₆	—	158	2	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	158	1	no	—
84	D₄	—	316	1	no	—
85	D₄	—	158	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	316	1	no	—
89	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
90	C₈	—	316	1	no	—
91	C₈	#11929611	1	1	yes	characteristic
92	C₈	#11929611	1	1	yes	characteristic
93	C₈	Q₈×D₇₉	1	1	yes	characteristic
94	C₈	#11929625	1	1	yes	characteristic
95	C₂²	—	316	1	no	—
96	C₂²	#14579187	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	8	2	no	—
99	C₄	—	316	1	no	—
100	C₄	#14579137	1	1	yes	characteristic
101	C₄	C₂×#11929611	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	632	1	no	—
104	C₂	#19098850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098681

C₂

yes

maximal, characteristic

#19098385

C₂

yes

maximal, complemented

#19098384

C₂

yes

maximal, characteristic, complemented

#19098372

C₂

yes

maximal

#498×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579183

C₂²

yes

characteristic

#14579167

—

#14579101

C₂²

yes

—

#14579097

C₂²

yes

characteristic

#14579089

—

C₄⋊Q₈×C₇₉

—

#106×C₇₉

—

#106×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929624

—

Dic₃₁₆

Q₈

yes

characteristic, complemented

#11929613

—

D₆₃₂

Q₈

yes

characteristic, complemented

#11929608

—

#11929607

—

C₈×D₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

SD₁₆

yes

characteristic

C₈×C₇₉

SD₁₆

yes

characteristic

C₈×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#476

yes

characteristic

D₇₉

—

#2230

—

maximal, Sylow

C₇₉

#2230

yes

characteristic, complemented, Sylow

#498

D₇₉

yes

characteristic, complemented

—

—

—

—

—

—

158

—

C₄⋊Q₈

—

M₃₂

—

#108

—

#106

—

#106

C₂×D₇₉

yes

—

#104

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

#41

—

158

—

C₂×Q₈

—

Q₁₆

—

SD₁₆

—

158

—

D₈

—

M₁₆

—

316

—

M₁₆

—

158

—

C₂×C₈

—

316

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₇₉

yes

characteristic

C₁₆

—

158

—

Q₈

—

Q₈

—

158

—

D₄

—

316

—

D₄

—

158

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₈

—

316

—

C₈

yes

C₈

yes

C₈

Q₈×D₇₉

yes

characteristic

C₈

#11929625

yes

characteristic

C₂²

—

316

—

C₂²

#14579187

yes

characteristic

C₄

—

C₄

—

C₄

—

316

—

100

C₄

#14579137

yes

characteristic

101

C₄

C₂×#11929611

yes

characteristic

102

C₂

—

103

C₂

—

632

—

104

C₂

#19098850

yes

characteristic

	subgroup	G/H
center	C₂	#19098850
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#498×C₇₉	C₂
socle	C₂×C₇₉	#476
prime radical	#14579183	C₂²
hypercenter	#498	D₇₉
hypocenter	C₇₉	#2230
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

subgroup

G/H

center

C₂

#19098850

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#498×C₇₉

C₂

C₂×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

N	H ≅ G/N	type
2. #19098385	C₂	semidirect
3. #19098384	C₂	semidirect
5. #498×C₇₉	C₂	semidirect
16. Dic₃₁₆	Q₈	semidirect
18. D₆₃₂	Q₈	semidirect
53. C₇₉	#2230	semidirect
54. #498	D₇₉	semidirect

H ≅ G/N

type

2. #19098385

C₂

semidirect

3. #19098384

C₂

semidirect

5. #498×C₇₉

C₂

semidirect

16. Dic₃₁₆

Q₈

semidirect

18. D₆₃₂

Q₈

semidirect

53. C₇₉

#2230

semidirect

54. #498

D₇₉

semidirect

Subgroups — #23765568

Conjugacy classes of subgroups

Complements