Group #37509993 — subgroups

Subgroups — #37509993

Order 22912.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#19153105	C₂	1	1	yes	maximal, characteristic
2	#26038463	C₂	1	1	yes	maximal, characteristic
3	#26038207	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668243	C₂²	1	2	yes	—
6	#19153105	—	2	1	no	—
7	#19153105	—	2	1	no	—
8	#19153105	—	2	2	no	—
9	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
10	C₂×Dic₁₇₉⋊C₄	C₄	1	4	yes	—
11	C₄⋊(C₁₇₉⋊C₈)	—	2	2	no	—
12	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	#97×C₁₇₉	—	2	1	no	—
15	C₄×C₈×C₁₇₉	—	2	1	no	—
16	C₂²×Dic₁₇₉	C₂×C₄	1	4	yes	—
17	#15668243	—	2	4	no	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₇₉	—	2	2	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	Dic₁₇₉⋊C₄	—	2	4	no	—
22	Dic₁₇₉⋊C₄	—	2	4	no	—
23	C₂²×C₄×C₁₇₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₇₉	—	2	2	no	—
26	M₁₆×C₁₇₉	Q₈	1	4	yes	—
27	C₂×C₈×C₁₇₉	—	2	2	no	—
28	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
29	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
30	C₄²×C₁₇₉	D₄	1	2	yes	—
31	C₂×Dic₁₇₉	—	2	4	no	—
32	C₂×Dic₁₇₉	—	4	4	no	—
33	C₁₇₉⋊C₈	—	4	2	no	—
34	C₂³×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	—	4	2	no	—
36	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₇₉	#41	1	2	yes	—
42	C₈×C₁₇₉	—	2	4	no	—
43	C₂²×C₁₇₉	—	2	2	no	—
44	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₇₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₇₉	—	2	2	no	—
47	C₄×C₁₇₉	—	2	2	no	—
48	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₇₉	#124	1	1	yes	characteristic
50	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₇₉	—	4	4	no	—
52	C₂×C₁₇₉	—	2	2	no	—
53	C₂×C₁₇₉	#391	1	1	yes	characteristic
54	C₂×C₁₇₉	#413	1	2	yes	—
55	C₁₇₉	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	179	1	no	maximal, Sylow
57	C₂×#125	—	179	1	no	—
58	#423	—	179	1	no	—
59	C₄×M₁₆	D₁₇₉	1	1	yes	characteristic
60	#343	—	179	4	no	—
61	C₂×M₁₆	—	179	2	no	—
62	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
63	#125	—	358	1	no	—
64	#125	—	358	1	no	—
65	#125	—	358	2	no	—
66	C₂×C₄⋊C₄	—	179	4	no	—
67	C₂×C₄⋊C₄	—	179	2	no	—
68	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
69	C₄⋊C₈	—	358	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	179	4	no	—
73	C₂²×C₄	C₄×D₁₇₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	358	4	no	—
77	M₁₆	Dic₃₅₈	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	358	2	no	—
80	C₄⋊C₄	—	358	4	no	—
81	C₄⋊C₄	—	358	4	no	—
82	C₄⋊C₄	—	358	2	no	—
83	C₄⋊C₄	—	358	2	no	—
84	C₄²	C₂²⋊D₁₇₉	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₇₉	1	1	yes	characteristic
86	C₄²	D₇₁₆	1	2	yes	—
87	C₂³	C₂×C₄×D₁₇₉	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	358	4	no	—
90	C₂×C₄	—	716	4	no	—
91	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
94	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₃₅₈	1	2	yes	—
96	C₂×C₄	#15668259	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	716	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
101	C₂²	C₄×Dic₃₅₈	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	716	4	no	—
105	C₄	#19153092	1	2	yes	—
106	C₄	#19153101	1	1	yes	characteristic
107	C₄	#19153113	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#26038345	1	1	yes	characteristic
110	C₂	#26038421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153105

C₂

yes

maximal, characteristic

#26038463

C₂

yes

maximal, characteristic

#26038207

C₂

yes

maximal

C₄×M₁₆×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

#19153105

—

#19153105

—

#19153105

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

C₂×Dic₁₇₉⋊C₄

C₄

yes

—

C₄⋊(C₁₇₉⋊C₈)

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#97×C₁₇₉

—

C₄×C₈×C₁₇₉

—

C₂²×Dic₁₇₉

C₂×C₄

yes

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

Dic₁₇₉⋊C₄

—

Dic₁₇₉⋊C₄

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

Q₈

yes

—

C₂×C₈×C₁₇₉

—

C₄²×C₁₇₉

D₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

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

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₄×Q₈

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#124

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#391

yes

characteristic

C₂×C₁₇₉

#413

yes

—

C₁₇₉

#1998

yes

characteristic, complemented, Sylow

#1998

—

179

maximal, Sylow

C₂×#125

—

179

—

#423

—

179

—

C₄×M₁₆

D₁₇₉

yes

characteristic

#343

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

#125

—

358

—

#125

—

358

—

#125

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

358

—

#97

—

C₄×C₈

—

C₂²×C₄

—

179

—

C₂²×C₄

C₄×D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

358

—

M₁₆

Dic₃₅₈

yes

—

C₂×C₈

—

C₂×C₈

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄²

C₂²⋊D₁₇₉

yes

characteristic

C₄²

C₂²⋊D₁₇₉

yes

characteristic

C₄²

D₇₁₆

yes

—

C₂³

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

716

—

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

C₂×D₇₁₆

yes

characteristic

C₂×C₄

C₂×Dic₃₅₈

yes

—

C₂×C₄

#15668259

yes

—

C₈

—

C₈

—

716

—

C₂²

—

100

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

101

C₂²

C₄×Dic₃₅₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

716

—

105

C₄

#19153092

yes

—

106

C₄

#19153101

yes

characteristic

107

C₄

#19153113

yes

characteristic

108

C₂

—

109

C₂

#26038345

yes

characteristic

110

C₂

#26038421

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₇₉)⋊C₄
commutator, prime radical	C₂³×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₄×M₁₆×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₄×M₁₆	D₁₇₉
hypocenter	C₁₇₉	#1998
LCS	G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

commutator, prime radical

C₂³×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₄×M₁₆×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

hypercenter

C₄×M₁₆

D₁₇₉

hypocenter

C₁₇₉

#1998

LCS

G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₇₉	#1998	semidirect

H ≅ G/N

type

55. C₁₇₉

#1998

semidirect

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

subgroup	index
C₂×#19153105	2
#26038463	2
#26038207	2
C₄×M₁₆×C₁₇₉	2
#1998	179

subgroup

index

C₄×M₁₆×C₁₇₉

179

Subgroups — #37509993

Conjugacy classes of subgroups

Complements

Maximal subgroups