Group #23773914 — subgroups

Subgroups — #23773914

Order 10112.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#19099120	C₂	1	2	yes	maximal
2	#19098968	C₂	1	2	yes	maximal, complemented
3	#19098965	C₂	1	4	yes	maximal, complemented
4	#19098961	C₂	1	2	yes	maximal, complemented
5	#19098601	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#14579161	C₂²	1	4	yes	—
8	#14579153	C₂²	1	4	yes	—
9	#14579140	—	2	4	no	—
10	C₄×Dic₁₅₈	—	2	4	no	—
11	#14579120	C₂²	1	4	yes	—
12	#14579095	C₂²	1	8	yes	—
13	#14579093	C₂²	1	8	yes	—
14	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₇₉	C₂²	1	2	yes	—
18	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₅₈	—	2	4	no	—
20	C₄⋊Dic₇₉	—	2	4	no	—
21	C₄⋊Dic₇₉	C₂³	1	8	yes	—
22	Dic₇₉⋊C₄	—	2	8	no	—
23	C₂²⋊Dic₇₉	—	2	4	no	—
24	C₄×Dic₇₉	—	2	4	no	—
25	C₂²×C₄×C₇₉	D₄	1	2	yes	—
26	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₉	Q₈	1	8	yes	complemented
28	C₂×C₈×C₇₉	C₂³	1	2	yes	—
29	C₂×C₈×C₇₉	C₂³	1	2	yes	—
30	C₄²×C₇₉	D₄	1	2	yes	—
31	C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	—	2	4	no	—
35	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₇₉	—	2	8	no	—
41	Dic₁₅₈	—	4	8	no	—
42	C₈×C₇₉	C₂×Q₈	1	4	yes	—
43	C₈×C₇₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₇₉	—	2	2	no	—
45	C₂²×C₇₉	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₇₉	—	2	2	no	—
48	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
49	C₄×C₇₉	#135	1	4	yes	—
50	C₄×C₇₉	C₂²×Q₈	1	2	yes	—
51	Dic₇₉	—	4	8	no	—
52	C₂×C₇₉	—	2	2	no	—
53	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₇₉	C₂×#135	1	2	yes	—
55	#3160	—	79	1	no	maximal, Sylow
56	C₇₉	#3160	1	1	yes	characteristic, complemented, Sylow
57	#533	—	79	2	no	—
58	#500	—	79	2	no	—
59	#499	—	79	4	no	—
60	#498	—	79	2	no	—
61	#427	—	79	4	no	—
62	C₂×C₄×C₈	D₇₉	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₇₉	1	2	yes	—
64	C₄⋊Q₈	—	79	4	no	—
65	#125	—	79	4	no	—
66	C₂²⋊Q₈	—	158	4	no	—
67	C₄×Q₈	—	158	4	no	—
68	#117	—	79	4	no	—
69	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
70	#107	—	79	8	no	—
71	#106	—	79	8	no	—
72	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₇₉	1	2	yes	—
74	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
75	C₂×Q₈	—	158	4	no	—
76	C₂²×C₄	D₃₁₆	1	2	yes	—
77	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
78	C₂×C₈	Dic₁₅₈	1	8	yes	complemented
79	C₂×C₈	C₂²×D₇₉	1	2	yes	—
80	C₂×C₈	C₂²×D₇₉	1	2	yes	—
81	C₄⋊C₄	—	79	8	no	—
82	C₄⋊C₄	—	158	8	no	—
83	C₄⋊C₄	—	158	4	no	—
84	C₂²⋊C₄	—	158	4	no	—
85	C₄²	—	158	4	no	—
86	C₄²	D₃₁₆	1	2	yes	—
87	C₄²	C₂²×D₇₉	1	1	yes	characteristic
88	C₄²	C₂²×D₇₉	1	1	yes	characteristic
89	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
90	Q₈	—	316	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	158	8	no	—
93	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
94	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
95	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
97	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
98	C₈	C₂×Dic₁₅₈	1	4	yes	—
99	C₈	C₂×Dic₁₅₈	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#14579161	1	2	yes	—
102	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	316	8	no	—
105	C₄	#14579161	1	2	yes	—
106	C₄	#14579183	1	4	yes	—
107	C₄	C₂²×Dic₁₅₈	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#14579161	1	1	yes	characteristic
110	C₂	C₂×#14579183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal

#19098968

C₂

yes

maximal, complemented

#19098965

C₂

yes

maximal, complemented

#19098961

C₂

yes

maximal, complemented

#19098601

C₂

yes

maximal

C₂×C₄×C₈×C₇₉

C₂

yes

maximal, characteristic

#14579161

C₂²

yes

—

#14579153

C₂²

yes

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

C₂²

yes

—

#14579095

C₂²

yes

—

#14579093

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

Q₈

yes

complemented

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

C₂×Q₈

yes

—

C₈×C₇₉

C₂×Q₈

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄⋊Q₈

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄⋊Q₈

yes

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂²×Q₈

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₉

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₇₉

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×C₄×C₈

D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₄×C₈

C₂×D₇₉

yes

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

158

—

C₂²×C₄

D₃₁₆

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

Dic₁₅₈

yes

complemented

C₂×C₈

C₂²×D₇₉

yes

—

C₂×C₈

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

158

—

C₄²

—

158

—

C₄²

D₃₁₆

yes

—

C₄²

C₂²×D₇₉

yes

characteristic

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

C₂×D₃₁₆

yes

characteristic

Q₈

—

316

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂×D₃₁₆

yes

characteristic

C₂×C₄

C₂×Dic₁₅₈

yes

—

C₂×C₄

C₂³×D₇₉

yes

characteristic

C₈

C₂×Dic₁₅₈

yes

—

C₈

C₂×Dic₁₅₈

yes

—

100

C₂²

—

101

C₂²

#14579161

yes

—

102

C₂²

C₂²×D₃₁₆

yes

characteristic

103

C₄

—

104

C₄

—

316

—

105

C₄

#14579161

yes

—

106

C₄

#14579183

yes

—

107

C₄

C₂²×Dic₁₅₈

yes

—

108

C₂

—

109

C₂

C₂×#14579161

yes

characteristic

110

C₂

C₂×#14579183

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄×C₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₇₉
hypocenter	C₇₉	#3160
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄×C₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₇₉

hypocenter

C₇₉

#3160

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19098968	C₂	semidirect
3. #19098965	C₂	semidirect
4. #19098961	C₂	semidirect
27. C₂×C₈×C₇₉	Q₈	semidirect
56. C₇₉	#3160	semidirect
78. C₂×C₈	Dic₁₅₈	semidirect

H ≅ G/N

type

2. #19098968

C₂

semidirect

3. #19098965

C₂

semidirect

4. #19098961

C₂

semidirect

27. C₂×C₈×C₇₉

Q₈

semidirect

56. C₇₉

#3160

semidirect

78. C₂×C₈

Dic₁₅₈

semidirect

Subgroups — #23773914

Conjugacy classes of subgroups

Complements