Group #23765174 — subgroups

Subgroups — #23765174

Order 10112.

144 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#14579101	C₂	1	1	yes	maximal, characteristic
2	#19098601	C₂	1	1	yes	maximal, characteristic
3	#19098387	C₂	1	4	yes	maximal, complemented
4	C₂×#108×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#14579120	—	2	1	no	—
6	#14579101	C₄	1	2	yes	complemented
7	#14579101	C₂²	1	2	yes	—
8	C₂×C₇₉⋊C₁₆	C₄	1	2	yes	complemented
9	#14579095	—	2	1	no	—
10	#14579093	C₂²	1	2	yes	—
11	C₂×M₁₆×C₇₉	—	2	1	no	—
12	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#108×C₇₉	—	2	1	no	—
14	#108×C₇₉	C₂²	1	2	yes	—
15	C₄⋊Dic₇₉	—	2	2	no	—
16	C₂²⋊Dic₇₉	—	4	1	no	—
17	C₄×Dic₇₉	—	4	1	no	—
18	C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
20	M₁₆×C₇₉	—	2	2	no	—
21	M₁₆×C₇₉	—	4	1	no	—
22	C₂×C₈×C₇₉	—	4	1	no	—
23	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₉	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₇₉	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂³×C₇₉	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₇₉	—	4	2	no	—
37	C₈×C₇₉	—	4	2	no	—
38	C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₇₉	—	4	1	no	—
43	C₂²×C₇₉	#106	1	1	yes	characteristic
44	C₂²×C₇₉	#106	1	1	yes	characteristic
45	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₄×C₇₉	#106	1	1	yes	characteristic
47	C₄×C₇₉	#106	1	1	yes	characteristic
48	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
50	Dic₇₉	—	8	2	no	—
51	C₂×C₇₉	—	2	1	no	—
52	C₂×C₇₉	—	2	1	no	—
53	C₂×C₇₉	—	2	1	no	—
54	C₂×C₇₉	C₂×#106	1	1	yes	characteristic
55	#2155	—	79	1	no	maximal, Sylow
56	C₇₉	#2155	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	79	1	no	—
58	C₂×#108	D₇₉	1	1	yes	characteristic
59	#427	—	79	1	no	—
60	#365	—	79	4	no	—
61	C₂×M₁₆	—	2	1	no	—
62	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
63	#117	—	158	1	no	—
64	M₃₂	—	79	2	no	—
65	M₃₂	—	79	2	no	—
66	C₂×C₁₆	—	79	2	no	—
67	#108	—	2	1	no	—
68	#108	C₂×D₇₉	1	2	yes	—
69	#107	—	158	1	no	—
70	#106	—	79	2	no	—
71	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
72	M₁₆	—	2	2	no	—
73	M₁₆	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	Dic₁₅₈	1	1	yes	characteristic
76	C₂×C₈	Dic₁₅₈	1	1	yes	characteristic
77	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
78	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
81	C₄⋊C₄	—	158	2	no	—
82	C₂²⋊C₄	—	316	1	no	—
83	C₄²	—	316	1	no	—
84	C₁₆	—	79	4	no	—
85	C₂³	#11929611	1	1	yes	characteristic
86	C₂×C₄	—	316	2	no	—
87	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
89	C₂×C₄	#11929611	1	1	yes	characteristic
90	C₂×C₄	#11929612	1	1	yes	characteristic
91	C₂×C₄	#11929612	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
95	C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
96	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
97	C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#14579092	1	1	yes	characteristic
100	C₂²	#14579092	1	1	yes	characteristic
101	C₂²	C₂×#11929611	1	1	yes	characteristic
102	C₄	—	632	2	no	—
103	C₄	#14579092	1	1	yes	characteristic
104	C₄	#14579092	1	1	yes	characteristic
105	C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
106	C₄	C₂×#11929612	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	C₂×#14579092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579101

C₂

yes

maximal, characteristic

#19098601

C₂

yes

maximal, characteristic

#19098387

C₂

yes

maximal, complemented

C₂×#108×C₇₉

C₂

yes

maximal, characteristic

#14579120

—

#14579101

C₄

yes

complemented

#14579101

C₂²

yes

—

C₂×C₇₉⋊C₁₆

C₄

yes

complemented

#14579095

—

#14579093

C₂²

yes

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#108×C₇₉

—

#108×C₇₉

C₂²

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

D₇₉

yes

characteristic

#427

—

#365

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

#117

—

158

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

C₂×D₇₉

yes

—

#107

—

158

—

#106

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

Dic₁₅₈

yes

characteristic

C₂×C₈

Dic₁₅₈

yes

characteristic

C₂×C₈

C₄×D₇₉

yes

characteristic

C₂×C₈

C₄×D₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₄²

—

316

—

C₁₆

—

C₂³

#11929611

yes

characteristic

C₂×C₄

—

316

—

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₈

—

C₈

Dic₇₉⋊C₄

yes

characteristic

C₈

Dic₇₉⋊C₄

yes

characteristic

C₈

C₂×C₄×D₇₉

yes

characteristic

C₈

C₂×Dic₁₅₈

yes

characteristic

C₂²

—

C₂²

#14579092

yes

characteristic

100

C₂²

#14579092

yes

characteristic

101

C₂²

C₂×#11929611

yes

characteristic

102

C₄

—

632

—

103

C₄

#14579092

yes

characteristic

104

C₄

#14579092

yes

characteristic

105

C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

106

C₄

C₂×#11929612

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

C₂×#14579092

yes

characteristic

	subgroup	G/H
center	C₄	#14579092
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₂×#108×C₇₉	C₂
socle	C₂×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	SD₁₆
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂×#108	D₇₉
hypocenter	C₇₉	#2155
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₄

#14579092

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₂×#108×C₇₉

C₂

socle

C₂×C₇₉

C₂×#106

prime radical

C₂³×C₇₉

SD₁₆

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
3. #19098387	C₂	semidirect
6. #14579101	C₄	semidirect
8. C₂×C₇₉⋊C₁₆	C₄	semidirect
56. C₇₉	#2155	semidirect

H ≅ G/N

type

3. #19098387

C₂

semidirect

6. #14579101

C₄

semidirect

8. C₂×C₇₉⋊C₁₆

C₄

semidirect

56. C₇₉

#2155

semidirect

Subgroups — #23765174

Conjugacy classes of subgroups

Complements