Group #18003191 — subgroups

Subgroups — #18003191

Order 4736.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#14558016	C₂	1	2	yes	maximal
2	#14557575	C₂	1	6	yes	maximal, complemented
3	#14557571	C₂	1	3	yes	maximal, complemented
4	C₂×#11924468	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₃₇	—	2	2	no	—
7	D₄×Dic₃₇	—	2	6	no	—
8	#11924495	C₂²	1	6	yes	—
9	C₂×C₄⋊Dic₃₇	C₂²	1	6	yes	—
10	#11924470	C₂²	1	6	yes	—
11	#11924468	C₂²	1	3	yes	—
12	#11924468	C₂²	1	6	yes	—
13	#11924468	C₂²	1	1	yes	characteristic
14	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
15	#131×C₃₇	C₄	1	8	yes	complemented
16	C₂×M₁₆×C₃₇	C₂²	1	3	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	3	yes	—
18	#41×C₃₇	C₂×C₄	1	4	yes	—
19	#41×C₃₇	Q₈	1	4	yes	—
20	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
21	C₂×D₄×C₃₇	D₄	1	3	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
23	C₂²×Dic₃₇	—	2	6	no	—
24	M₁₆×C₃₇	C₂×C₄	1	12	yes	complemented
25	C₂×C₈×C₃₇	C₂×C₄	1	12	yes	complemented
26	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₃₇	C₂³	1	3	yes	—
28	C₂²⋊Dic₃₇	—	2	6	no	—
29	C₄⋊Dic₃₇	—	2	6	no	—
30	C₄⋊Dic₃₇	C₂³	1	6	yes	—
31	C₄⋊Dic₃₇	C₂³	1	2	yes	—
32	C₄×Dic₃₇	—	2	6	no	—
33	C₂³×C₃₇	C₂×D₄	1	3	yes	—
34	Q₈×C₃₇	C₄⋊C₄	1	4	yes	—
35	D₄×C₃₇	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₃₇	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₃₇	—	2	2	no	—
41	C₂×Dic₃₇	—	2	6	no	—
42	C₂×Dic₃₇	—	8	6	no	—
43	C₈×C₃₇	C₂²×C₄	1	2	yes	—
44	C₈×C₃₇	C₂²×C₄	1	6	yes	—
45	C₂²×C₃₇	—	2	6	no	—
46	C₂²×C₃₇	C₂×C₄⋊C₄	1	6	yes	—
47	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₃₇	C₂×C₄⋊C₄	1	6	yes	—
49	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
50	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
51	Dic₃₇	—	4	2	no	—
52	Dic₃₇	—	4	6	no	—
53	#2912	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	—	2	6	no	—
55	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃₇	#577	1	2	yes	—
57	C₂×#131	D₃₇	1	1	yes	characteristic
58	#520	—	37	2	no	—
59	#428	—	37	6	no	—
60	#427	—	37	3	no	—
61	C₂×#106	—	37	3	no	—
62	C₃₇	#2912	1	1	yes	characteristic, complemented, Sylow
63	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
64	#131	Dic₃₇	1	8	yes	complemented
65	C₂×M₁₆	C₂×D₃₇	1	3	yes	—
66	C₂²×C₈	C₂×D₃₇	1	3	yes	—
67	C₄×Q₈	—	74	2	no	—
68	C₄×D₄	—	74	6	no	—
69	#117	—	37	6	no	—
70	C₂×C₄⋊C₄	—	37	6	no	—
71	#107	—	37	6	no	—
72	#106	—	37	3	no	—
73	#106	—	37	6	no	—
74	#106	—	37	1	no	—
75	#41	Dic₇₄	1	4	yes	—
76	#41	C₂×Dic₃₇	1	4	yes	—
77	C₂×Q₈	D₁₄₈	1	1	yes	characteristic
78	C₂×D₄	D₁₄₈	1	3	yes	—
79	C₂²×C₄	—	74	6	no	—
80	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
81	M₁₆	C₂×Dic₃₇	1	12	yes	complemented
82	C₂×C₈	C₂×Dic₃₇	1	12	yes	complemented
83	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₃₇	1	3	yes	—
85	C₄⋊C₄	—	37	6	no	—
86	C₄⋊C₄	—	37	2	no	—
87	C₄⋊C₄	—	74	6	no	—
88	C₂²⋊C₄	—	74	6	no	—
89	C₄²	—	74	6	no	—
90	C₂³	C₂×D₁₄₈	1	3	yes	—
91	Q₈	C₄⋊Dic₃₇	1	4	yes	—
92	D₄	C₄⋊Dic₃₇	1	12	yes	—
93	C₂×C₄	—	296	6	no	—
94	C₂×C₄	—	74	2	no	—
95	C₂×C₄	—	74	6	no	—
96	C₂×C₄	C₂×Dic₇₄	1	6	yes	—
97	C₂×C₄	C₂×D₁₄₈	1	3	yes	—
98	C₂×C₄	C₂²×Dic₃₇	1	6	yes	—
99	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
100	C₈	C₂²×Dic₃₇	1	2	yes	—
101	C₈	C₂²×Dic₃₇	1	6	yes	—
102	C₂²	—	2	6	no	—
103	C₂²	C₂×C₄⋊Dic₃₇	1	6	yes	—
104	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
105	C₄	—	148	6	no	—
106	C₄	—	148	2	no	—
107	C₄	C₂×C₄⋊Dic₃₇	1	6	yes	—
108	C₄	C₂³×Dic₃₇	1	1	yes	characteristic
109	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
110	C₂	—	2	6	no	—
111	C₂	C₂²×C₄⋊Dic₃₇	1	1	yes	characteristic
112	C₂	#14558345	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558016

C₂

yes

maximal

#14557575

C₂

yes

maximal, complemented

#14557571

C₂

yes

maximal, complemented

C₂×#11924468

C₂

yes

maximal, complemented

C₂×#131×C₃₇

C₂

yes

maximal, characteristic

Q₈×Dic₃₇

—

D₄×Dic₃₇

—

#11924495

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

#11924468

C₂²

yes

—

#11924468

C₂²

yes

characteristic

C₂×#41×C₃₇

C₂²

yes

characteristic

#131×C₃₇

C₄

yes

complemented

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

—

#41×C₃₇

C₂×C₄

yes

—

#41×C₃₇

Q₈

yes

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×Dic₃₇

—

M₁₆×C₃₇

C₂×C₄

yes

complemented

C₂×C₈×C₃₇

C₂×C₄

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

—

Q₈×C₃₇

C₄⋊C₄

yes

—

D₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

C₂²×C₄

yes

—

C₈×C₃₇

C₂²×C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#2912

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₇

#577

yes

—

C₂×#131

D₃₇

yes

—

—

—

—

C₃₇

#2912

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

#131

Dic₃₇

yes

complemented

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

Dic₇₄

yes

—

#41

C₂×Dic₃₇

yes

—

C₂×Q₈

D₁₄₈

yes

characteristic

C₂×D₄

D₁₄₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

—

M₁₆

C₂×Dic₃₇

yes

complemented

C₂×C₈

C₂×Dic₃₇

yes

complemented

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

C₂×D₁₄₈

yes

—

Q₈

C₄⋊Dic₃₇

yes

—

D₄

C₄⋊Dic₃₇

yes

—

C₂×C₄

—

296

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₇₄

yes

—

C₂×C₄

C₂×D₁₄₈

yes

—

C₂×C₄

C₂²×Dic₃₇

yes

—

C₂×C₄

C₂³×D₃₇

yes

characteristic

100

C₈

C₂²×Dic₃₇

yes

—

101

C₈

C₂²×Dic₃₇

yes

—

102

C₂²

—

103

C₂²

C₂×C₄⋊Dic₃₇

yes

—

104

C₂²

C₂²×D₁₄₈

yes

characteristic

105

C₄

—

148

—

106

C₄

—

148

—

107

C₄

C₂×C₄⋊Dic₃₇

yes

—

108

C₄

C₂³×Dic₃₇

yes

characteristic

109

C₄

C₂²×Dic₇₄

yes

characteristic

110

C₂

—

111

C₂

C₂²×C₄⋊Dic₃₇

yes

characteristic

112

C₂

#14558345

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#131×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#131	D₃₇
hypocenter	C₃₇	#2912
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₁₄₈

commutator

C₄×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#131×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂×#41×C₃₇

C₂²

cosocle

C₂×C₄×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #14557575	C₂	semidirect
3. #14557571	C₂	semidirect
4. C₂×#11924468	C₂	semidirect
15. #131×C₃₇	C₄	semidirect
24. M₁₆×C₃₇	C₂×C₄	semidirect
25. C₂×C₈×C₃₇	C₂×C₄	semidirect
62. C₃₇	#2912	semidirect
64. #131	Dic₃₇	semidirect
81. M₁₆	C₂×Dic₃₇	semidirect
82. C₂×C₈	C₂×Dic₃₇	semidirect

H ≅ G/N

type

2. #14557575

C₂

semidirect

3. #14557571

C₂

semidirect

4. C₂×#11924468

C₂

semidirect

15. #131×C₃₇

C₄

semidirect

24. M₁₆×C₃₇

C₂×C₄

semidirect

25. C₂×C₈×C₃₇

C₂×C₄

semidirect

62. C₃₇

#2912

semidirect

64. #131

Dic₃₇

semidirect

81. M₁₆

C₂×Dic₃₇

semidirect

82. C₂×C₈

C₂×Dic₃₇

semidirect

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

subgroup	index
#14558016	2
#14557575	2
#14557571	2
C₂×#11924468	2
C₂×#131×C₃₇	2
#2912	37

subgroup

index

C₂×#131×C₃₇

Subgroups — #18003191

Conjugacy classes of subgroups

Complements

Maximal subgroups