Group #23761237 — subgroups

Subgroups — #23761237

Order 10112.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×#11929608	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11929608	C₄	1	2	yes	—
5	C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
6	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
7	#14579072	—	2	1	no	—
8	C₄×C₇₉⋊C₈	—	2	1	no	—
9	C₄×C₇₉⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₇₉	—	2	1	no	—
11	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₇₉	—	2	1	no	—
13	C₄×C₈×C₇₉	C₄	1	2	yes	—
14	#11929608	—	2	2	no	—
15	#11929608	C₈	1	4	yes	complemented
16	C₂×C₇₉⋊C₈	—	2	2	no	—
17	C₂×C₇₉⋊C₈	—	2	2	no	—
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	C₂×C₈×C₇₉	—	2	4	no	—
22	C₂×C₈×C₇₉	—	2	2	no	—
23	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₇₉	D₄	1	1	yes	characteristic
26	C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₂³×C₇₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₉	—	2	2	no	—
29	C₂×C₄×C₇₉	—	2	2	no	—
30	C₂×C₄×C₇₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₇₉	M₁₆	1	2	yes	—
38	C₈×C₇₉	—	2	4	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	#95	1	1	yes	characteristic
44	C₂²×C₇₉	C₄⋊C₈	1	2	yes	—
45	C₄×C₇₉	—	2	1	no	—
46	C₄×C₇₉	—	2	1	no	—
47	C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
48	C₄×C₇₉	#97	1	1	yes	characteristic
49	C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
50	C₄×C₇₉	#104	1	2	yes	—
51	C₄×C₇₉	#108	1	2	yes	—
52	C₂×C₇₉	—	2	2	no	—
53	C₂×C₇₉	#336	1	1	yes	characteristic
54	C₂×C₇₉	#339	1	1	yes	characteristic
55	C₂×C₇₉	#341	1	1	yes	characteristic
56	#1278	—	79	1	no	maximal, Sylow
57	C₇₉	#1278	1	1	yes	characteristic, complemented, Sylow
58	#431	—	79	1	no	—
59	C₄×M₁₆	—	79	1	no	—
60	C₂×C₄×C₈	D₇₉	1	1	yes	characteristic
61	C₂×M₁₆	—	79	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
64	C₄⋊C₈	—	79	2	no	—
65	C₂²⋊C₈	—	158	1	no	—
66	#97	—	158	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	158	1	no	—
69	C₄×C₈	—	158	1	no	—
70	C₄×C₈	Dic₇₉	1	2	yes	—
71	C₂²×C₄	Dic₁₅₈	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
74	M₁₆	—	79	4	no	—
75	M₁₆	—	158	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	158	2	no	—
79	C₂×C₈	—	158	2	no	—
80	C₄²	C₄×D₇₉	1	1	yes	characteristic
81	C₄²	D₃₁₆	1	1	yes	characteristic
82	C₄²	C₂×Dic₇₉	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	—	2	2	no	—
87	C₂×C₄	C₄×Dic₇₉	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
91	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
93	C₂×C₄	C₈×D₇₉	1	2	yes	—
94	C₂×C₄	#11929607	1	2	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	316	2	no	—
97	C₈	—	158	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₇₉⋊#95	1	1	yes	characteristic
101	C₂²	Dic₇₉⋊C₈	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₇₉	1	1	yes	characteristic
105	C₄	#14579071	1	1	yes	characteristic
106	C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
107	C₄	#14579089	1	2	yes	—
108	C₄	#14579097	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#19098317	1	1	yes	characteristic
111	C₂	#19098323	1	1	yes	characteristic
112	C₂	#19098328	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098616

C₂

yes

maximal, characteristic

C₄×#11929608

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₄

yes

—

#11929608

—

#11929608

C₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

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

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₈

yes

—

C₂×C₄×C₇₉

M₁₆

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#95

yes

characteristic

C₂²×C₇₉

C₄⋊C₈

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×C₈

yes

characteristic

C₄×C₇₉

#97

yes

characteristic

C₄×C₇₉

C₂²⋊C₈

yes

—

C₄×C₇₉

#104

yes

—

C₄×C₇₉

#108

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

D₇₉

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

158

—

#97

—

158

—

C₄×C₈

—

C₄×C₈

—

158

—

C₄×C₈

—

158

—

C₄×C₈

Dic₇₉

yes

—

C₂²×C₄

Dic₁₅₈

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄²

C₄×D₇₉

yes

characteristic

C₄²

D₃₁₆

yes

characteristic

C₄²

C₂×Dic₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₂³

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₉

yes

characteristic

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

C₈×D₇₉

yes

—

C₂×C₄

#11929607

yes

—

C₈

—

C₈

—

316

—

C₈

—

158

—

C₂²

—

C₂²

—

100

C₂²

C₇₉⋊#95

yes

characteristic

101

C₂²

Dic₇₉⋊C₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₇₉

yes

characteristic

105

C₄

#14579071

yes

characteristic

106

C₄

(C₂×D₇₉)⋊C₈

yes

—

107

C₄

#14579089

yes

—

108

C₄

#14579097

yes

—

109

C₂

—

110

C₂

#19098317

yes

characteristic

111

C₂

#19098323

yes

characteristic

112

C₂

#19098328

yes

characteristic

	subgroup	G/H
center	C₄²	D₃₁₆
commutator	C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₇₉
Fitting	C₂×C₄×C₈×C₇₉	C₂
socle	C₂²×C₇₉	#95
prime radical	C₂³×C₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂²
hypercenter	C₂×C₄×C₈	D₇₉
hypocenter	C₇₉	#1278
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₃₁₆

commutator

C₄×C₇₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₇₉

Fitting

C₂×C₄×C₈×C₇₉

C₂

socle

C₂²×C₇₉

#95

prime radical

C₂³×C₇₉

C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂²

hypercenter

C₂×C₄×C₈

D₇₉

hypocenter

C₇₉

#1278

LCS

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
15. #11929608	C₈	semidirect
57. C₇₉	#1278	semidirect

H ≅ G/N

type

15. #11929608

C₈

semidirect

57. C₇₉

#1278

semidirect

Subgroups — #23761237

Conjugacy classes of subgroups

Complements