Group #37509147 — subgroups

Subgroups — #37509147

Order 22912.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
2	#26038209	C₂	1	2	yes	maximal
3	#26038198	C₂	1	2	yes	maximal
4	C₁₇₉⋊#338	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂×#15668243	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153023	C₄	1	2	yes	complemented
10	C₈×Dic₁₇₉	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
12	#117×C₁₇₉	C₂²	1	2	yes	—
13	C₂²×Dic₁₇₉	—	2	1	no	—
14	#15668243	—	2	2	no	—
15	#15668243	—	2	2	no	—
16	#15668243	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₇₉	—	2	1	no	—
20	C₄×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₇₉	—	2	2	no	—
24	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
25	C₄⋊C₄×C₁₇₉	—	2	2	no	—
26	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	—	2	2	no	—
28	C₂×Dic₁₇₉	—	2	1	no	—
29	C₂×Dic₁₇₉	—	2	1	no	—
30	C₂×Dic₁₇₉	—	4	2	no	—
31	C₁₇₉⋊C₈	—	2	2	no	—
32	C₁₇₉⋊C₈	—	4	2	no	—
33	C₂³×C₁₇₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	—	4	2	no	—
35	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₉	#41	1	2	yes	—
37	C₂×C₄×C₁₇₉	#41	1	2	yes	—
38	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
39	C₈×C₁₇₉	—	2	2	no	—
40	C₂²×C₁₇₉	—	4	1	no	—
41	C₂²×C₁₇₉	#117	1	1	yes	characteristic
42	C₂²×C₁₇₉	#126	1	2	yes	—
43	C₄×C₁₇₉	—	8	2	no	—
44	C₄×C₁₇₉	#117	1	1	yes	characteristic
45	C₄×C₁₇₉	#117	1	1	yes	characteristic
46	C₄×C₁₇₉	#126	1	2	yes	—
47	Dic₁₇₉	—	2	2	no	—
48	Dic₁₇₉	—	4	1	no	—
49	C₂×C₁₇₉	—	2	1	no	—
50	C₂×C₁₇₉	—	2	2	no	—
51	C₂×C₁₇₉	#383	1	1	yes	characteristic
52	C₁₇₉	#1841	1	1	yes	characteristic, complemented, Sylow
53	#1841	—	179	1	no	maximal, Sylow
54	C₄×M₁₆	—	179	1	no	—
55	#344	—	179	2	no	—
56	#339	—	179	2	no	—
57	#338	—	179	1	no	—
58	H(3,ℤ/4)	D₁₇₉	1	1	yes	characteristic
59	C₂×M₁₆	—	179	2	no	—
60	C₂×M₁₆	—	179	1	no	—
61	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
62	#117	C₂×D₁₇₉	1	2	yes	—
63	C₂×C₄²	—	179	1	no	—
64	#97	—	179	2	no	—
65	C₄×C₈	—	179	2	no	—
66	C₂²×C₄	—	358	1	no	—
67	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	358	2	no	—
70	M₁₆	—	358	2	no	—
71	M₁₆	—	358	2	no	—
72	C₂×C₈	—	358	2	no	—
73	C₂×C₈	—	179	2	no	—
74	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	358	1	no	—
79	C₄²	—	179	1	no	—
80	C₄²	—	179	1	no	—
81	C₂³	#15668258	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	358	1	no	—
84	C₂×C₄	—	358	1	no	—
85	C₂×C₄	—	716	2	no	—
86	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
87	C₂×C₄	#15668258	1	2	yes	—
88	C₂×C₄	#15668259	1	2	yes	—
89	C₂×C₄	#15668260	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	358	2	no	—
92	C₈	—	716	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#19153071	1	1	yes	characteristic
95	C₂²	#19153106	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	358	2	no	—
98	C₄	—	716	1	no	—
99	C₄	#19153069	1	1	yes	characteristic
100	C₄	#19153070	1	1	yes	characteristic
101	C₄	#19153108	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#26038306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₁₇₉

C₂

yes

maximal, characteristic

#26038209

C₂

yes

maximal

#26038198

C₂

yes

maximal

C₁₇₉⋊#338

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂×#15668243

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

#19153023

C₄

yes

complemented

C₈×Dic₁₇₉

C₄

yes

complemented

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

—

C₂²×Dic₁₇₉

—

#15668243

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

C₂×C₄

yes

characteristic

C₄×Dic₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#117

yes

characteristic

C₂²×C₁₇₉

#126

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#383

yes

characteristic

C₁₇₉

#1841

yes

characteristic, complemented, Sylow

#1841

—

179

maximal, Sylow

C₄×M₁₆

—

179

—

#344

—

179

—

#339

—

179

—

#338

—

179

—

H(3,ℤ/4)

D₁₇₉

yes

characteristic

C₂×M₁₆

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

#117

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

#97

—

179

—

C₄×C₈

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

358

—

C₄²

—

179

—

C₄²

—

179

—

C₂³

#15668258

yes

characteristic

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

358

—

C₂×C₄

—

716

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

#15668258

yes

—

C₂×C₄

#15668259

yes

—

C₂×C₄

#15668260

yes

characteristic

C₈

—

C₈

—

358

—

C₈

—

716

—

C₂²

—

C₂²

#19153071

yes

characteristic

C₂²

#19153106

yes

—

C₄

—

C₄

—

358

—

C₄

—

716

—

C₄

#19153069

yes

characteristic

100

C₄

#19153070

yes

characteristic

101

C₄

#19153108

yes

—

102

C₂

—

103

C₂

—

104

C₂

#26038306

yes

characteristic

	subgroup	G/H
center	C₄	#19153070
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	H(3,ℤ/4)×C₁₇₉	C₂
socle	C₂×C₁₇₉	#383
prime radical	C₂³×C₁₇₉	#41
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	H(3,ℤ/4)	D₁₇₉
hypocenter	C₁₇₉	#1841
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19153070

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

H(3,ℤ/4)×C₁₇₉

C₂

C₂×C₁₇₉

C₂³×C₁₇₉

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #19153023	C₄	semidirect
10. C₈×Dic₁₇₉	C₄	semidirect
52. C₁₇₉	#1841	semidirect

H ≅ G/N

type

9. #19153023

C₄

semidirect

10. C₈×Dic₁₇₉

C₄

semidirect

52. C₁₇₉

#1841

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
M₁₆×Dic₁₇₉	2
#26038209	2
#26038198	2
C₁₇₉⋊#338	2
H(3,ℤ/4)×C₁₇₉	2
#1841	179

subgroup

index

M₁₆×Dic₁₇₉

#26038209

#26038198

C₁₇₉⋊#338

H(3,ℤ/4)×C₁₇₉

#1841

179

Subgroups — #37509147

Conjugacy classes of subgroups

Complements

Maximal subgroups