Group #23762183 — subgroups

Subgroups — #23762183

Order 10112.

160 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19098620	C₂	1	1	yes	maximal, characteristic
2	#19098616	C₂	1	1	yes	maximal, characteristic
3	#19098315	C₂	1	2	yes	maximal, complemented
4	#19098313	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
7	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
10	C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
11	#14579072	—	2	1	no	—
12	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₈×C₇₉	—	2	1	no	—
14	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₉	—	2	1	no	—
16	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×C₇₉⋊C₈	—	2	1	no	—
18	C₂×C₇₉⋊C₈	—	2	1	no	—
19	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₉	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₇₉	—	2	2	no	—
24	C₂×C₈×C₇₉	—	4	2	no	—
25	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₄²×C₇₉	Q₈	1	1	yes	characteristic
28	C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₂³×C₇₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₉	—	2	2	no	—
31	C₂×C₄×C₇₉	—	2	2	no	—
32	C₂×C₄×C₇₉	—	2	1	no	—
33	C₂×C₄×C₇₉	—	2	1	no	—
34	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₇₉	—	4	2	no	—
41	C₇₉⋊C₈	—	2	4	no	—
42	C₇₉⋊C₈	—	4	1	no	—
43	C₇₉⋊C₈	—	4	1	no	—
44	C₂²×C₇₉	—	2	1	no	—
45	C₂²×C₇₉	—	2	1	no	—
46	C₂²×C₇₉	#108	1	2	yes	—
47	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₇₉	—	2	1	no	—
49	C₄×C₇₉	—	2	1	no	—
50	C₄×C₇₉	—	2	1	no	—
51	C₄×C₇₉	—	2	1	no	—
52	C₄×C₇₉	#131	1	2	yes	—
53	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₇₉	#137	1	1	yes	characteristic
55	C₂×C₇₉	—	2	2	no	—
56	C₂×C₇₉	#424	1	1	yes	characteristic
57	C₂×C₇₉	#428	1	1	yes	characteristic
58	C₂×C₇₉	C₂×#108	1	1	yes	characteristic
59	#1573	—	79	1	no	maximal, Sylow
60	C₇₉	#1573	1	1	yes	characteristic, complemented, Sylow
61	#432	—	79	1	no	—
62	#431	—	79	1	no	—
63	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
64	#335	—	79	2	no	—
65	#334	—	79	2	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	79	1	no	—
70	C₄⋊C₈	—	79	1	no	—
71	C₄⋊C₈	—	79	2	no	—
72	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
73	C₂²⋊C₈	—	79	2	no	—
74	C₂²⋊C₈	—	158	1	no	—
75	#97	—	158	1	no	—
76	C₄×C₈	—	79	2	no	—
77	C₂²×C₄	Dic₁₅₈	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	79	4	no	—
83	C₂×C₈	—	158	1	no	—
84	C₂×C₈	—	158	1	no	—
85	C₄²	Dic₁₅₈	1	1	yes	characteristic
86	C₄²	C₄×D₇₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₄²	C₂²×D₇₉	1	1	yes	characteristic
89	C₂³	Dic₇₉⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	316	1	no	—
102	C₈	—	316	1	no	—
103	C₈	—	158	4	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#14579096	1	2	yes	—
107	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14579167	1	2	yes	—
113	C₄	#14579187	1	1	yes	characteristic
114	C₄	#14579192	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#19098589	1	1	yes	characteristic
117	C₂	#19098603	1	1	yes	characteristic
118	C₂	C₂×#14579096	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098620

C₂

yes

maximal, characteristic

#19098616

C₂

yes

maximal, characteristic

#19098315

C₂

yes

maximal, complemented

#19098313

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₈)

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

C₄

yes

—

#14579072

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂²×C₈×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

C₂×C₈×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

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#108

yes

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#131

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#432

—

#431

—

C₂×C₄⋊C₈

D₇₉

yes

characteristic

#335

—

#334

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

158

—

#97

—

158

—

C₄×C₈

—

C₂²×C₄

Dic₁₅₈

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

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

—

C₂×C₄

—

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

100

C₈

—

101

C₈

—

316

—

102

C₈

—

316

—

103

C₈

—

158

—

104

C₂²

—

105

C₂²

—

106

C₂²

#14579096

yes

—

107

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#14579167

yes

—

113

C₄

#14579187

yes

characteristic

114

C₄

#14579192

yes

characteristic

115

C₂

—

116

C₂

#19098589

yes

characteristic

117

C₂

#19098603

yes

characteristic

118

C₂

C₂×#14579096

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₄
prime radical	C₂³×C₇₉	C₄⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#1573
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₄⋊C₄

prime radical

C₂³×C₇₉

C₄⋊C₄

cosocle

C₄²×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#1573

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19098315	C₂	semidirect
4. #19098313	C₂	semidirect
60. C₇₉	#1573	semidirect

H ≅ G/N

type

3. #19098315

C₂

semidirect

4. #19098313

C₂

semidirect

60. C₇₉

#1573

semidirect

Subgroups — #23762183

Conjugacy classes of subgroups

Complements