Group #23762176 — subgroups

Subgroups — #23762176

Order 10112.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19098616	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19098315	C₂	1	4	yes	maximal, complemented
4	#423×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₈	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
8	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
10	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
11	C₂×M₁₆×C₇₉	—	2	1	no	—
12	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₉	—	2	1	no	—
14	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₇₉⋊C₈	—	2	2	no	—
16	C₂×C₇₉⋊C₈	—	4	2	no	—
17	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₉	Q₈	1	1	yes	characteristic
21	M₁₆×C₇₉	—	4	2	no	—
22	C₂×C₈×C₇₉	—	2	2	no	—
23	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₄²×C₇₉	Q₈	1	1	yes	characteristic
26	C₄²×C₇₉	C₂³	1	1	yes	characteristic
27	C₂³×C₇₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₉	—	2	2	no	—
29	C₂×C₄×C₇₉	—	4	1	no	—
30	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	D₈	1	2	yes	—
34	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
35	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₇₉	—	4	2	no	—
39	C₇₉⋊C₈	—	2	4	no	—
40	C₇₉⋊C₈	—	4	2	no	—
41	C₂²×C₇₉	—	2	1	no	—
42	C₂²×C₇₉	—	2	1	no	—
43	C₂²×C₇₉	#107	1	2	yes	—
44	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₇₉	—	2	1	no	—
46	C₄×C₇₉	—	2	1	no	—
47	C₄×C₇₉	—	2	1	no	—
48	C₄×C₇₉	#107	1	2	yes	—
49	C₄×C₇₉	#131	1	2	yes	—
50	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₇₉	—	2	2	no	—
53	C₂×C₇₉	#424	1	1	yes	characteristic
54	C₂×C₇₉	C₂×#107	1	1	yes	characteristic
55	C₂×C₇₉	#430	1	1	yes	characteristic
56	#1571	—	79	1	no	maximal, Sylow
57	C₇₉	#1571	1	1	yes	characteristic, complemented, Sylow
58	#431	—	79	1	no	—
59	#423	D₇₉	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	79	1	no	—
61	#335	—	79	4	no	—
62	C₂×M₁₆	—	2	1	no	—
63	C₂²×C₈	—	158	1	no	—
64	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	79	2	no	—
67	C₄⋊C₈	—	79	2	no	—
68	C₄⋊C₈	—	158	1	no	—
69	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
70	C₂²⋊C₈	—	79	2	no	—
71	C₄×C₈	—	79	2	no	—
72	C₂²×C₄	Dic₁₅₈	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	316	2	no	—
78	C₂×C₈	—	79	4	no	—
79	C₂×C₈	—	158	2	no	—
80	C₄²	Dic₁₅₈	1	1	yes	characteristic
81	C₄²	C₄×D₇₉	1	1	yes	characteristic
82	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
83	C₄²	C₂²×D₇₉	1	1	yes	characteristic
84	C₂³	Dic₇₉⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
89	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#11929609	1	2	yes	—
91	C₂×C₄	#11929614	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	316	2	no	—
97	C₈	—	158	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14579094	1	2	yes	—
101	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14579094	1	2	yes	—
106	C₄	#14579169	1	2	yes	—
107	C₄	C₂×#11929609	1	1	yes	characteristic
108	C₄	C₂×#11929614	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#19098591	1	1	yes	characteristic
111	C₂	C₂×#14579094	1	1	yes	characteristic
112	C₂	#19098611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098616

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#19098315

C₂

yes

maximal, complemented

#423×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

C₄

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×M₁₆×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

Q₈

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

Q₈

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#107

yes

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#107

yes

—

C₄×C₇₉

#131

yes

—

C₄×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#431

—

#423

D₇₉

yes

characteristic

C₂×C₄⋊C₈

—

#335

—

C₂×M₁₆

—

C₂²×C₈

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₅₈

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

C₂×C₈

—

158

—

C₄²

Dic₁₅₈

yes

characteristic

C₄²

C₄×D₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

#11929609

yes

—

C₂×C₄

#11929614

yes

—

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

C₈

—

C₈

—

316

—

C₈

—

158

—

C₂²

—

C₂²

—

100

C₂²

#14579094

yes

—

101

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14579094

yes

—

106

C₄

#14579169

yes

—

107

C₄

C₂×#11929609

yes

characteristic

108

C₄

C₂×#11929614

yes

characteristic

109

C₂

—

110

C₂

#19098591

yes

characteristic

111

C₂

C₂×#14579094

yes

characteristic

112

C₂

#19098611

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₄
prime radical	C₂³×C₇₉	C₄⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1571
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#423×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₄⋊C₄

prime radical

C₂³×C₇₉

C₄⋊C₄

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #19098315	C₂	semidirect
57. C₇₉	#1571	semidirect

H ≅ G/N

type

3. #19098315

C₂

semidirect

57. C₇₉

#1571

semidirect

Subgroups — #23762176

Conjugacy classes of subgroups

Complements