Group #37509981 — subgroups

Subgroups — #37509981

Order 22912.

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

normal subgroups only

This group has 244 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	#26038996	C₂	1	1	yes	maximal, characteristic
2	#26038463	C₂	1	1	yes	maximal, characteristic
3	#26038199	C₂	1	4	yes	maximal
4	#376×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668243	C₂²	1	2	yes	—
6	#19153113	C₄	1	2	yes	complemented
7	#19153105	C₄	1	2	yes	complemented
8	#19153092	—	2	2	no	—
9	C₄×Dic₃₅₈	—	2	2	no	—
10	#19153072	C₂²	1	2	yes	—
11	C₄⋊(C₁₇₉⋊C₈)	—	2	2	no	—
12	C₂×C₄²×C₁₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	#95×C₁₇₉	—	2	2	no	—
15	C₂×Dic₃₅₈	—	2	2	no	—
16	#15668243	—	2	4	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₄⋊Dic₁₇₉	—	2	2	no	—
19	C₄⋊Dic₁₇₉	C₂×C₄	1	4	yes	—
20	Dic₁₇₉⋊C₄	—	2	4	no	—
21	C₂²⋊Dic₁₇₉	—	2	2	no	—
22	C₄×Dic₁₇₉	—	2	2	no	—
23	C₂²×C₄×C₁₇₉	—	2	2	no	—
24	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
26	C₄²×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₇₉	D₄	1	2	yes	—
30	C₄²×C₁₇₉	Q₈	1	4	yes	complemented
31	Dic₃₅₈	—	4	4	no	—
32	C₂×Dic₁₇₉	—	2	4	no	—
33	C₁₇₉⋊C₈	—	4	2	no	—
34	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	—	2	4	no	—
36	C₂×C₄×C₁₇₉	—	4	2	no	—
37	C₂×C₄×C₁₇₉	—	4	4	no	—
38	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₇₉	#41	1	2	yes	—
44	C₂²×C₁₇₉	—	2	2	no	—
45	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₇₉	—	2	2	no	—
48	C₄×C₁₇₉	—	2	2	no	—
49	C₄×C₁₇₉	—	4	4	no	—
50	C₄×C₁₇₉	C₄×Q₈	1	2	yes	—
51	C₄×C₁₇₉	#124	1	1	yes	characteristic
52	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₇₉	—	4	4	no	—
54	C₂×C₁₇₉	—	2	2	no	—
55	C₂×C₁₇₉	#391	1	1	yes	characteristic
56	C₂×C₁₇₉	#421	1	2	yes	—
57	C₁₇₉	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	179	1	no	maximal, Sylow
59	#533	—	179	1	no	—
60	#423	—	179	1	no	—
61	#376	D₁₇₉	1	1	yes	characteristic
62	#339	—	179	4	no	—
63	C₂×M₁₆	—	179	2	no	—
64	C₄⋊Q₈	—	179	2	no	—
65	#125	—	179	2	no	—
66	C₂²⋊Q₈	—	358	2	no	—
67	C₄×Q₈	—	358	2	no	—
68	#117	—	179	2	no	—
69	C₂×C₄²	C₂×D₁₇₉	1	2	yes	—
70	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
71	C₄⋊C₈	—	358	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	358	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
77	M₁₆	—	358	4	no	—
78	C₂×C₈	—	358	2	no	—
79	C₄⋊C₄	—	358	4	no	—
80	C₄⋊C₄	—	358	2	no	—
81	C₄⋊C₄	—	179	4	no	—
82	C₂²⋊C₄	—	358	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	358	2	no	—
85	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
86	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
87	C₄²	D₇₁₆	1	2	yes	—
88	C₄²	Dic₃₅₈	1	4	yes	complemented
89	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
90	Q₈	—	716	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	358	4	no	—
95	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
98	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₃₅₈	1	2	yes	—
100	C₂×C₄	#15668259	1	2	yes	—
101	C₈	—	716	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
104	C₂²	#19153092	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	716	4	no	—
109	C₄	C₄×Dic₃₅₈	1	2	yes	—
110	C₄	#19153101	1	1	yes	characteristic
111	C₄	#19153113	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#26038345	1	1	yes	characteristic
114	C₂	#26038457	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038996

C₂

yes

maximal, characteristic

#26038463

C₂

yes

maximal, characteristic

#26038199

C₂

yes

maximal

#376×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

#19153113

C₄

yes

complemented

#19153105

C₄

yes

complemented

#19153092

—

C₄×Dic₃₅₈

—

#19153072

C₂²

yes

—

C₄⋊(C₁₇₉⋊C₈)

—

C₂×C₄²×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#95×C₁₇₉

—

C₂×Dic₃₅₈

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

C₂×C₄

yes

—

Dic₁₇₉⋊C₄

—

C₂²⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄²×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

Q₈

yes

complemented

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×Q₈

yes

—

C₄×C₁₇₉

#124

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#391

yes

characteristic

C₂×C₁₇₉

#421

yes

—

C₁₇₉

#1995

yes

characteristic, complemented, Sylow

#1995

—

179

maximal, Sylow

#533

—

179

—

#423

—

179

—

#376

D₁₇₉

yes

characteristic

#339

—

179

—

C₂×M₁₆

—

179

—

C₄⋊Q₈

—

179

—

#125

—

179

—

C₂²⋊Q₈

—

358

—

C₄×Q₈

—

358

—

#117

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

358

—

#95

—

C₂×Q₈

—

358

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

358

—

C₂×C₈

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

179

—

C₂²⋊C₄

—

358

—

C₄²

—

C₄²

—

358

—

C₄²

C₄×D₁₇₉

yes

characteristic

C₄²

C₄×D₁₇₉

yes

characteristic

C₄²

D₇₁₆

yes

—

C₄²

Dic₃₅₈

yes

complemented

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

Q₈

—

716

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×D₇₁₆

yes

characteristic

C₂×C₄

C₂×Dic₃₅₈

yes

—

100

C₂×C₄

#15668259

yes

—

101

C₈

—

716

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

104

C₂²

#19153092

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

716

—

109

C₄

C₄×Dic₃₅₈

yes

—

110

C₄

#19153101

yes

characteristic

111

C₄

#19153113

yes

characteristic

112

C₂

—

113

C₂

#26038345

yes

characteristic

114

C₂

#26038457

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₆
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#376×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#376	D₁₇₉
hypocenter	C₁₇₉	#1995
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₆

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#376×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
6. #19153113	C₄	semidirect
7. #19153105	C₄	semidirect
30. C₄²×C₁₇₉	Q₈	semidirect
57. C₁₇₉	#1995	semidirect
88. C₄²	Dic₃₅₈	semidirect

H ≅ G/N

type

6. #19153113

C₄

semidirect

7. #19153105

C₄

semidirect

30. C₄²×C₁₇₉

Q₈

semidirect

57. C₁₇₉

#1995

semidirect

88. C₄²

Dic₃₅₈

semidirect

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

subgroup	index
#26038996	2
#26038463	2
#26038199	2
#376×C₁₇₉	2
#1995	179

subgroup

index

#376×C₁₇₉

179

Subgroups — #37509981

Conjugacy classes of subgroups

Complements

Maximal subgroups