Group #37510599 — subgroups

Subgroups — #37510599

Order 22912.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂²×C₁₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#26038394	C₂	1	1	yes	maximal, characteristic
3	#26038214	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₇₉	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₄	1	1	yes	characteristic
7	#19153072	C₄	1	1	yes	characteristic
8	C₂×C₁₇₉⋊C₁₆	C₄	1	4	yes	complemented
9	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19153023	C₄	1	2	yes	—
12	C₈×Dic₁₇₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₇₉	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₁₇₉	C₂²	1	2	yes	—
16	#15668243	C₈	1	4	yes	—
17	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊Dic₁₇₉	C₈	1	2	yes	—
19	C₂²⋊Dic₁₇₉	C₈	1	2	yes	—
20	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
21	C₁₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₁₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₇₉	C₂×C₄	1	4	yes	—
29	C₂×Dic₁₇₉	C₂×C₈	1	4	yes	—
30	C₁₇₉⋊C₈	C₂×C₈	1	4	yes	—
31	C₂³×C₁₇₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₇₉	C₄²	1	2	yes	—
33	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₇₉	C₄²	1	2	yes	—
37	C₈×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₇₉	—	2	2	no	—
40	C₂²×C₁₇₉	C₄×C₈	1	2	yes	—
41	C₂²×C₁₇₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₇₉	C₄×C₈	1	2	yes	—
43	C₄×C₁₇₉	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₇₉	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₇₉	—	2	4	no	—
46	C₂×C₁₇₉	—	2	2	no	—
47	C₂×C₁₇₉	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₇₉	#504	1	2	yes	—
49	C₁₇₉	#2108	1	1	yes	characteristic, complemented, Sylow
50	#2108	—	179	1	no	maximal, Sylow
51	C₂×M₃₂	D₁₇₉	1	1	yes	characteristic
52	C₂²×C₁₆	—	179	1	no	—
53	#405	—	179	1	no	—
54	#346	—	179	2	no	—
55	C₄×C₁₆	—	179	2	no	—
56	C₂×M₁₆	—	179	1	no	—
57	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
58	#117	—	179	1	no	—
59	M₃₂	Dic₁₇₉	1	4	yes	complemented
60	C₂×C₁₆	—	179	4	no	—
61	C₂×C₁₆	—	179	1	no	—
62	C₂×C₁₆	—	179	1	no	—
63	C₂×C₁₆	C₂×D₁₇₉	1	2	yes	—
64	#97	—	179	2	no	—
65	C₄×C₈	—	179	2	no	—
66	C₂²×C₄	C₄×D₁₇₉	1	1	yes	characteristic
67	M₁₆	—	179	4	no	—
68	C₂×C₈	—	179	2	no	—
69	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
70	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
71	C₂×C₈	C₂×Dic₁₇₉	1	2	yes	—
72	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
73	C₄⋊C₄	—	179	2	no	—
74	C₂²⋊C₄	—	179	2	no	—
75	C₄²	—	179	2	no	—
76	C₁₆	—	179	2	no	—
77	C₁₆	—	179	2	no	—
78	C₁₆	C₂×Dic₁₇₉	1	4	yes	—
79	C₂³	C₈×D₁₇₉	1	1	yes	characteristic
80	C₂×C₄	—	179	4	no	—
81	C₂×C₄	C₄×Dic₁₇₉	1	2	yes	—
82	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
83	C₂×C₄	C₈×D₁₇₉	1	2	yes	—
84	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
85	C₈	—	179	4	no	—
86	C₈	C₄×Dic₁₇₉	1	2	yes	—
87	C₈	C₂×C₄×D₁₇₉	1	1	yes	characteristic
88	C₈	C₂²×Dic₁₇₉	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₁₇₉	1	2	yes	—
91	C₂²	C₂×C₈×D₁₇₉	1	1	yes	characteristic
92	C₄	—	358	4	no	—
93	C₄	C₈×Dic₁₇₉	1	2	yes	—
94	C₄	C₂×C₄×Dic₁₇₉	1	1	yes	characteristic
95	C₄	C₂×C₈×D₁₇₉	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₁₇₉	1	1	yes	characteristic
98	C₂	#26038857	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₇₉⋊C₁₆

C₂

yes

maximal, characteristic

#26038394

C₂

yes

maximal, characteristic

#26038214

C₂

yes

maximal, complemented

C₁₆×Dic₁₇₉

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₄

yes

characteristic

#19153072

C₄

yes

characteristic

C₂×C₁₇₉⋊C₁₆

C₄

yes

complemented

C₂×C₁₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₇₉⋊C₁₆

C₂²

yes

characteristic

#19153023

C₄

yes

—

C₈×Dic₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

M₃₂×C₁₇₉

C₄

yes

complemented

C₂×C₁₆×C₁₇₉

C₂²

yes

—

#15668243

C₈

yes

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₇₉

C₈

yes

—

C₂²⋊Dic₁₇₉

C₈

yes

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₁₇₉⋊C₁₆

C₂×C₄

yes

—

C₁₇₉⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₂×C₈×C₁₇₉

C₂³

yes

characteristic

C₁₆×C₁₇₉

C₂×C₄

yes

—

C₂×Dic₁₇₉

C₂×C₈

yes

—

C₁₇₉⋊C₈

C₂×C₈

yes

—

C₂³×C₁₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₉

C₄²

yes

—

C₂×C₄×C₁₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂×C₈

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₈×C₁₇₉

C₄²

yes

—

C₈×C₁₇₉

C₂²×C₄

yes

characteristic

C₈×C₁₇₉

C₂²×C₄

yes

characteristic

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₄×C₈

yes

—

C₂²×C₁₇₉

C₂²×C₈

yes

characteristic

C₄×C₁₇₉

C₄×C₈

yes

—

C₄×C₁₇₉

C₂×C₄²

yes

characteristic

C₄×C₁₇₉

C₂²×C₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₇₉

#504

yes

—

C₁₇₉

#2108

yes

characteristic, complemented, Sylow

#2108

—

179

maximal, Sylow

C₂×M₃₂

D₁₇₉

yes

characteristic

C₂²×C₁₆

—

179

—

#405

—

179

—

#346

—

179

—

C₄×C₁₆

—

179

—

C₂×M₁₆

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

—

179

—

M₃₂

Dic₁₇₉

yes

complemented

C₂×C₁₆

—

179

—

C₂×C₁₆

—

179

—

C₂×C₁₆

—

179

—

C₂×C₁₆

C₂×D₁₇₉

yes

—

#97

—

179

—

C₄×C₈

—

179

—

C₂²×C₄

C₄×D₁₇₉

yes

characteristic

M₁₆

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₄×D₁₇₉

yes

—

C₂×C₈

C₂×Dic₁₇₉

yes

—

C₂×C₈

C₂²×D₁₇₉

yes

characteristic

C₄⋊C₄

—

179

—

C₂²⋊C₄

—

179

—

C₄²

—

179

—

C₁₆

—

179

—

C₁₆

—

179

—

C₁₆

C₂×Dic₁₇₉

yes

—

C₂³

C₈×D₁₇₉

yes

characteristic

C₂×C₄

—

179

—

C₂×C₄

C₄×Dic₁₇₉

yes

—

C₂×C₄

C₈×D₁₇₉

yes

characteristic

C₂×C₄

C₈×D₁₇₉

yes

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₈

—

179

—

C₈

C₄×Dic₁₇₉

yes

—

C₈

C₂×C₄×D₁₇₉

yes

characteristic

C₈

C₂²×Dic₁₇₉

yes

characteristic

C₂²

—

C₂²

C₈×Dic₁₇₉

yes

—

C₂²

C₂×C₈×D₁₇₉

yes

characteristic

C₄

—

358

—

C₄

C₈×Dic₁₇₉

yes

—

C₄

C₂×C₄×Dic₁₇₉

yes

characteristic

C₄

C₂×C₈×D₁₇₉

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₁₇₉

yes

characteristic

C₂

#26038857

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇₉
commutator	C₂×C₁₇₉	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂²×C₈
prime radical	C₂³×C₁₇₉	C₂×C₈
cosocle	C₂×C₈×C₁₇₉	C₂³
hypercenter	C₂×M₃₂	D₁₇₉
hypocenter	C₁₇₉	#2108
LCS	G ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₇₉

commutator

C₂×C₁₇₉

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂²×C₈

prime radical

C₂³×C₁₇₉

C₂×C₈

cosocle

C₂×C₈×C₁₇₉

C₂³

hypercenter

C₂×M₃₂

D₁₇₉

C₁₇₉

G ▷ C₂×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #26038214	C₂	semidirect
4. C₁₆×Dic₁₇₉	C₂	semidirect
8. C₂×C₁₇₉⋊C₁₆	C₄	semidirect
14. M₃₂×C₁₇₉	C₄	semidirect
49. C₁₇₉	#2108	semidirect
59. M₃₂	Dic₁₇₉	semidirect

H ≅ G/N

type

3. #26038214

C₂

semidirect

4. C₁₆×Dic₁₇₉

C₂

semidirect

8. C₂×C₁₇₉⋊C₁₆

C₄

semidirect

14. M₃₂×C₁₇₉

C₄

semidirect

49. C₁₇₉

#2108

semidirect

59. M₃₂

Dic₁₇₉

semidirect

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

subgroup	index
C₂²×C₁₇₉⋊C₁₆	2
#26038394	2
#26038214	2
C₁₆×Dic₁₇₉	2
C₂×M₃₂×C₁₇₉	2
#2108	179

subgroup

index

C₂²×C₁₇₉⋊C₁₆

#26038394

#26038214

C₁₆×Dic₁₇₉

C₂×M₃₂×C₁₇₉

#2108

179

Subgroups — #37510599

Conjugacy classes of subgroups

Complements

Maximal subgroups