Group #23765549 — subgroups

Subgroups — #23765549

Order 10112.

130 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#19098388	C₂	1	2	yes	maximal
3	C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#19098366	C₂	1	2	yes	maximal
5	#500×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₅₈	—	2	1	no	—
8	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#14579095	C₂²	1	1	yes	characteristic
10	#14579086	—	2	1	no	—
11	C₄⋊Q₈×C₇₉	—	2	1	no	—
12	#107×C₇₉	—	2	1	no	—
13	#107×C₇₉	C₂²	1	2	yes	—
14	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₅₈	—	2	1	no	—
16	Dic₃₁₆	—	2	1	no	—
17	Dic₃₁₆	Q₈	1	2	yes	complemented
18	C₄⋊Dic₇₉	—	2	1	no	—
19	Dic₇₉⋊C₄	—	4	1	no	—
20	C₄×Dic₇₉	—	4	1	no	—
21	C₇₉⋊C₁₆	—	2	2	no	—
22	C₂×Q₈×C₇₉	—	4	1	no	—
23	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₇₉	—	2	2	no	—
26	C₄⋊C₄×C₇₉	—	4	1	no	—
27	C₄²×C₇₉	D₄	1	1	yes	characteristic
28	Q₈×C₇₉	—	8	2	no	—
29	C₂×C₄×C₇₉	—	4	2	no	—
30	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
31	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇₉	—	4	1	no	—
34	C₂×Dic₇₉	—	4	1	no	—
35	Dic₁₅₈	—	2	2	no	—
36	Dic₁₅₈	—	4	1	no	—
37	C₈×C₇₉	—	2	1	no	—
38	C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₇₉	#41	1	1	yes	characteristic
40	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
41	C₄×C₇₉	—	2	1	no	—
42	C₄×C₇₉	—	8	2	no	—
43	C₄×C₇₉	Q₃₂	1	2	yes	—
44	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₇₉	#137	1	1	yes	characteristic
46	Dic₇₉	—	4	2	no	—
47	Dic₇₉	—	8	1	no	—
48	C₂×C₇₉	#474	1	1	yes	characteristic
49	C₂×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×C₇₉	#509	1	1	yes	characteristic
51	#2227	—	79	1	no	maximal, Sylow
52	C₇₉	#2227	1	1	yes	characteristic, complemented, Sylow
53	#500	D₇₉	1	1	yes	characteristic
54	C₄×Q₁₆	—	79	1	no	—
55	#366	—	79	2	no	—
56	C₄⋊C₁₆	—	79	1	no	—
57	#358	—	79	2	no	—
58	C₂×Q₁₆	—	79	1	no	—
59	C₄⋊Q₈	—	2	1	no	—
60	C₄×Q₈	—	158	1	no	—
61	C₂×C₁₆	—	79	2	no	—
62	#107	—	2	1	no	—
63	#107	—	79	1	no	—
64	#107	C₂×D₇₉	1	2	yes	—
65	#103	—	158	1	no	—
66	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	158	1	no	—
69	Q₁₆	—	79	2	no	—
70	Q₁₆	—	158	1	no	—
71	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	316	1	no	—
76	C₄⋊C₄	—	158	1	no	—
77	C₄²	—	316	1	no	—
78	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
79	C₁₆	—	158	2	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	316	1	no	—
82	Q₈	—	158	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	316	1	no	—
85	C₂×C₄	—	316	1	no	—
86	C₂×C₄	#11929609	1	1	yes	characteristic
87	C₂×C₄	#11929609	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	Q₈×D₇₉	1	1	yes	characteristic
91	C₈	#11929625	1	1	yes	characteristic
92	C₂²	C₂×#11929609	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	8	2	no	—
95	C₄	—	316	2	no	—
96	C₄	—	632	1	no	—
97	C₄	#14579107	1	2	yes	—
98	C₄	#14579137	1	1	yes	characteristic
99	C₄	#14579192	1	1	yes	characteristic
100	C₂	#19098840	1	1	yes	characteristic
101	C₂	C₂×#14579107	1	1	yes	characteristic
102	C₂	#19098999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic

#19098388

C₂

yes

maximal

C₄⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic

#19098366

C₂

yes

maximal

#500×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₄×Dic₁₅₈

—

C₂×C₇₉⋊C₁₆

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579086

—

C₄⋊Q₈×C₇₉

—

#107×C₇₉

—

#107×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

Q₈

yes

complemented

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

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

D₈

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

Q₃₂

yes

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#137

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#474

yes

characteristic

C₂×C₇₉

C₂×Q₃₂

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#500

D₇₉

yes

characteristic

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

158

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₇₉

yes

—

#103

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

158

—

Q₁₆

—

Q₁₆

—

158

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄²

—

316

—

C₄²

C₂²⋊D₇₉

yes

characteristic

C₁₆

—

158

—

Q₈

—

Q₈

—

316

—

Q₈

—

158

—

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₈

—

C₈

Q₈×D₇₉

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

316

—

C₄

—

632

—

C₄

#14579107

yes

—

C₄

yes

C₄

yes

100

C₂

#19098840

yes

characteristic

101

C₂

C₂×#14579107

yes

characteristic

102

C₂

#19098999

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#500×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₈
hypercenter	#500	D₇₉
hypocenter	C₇₉	#2227
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11929609

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#500×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
17. Dic₃₁₆	Q₈	semidirect
52. C₇₉	#2227	semidirect

H ≅ G/N

type

17. Dic₃₁₆

Q₈

semidirect

52. C₇₉

#2227

semidirect

Subgroups — #23765549

Conjugacy classes of subgroups

Complements