Group #37510505 — subgroups

Subgroups — #37510505

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×#19153047	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	#19153047	—	2	2	no	—
9	#19153046	—	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	1	yes	characteristic
27	C₄⋊C₄×C₁₇₉	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₁₇₉	—	2	1	no	—
29	C₄²×C₁₇₉	—	2	1	no	—
30	C₂×Dic₁₇₉	—	4	2	no	—
31	C₂×Dic₁₇₉	—	8	2	no	—
32	C₁₇₉⋊C₈	—	2	4	no	—
33	C₂³×C₁₇₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	—	2	1	no	—
35	C₂×C₄×C₁₇₉	—	2	1	no	—
36	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₇₉	#41	1	2	yes	—
40	C₈×C₁₇₉	—	2	2	no	—
41	C₈×C₁₇₉	—	4	1	no	—
42	C₂²×C₁₇₉	—	2	1	no	—
43	C₂²×C₁₇₉	—	2	1	no	—
44	C₂²×C₁₇₉	#121	1	1	yes	characteristic
45	C₂²×C₁₇₉	#125	1	2	yes	—
46	C₄×C₁₇₉	—	4	1	no	—
47	C₄×C₁₇₉	—	4	1	no	—
48	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₇₉	#123	1	1	yes	characteristic
50	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₇₉	—	8	2	no	—
52	C₂×C₁₇₉	—	2	2	no	—
53	C₂×C₁₇₉	#398	1	1	yes	characteristic
54	C₂×C₁₇₉	#472	1	1	yes	characteristic
55	C₂×C₁₇₉	#473	1	1	yes	characteristic
56	C₁₇₉	#2089	1	1	yes	characteristic, complemented, Sylow
57	#2089	—	179	1	no	maximal, Sylow
58	#428	—	179	2	no	—
59	C₂×#107	—	179	1	no	—
60	#424	D₁₇₉	1	1	yes	characteristic
61	#343	—	179	2	no	—
62	#341	—	179	1	no	—
63	C₂×M₁₆	—	179	2	no	—
64	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
66	#117	C₂×D₁₇₉	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	179	2	no	—
68	#107	—	358	2	no	—
69	#107	—	358	2	no	—
70	#106	—	358	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	358	2	no	—
73	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	358	2	no	—
76	M₁₆	—	179	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	358	2	no	—
80	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
81	C₄⋊C₄	—	358	2	no	—
82	C₄⋊C₄	—	358	2	no	—
83	C₄⋊C₄	D₇₁₆	1	1	yes	characteristic
84	C₄⋊C₄	D₇₁₆	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#15668259	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	1432	2	no	—
91	C₂×C₄	—	716	2	no	—
92	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
93	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
94	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
95	C₂×C₄	#15668258	1	2	yes	—
96	C₈	—	2	2	no	—
97	C₈	—	4	1	no	—
98	C₈	—	358	4	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#19153087	1	1	yes	characteristic
102	C₂²	#19153103	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	1432	2	no	—
106	C₄	#19153093	1	2	yes	—
107	C₄	#19153094	1	1	yes	characteristic
108	C₄	#19153111	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#26038369	1	1	yes	characteristic
111	C₂	#26038709	1	1	yes	characteristic
112	C₂	#26038714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038482

C₂

yes

maximal

C₂×#19153047

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

—

#19153047

—

#19153046

—

#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

characteristic

C₄⋊C₄×C₁₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

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

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂²×C₁₇₉

#125

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#123

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2089

—

179

maximal, Sylow

#428

—

179

—

C₂×#107

—

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

—

#107

—

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

characteristic

C₄⋊C₄

D₇₁₆

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#15668259

yes

characteristic

C₂×C₄

—

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

—

100

C₂²

—

101

C₂²

#19153087

yes

characteristic

102

C₂²

#19153103

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

1432

—

106

C₄

#19153093

yes

—

107

C₄

#19153094

yes

characteristic

108

C₄

#19153111

yes

characteristic

109

C₂

—

110

C₂

#26038369

yes

characteristic

111

C₂

#26038709

yes

characteristic

112

C₂

#26038714

yes

characteristic

	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	#2089
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
56. C₁₇₉	#2089	semidirect

H ≅ G/N

type

56. C₁₇₉

#2089

semidirect

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

subgroup	index
#26038482	2
C₂×#19153047	2
#26038207	2
#26038204	2
#424×C₁₇₉	2
#2089	179

subgroup

index

#424×C₁₇₉

179

Subgroups — #37510505

Conjugacy classes of subgroups

Complements

Maximal subgroups