Group #23762801 — subgroups

Subgroups — #23762801

Order 10112.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19098855	C₂	1	2	yes	maximal
2	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#19098316	C₂	1	1	yes	maximal, characteristic
4	#19098312	C₂	1	2	yes	maximal
5	#500×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	#14579094	—	2	2	no	—
8	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₇₉⋊C₈	—	2	1	no	—
11	#14579084	—	2	2	no	—
12	C₈×Dic₇₉	—	2	1	no	—
13	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
14	#107×C₇₉	—	2	1	no	—
15	#107×C₇₉	—	2	1	no	—
16	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
18	C₂×C₇₉⋊C₈	—	2	1	no	—
19	C₂×C₇₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
21	Dic₇₉⋊C₄	—	2	1	no	—
22	C₄×Dic₇₉	—	2	1	no	—
23	C₂×Q₈×C₇₉	—	2	2	no	—
24	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	Q₈×C₇₉	—	8	2	no	—
30	C₂×C₄×C₇₉	—	4	2	no	—
31	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
34	C₂×Dic₇₉	—	2	1	no	—
35	C₂×Dic₇₉	—	2	1	no	—
36	Dic₁₅₈	—	2	1	no	—
37	Dic₁₅₈	Q₁₆	1	2	yes	—
38	C₈×C₇₉	—	2	1	no	—
39	C₈×C₇₉	Q₁₆	1	2	yes	—
40	C₇₉⋊C₈	—	4	2	no	—
41	C₇₉⋊C₈	—	4	1	no	—
42	C₂²×C₇₉	#121	1	1	yes	characteristic
43	C₄×C₇₉	—	2	1	no	—
44	C₄×C₇₉	—	8	2	no	—
45	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₇₉	#135	1	1	yes	characteristic
48	C₄×C₇₉	#137	1	1	yes	characteristic
49	Dic₇₉	—	2	2	no	—
50	Dic₇₉	—	4	1	no	—
51	C₂×C₇₉	#462	1	1	yes	characteristic
52	C₂×C₇₉	#467	1	1	yes	characteristic
53	C₂×C₇₉	#472	1	1	yes	characteristic
54	#1676	—	79	1	no	maximal, Sylow
55	C₇₉	#1676	1	1	yes	characteristic, complemented, Sylow
56	#500	D₇₉	1	1	yes	characteristic
57	#477	—	79	2	no	—
58	C₈×Q₈	—	79	1	no	—
59	#335	—	79	1	no	—
60	#333	—	79	2	no	—
61	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
62	C₄×Q₈	—	79	1	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	—	158	2	no	—
66	C₄⋊C₈	—	79	1	no	—
67	C₄⋊C₈	—	79	2	no	—
68	C₄⋊C₈	—	158	1	no	—
69	#103	—	158	2	no	—
70	C₄×C₈	—	158	1	no	—
71	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	79	1	no	—
74	C₂×C₈	—	158	2	no	—
75	C₂×C₈	—	158	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	—	2	2	no	—
80	C₄⋊C₄	—	79	1	no	—
81	C₄⋊C₄	—	158	1	no	—
82	C₄²	—	158	1	no	—
83	C₄²	C₂²×D₇₉	1	1	yes	characteristic
84	Q₈	—	8	2	no	—
85	Q₈	—	79	2	no	—
86	Q₈	—	158	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	158	1	no	—
89	C₂×C₄	—	158	1	no	—
90	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
91	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
92	C₂×C₄	#11929623	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	316	2	no	—
95	C₈	—	316	1	no	—
96	C₈	#11929614	1	2	yes	—
97	C₂²	#14579132	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	8	2	no	—
100	C₄	—	316	1	no	—
101	C₄	—	158	2	no	—
102	C₄	Q₁₆×D₇₉	1	1	yes	characteristic
103	C₄	C₂×#11929614	1	1	yes	characteristic
104	C₄	#14579184	1	1	yes	characteristic
105	C₄	#14579192	1	1	yes	characteristic
106	C₂	#19098770	1	1	yes	characteristic
107	C₂	#19098801	1	1	yes	characteristic
108	C₂	#19098830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098855

C₂

yes

maximal

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098316

C₂

yes

maximal, characteristic

#19098312

C₂

yes

maximal

#500×C₇₉

C₂

yes

maximal, characteristic

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

Dic₇₉⋊C₈

—

#14579084

—

C₈×Dic₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

Q₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

Q₁₆

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#121

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#500

D₇₉

yes

characteristic

#477

—

C₈×Q₈

—

#335

—

#333

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₄×Q₈

—

#107

—

#107

—

#107

—

158

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

#103

—

158

—

C₄×C₈

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄²

—

158

—

C₄²

C₂²×D₇₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

158

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

—

158

—

C₂×C₄

D₄×D₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₂×C₄

#11929623

yes

characteristic

C₈

—

C₈

—

316

—

C₈

—

316

—

C₈

#11929614

yes

—

C₂²

#14579132

yes

characteristic

C₄

—

C₄

—

100

C₄

—

316

—

101

C₄

—

158

—

102

C₄

Q₁₆×D₇₉

yes

characteristic

103

C₄

C₂×#11929614

yes

characteristic

104

C₄

#14579184

yes

characteristic

105

C₄

#14579192

yes

characteristic

106

C₂

#19098770

yes

characteristic

107

C₂

#19098801

yes

characteristic

108

C₂

#19098830

yes

characteristic

	subgroup	G/H
center	C₂²	#14579132
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#500×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#121
hypercenter	#500	D₇₉
hypocenter	C₇₉	#1676
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14579132

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#500×C₇₉

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
55. C₇₉	#1676	semidirect

H ≅ G/N

type

55. C₇₉

#1676

semidirect

Subgroups — #23762801

Conjugacy classes of subgroups

Complements