Group #37510510 — subgroups

Subgroups — #37510510

Order 22912.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#26038482	C₂	1	2	yes	maximal
2	C₂×#19153045	C₂	1	1	yes	maximal, characteristic
3	#26038207	C₂	1	2	yes	maximal
4	#26038204	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
8	#19153046	—	2	2	no	—
9	#19153045	—	2	2	no	—
10	#19153044	—	2	2	no	—
11	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
13	#117×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₇₉	—	2	2	no	—
15	C₂²×Dic₁₇₉	—	2	2	no	—
16	#15668243	—	2	2	no	—
17	#15668243	Q₈	1	4	yes	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₇₉	—	2	2	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₉	—	2	2	no	—
23	C₂×C₈×C₁₇₉	—	2	2	no	—
24	C₂×C₈×C₁₇₉	—	2	1	no	—
25	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₇₉	—	2	1	no	—
28	C₄²×C₁₇₉	—	2	1	no	—
29	C₂×Dic₁₇₉	—	4	2	no	—
30	C₂×Dic₁₇₉	—	8	2	no	—
31	C₁₇₉⋊C₈	—	2	4	no	—
32	C₂³×C₁₇₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₇₉	—	2	2	no	—
34	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₇₉	#41	1	2	yes	—
38	C₈×C₁₇₉	—	2	2	no	—
39	C₈×C₁₇₉	—	4	1	no	—
40	C₂²×C₁₇₉	—	2	2	no	—
41	C₂²×C₁₇₉	#121	1	1	yes	characteristic
42	C₂²×C₁₇₉	#125	1	2	yes	—
43	C₄×C₁₇₉	—	4	2	no	—
44	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₇₉	#123	1	1	yes	characteristic
46	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₇₉	—	8	2	no	—
48	C₂×C₁₇₉	—	2	2	no	—
49	C₂×C₁₇₉	#398	1	1	yes	characteristic
50	C₂×C₁₇₉	#471	1	2	yes	—
51	C₁₇₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	179	1	no	maximal, Sylow
53	#428	—	179	2	no	—
54	C₂×#106	—	179	1	no	—
55	#424	D₁₇₉	1	1	yes	characteristic
56	#343	—	179	2	no	—
57	#341	—	179	1	no	—
58	C₂×M₁₆	—	179	2	no	—
59	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
61	#117	C₂×D₁₇₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	179	2	no	—
63	#107	—	358	2	no	—
64	#106	—	358	2	no	—
65	#106	—	358	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	358	2	no	—
68	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	358	2	no	—
71	M₁₆	—	179	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	358	2	no	—
75	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
76	C₄⋊C₄	—	358	2	no	—
77	C₄⋊C₄	—	358	2	no	—
78	C₄⋊C₄	D₇₁₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#15668259	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	1432	2	no	—
84	C₂×C₄	—	716	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
88	C₂×C₄	#15668258	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	358	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#19153087	1	1	yes	characteristic
94	C₂²	#19153103	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	1432	2	no	—
97	C₄	#19153093	1	2	yes	—
98	C₄	#19153094	1	1	yes	characteristic
99	C₄	#19153111	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#26038369	1	1	yes	characteristic
102	C₂	#26038702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038482

C₂

yes

maximal

C₂×#19153045

C₂

yes

maximal, characteristic

#26038207

C₂

yes

maximal

#26038204

C₂

yes

maximal, characteristic

#424×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153046

—

#19153045

—

#19153044

—

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₉

—

C₂²×Dic₁₇₉

—

#15668243

—

#15668243

Q₈

yes

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

D₄

yes

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂²×C₁₇₉

#125

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#123

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#398

yes

characteristic

C₂×C₁₇₉

#471

yes

—

C₁₇₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

179

maximal, Sylow

#428

—

179

—

C₂×#106

—

179

—

#424

D₁₇₉

yes

characteristic

#343

—

179

—

#341

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

—

179

—

#107

—

358

—

#106

—

358

—

#106

—

358

—

C₄⋊C₈

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

358

—

M₁₆

—

179

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

D₇₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#15668259

yes

characteristic

C₂×C₄

—

C₂×C₄

—

1432

—

C₂×C₄

—

716

—

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

C₂×D₇₁₆

yes

characteristic

C₂×C₄

Q₈×D₁₇₉

yes

—

C₂×C₄

#15668258

yes

—

C₈

—

C₈

—

C₈

—

358

—

C₂²

—

C₂²

#19153087

yes

characteristic

C₂²

#19153103

yes

—

C₄

—

C₄

—

1432

—

C₄

#19153093

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#26038369

yes

characteristic

102

C₂

#26038702

yes

—

	subgroup	G/H
center	C₂²	#19153087
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#424×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#121
prime radical	C₂³×C₁₇₉	#41
hypercenter	#424	D₁₇₉
hypocenter	C₁₇₉	#2090
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#19153087

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#424×C₁₇₉

C₂

C₂²×C₁₇₉

C₂³×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
51. C₁₇₉	#2090	semidirect

H ≅ G/N

type

51. C₁₇₉

#2090

semidirect

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

subgroup	index
#26038482	2
C₂×#19153045	2
#26038207	2
#26038204	2
#424×C₁₇₉	2
#2090	179

subgroup

index

#424×C₁₇₉

179

Subgroups — #37510510

Conjugacy classes of subgroups

Complements

Maximal subgroups