Group #36340884 — subgroups

Subgroups — #447×C₃₃₇ #36340884

Order 21568.

198 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₂²×D₄×C₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×C₃₃₇	C₂	1	2	yes	maximal, complemented
3	#121×C₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#102×C₃₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₃₃₇	—	2	1	no	—
7	C₂×D₄×C₃₃₇	—	2	4	no	—
8	C₂×D₄×C₃₃₇	—	2	1	no	—
9	C₂×D₄×C₃₃₇	—	2	1	no	—
10	C₂×D₄×C₃₃₇	C₂²	1	2	yes	—
11	C₂×D₄×C₃₃₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₄×C₃₃₇	C₂²	1	1	yes	characteristic
13	D₈×C₃₃₇	—	2	4	no	—
14	C₂×C₈×C₃₃₇	C₂²	1	2	yes	complemented
15	C₄⋊C₄×C₃₃₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊C₄×C₃₃₇	—	2	1	no	—
17	C₂³×C₃₃₇	—	2	4	no	—
18	C₂³×C₃₃₇	—	2	2	no	—
19	C₂³×C₃₃₇	—	2	1	no	—
20	C₂³×C₃₃₇	—	4	2	no	—
21	C₂³×C₃₃₇	—	4	2	no	—
22	C₂³×C₃₃₇	D₄	1	1	yes	characteristic
23	D₄×C₃₃₇	—	2	4	no	—
24	D₄×C₃₃₇	—	2	2	no	—
25	D₄×C₃₃₇	—	2	2	no	—
26	D₄×C₃₃₇	—	4	2	no	—
27	D₄×C₃₃₇	D₄	1	4	yes	complemented
28	C₂×C₄×C₃₃₇	—	2	2	no	—
29	C₂×C₄×C₃₃₇	—	2	1	no	—
30	C₂×C₄×C₃₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₃₇	C₂³	1	1	yes	characteristic
32	C₈×C₃₃₇	—	2	2	no	—
33	C₂²×C₃₃₇	—	2	1	no	—
34	C₂²×C₃₃₇	—	2	1	no	—
35	C₂²×C₃₃₇	—	2	4	no	—
36	C₂²×C₃₃₇	—	4	8	no	—
37	C₂²×C₃₃₇	—	4	2	no	—
38	C₂²×C₃₃₇	—	4	2	no	—
39	C₂²×C₃₃₇	—	4	1	no	—
40	C₂²×C₃₃₇	—	4	1	no	—
41	C₂²×C₃₃₇	—	4	1	no	—
42	C₂²×C₃₃₇	D₈	1	2	yes	complemented
43	C₂²×C₃₃₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₃₃₇	—	2	1	no	—
45	C₄×C₃₃₇	—	4	1	no	—
46	C₄×C₃₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₃₃₇	—	2	2	no	—
48	C₂×C₃₃₇	—	4	4	no	—
49	C₂×C₃₃₇	—	8	1	no	—
50	C₂×C₃₃₇	#120	1	1	yes	characteristic
51	C₂×C₃₃₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₃₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₃₃₇	#447	1	1	yes	characteristic, direct product factor, Sylow
54	#447	C₃₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×D₄	C₂×C₃₃₇	1	1	yes	characteristic, complemented
56	C₂×D₈	C₂×C₃₃₇	1	2	yes	complemented
57	#121	C₂×C₃₃₇	1	1	yes	characteristic, complemented
58	#102	C₂×C₃₃₇	1	2	yes	complemented
59	C₂²⋊C₈	C₂×C₃₃₇	1	1	yes	characteristic, complemented
60	C₂⁴	—	2	1	no	—
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	C₂²×C₃₃₇	1	2	yes	—
65	C₂×D₄	C₂²×C₃₃₇	1	1	yes	characteristic, complemented
66	C₂²×C₄	C₂²×C₃₃₇	1	1	yes	characteristic
67	D₈	—	2	4	no	—
68	C₂×C₈	C₂²×C₃₃₇	1	2	yes	complemented
69	C₄⋊C₄	C₂²×C₃₃₇	1	1	yes	characteristic, complemented
70	C₂²⋊C₄	—	2	1	no	—
71	C₂³	—	2	4	no	—
72	C₂³	—	2	2	no	—
73	C₂³	—	2	1	no	—
74	C₂³	—	4	2	no	—
75	C₂³	—	4	2	no	—
76	C₂³	D₄×C₃₃₇	1	1	yes	characteristic
77	D₄	—	2	4	no	—
78	D₄	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	D₄	D₄×C₃₃₇	1	4	yes	complemented
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	D₄×C₃₃₇	1	1	yes	characteristic
85	C₂×C₄	C₂³×C₃₃₇	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	4	8	no	—
91	C₂²	—	4	2	no	—
92	C₂²	—	4	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	D₈×C₃₃₇	1	2	yes	complemented
97	C₂²	C₂×D₄×C₃₃₇	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	C₂×D₄×C₃₃₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	4	4	no	—
103	C₂	—	8	1	no	—
104	C₂	#120×C₃₃₇	1	1	yes	characteristic
105	C₂	C₂×D₈×C₃₃₇	1	1	yes	characteristic
106	C₂	AGL(1,ℤ/8)×C₃₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄×C₃₃₇

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₃₃₇

C₂

yes

maximal, complemented

#121×C₃₃₇

C₂

yes

maximal, characteristic, complemented

#102×C₃₃₇

C₂

yes

maximal, complemented

C₂²⋊C₈×C₃₃₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₃₃₇

—

C₂×D₄×C₃₃₇

—

C₂×D₄×C₃₃₇

—

C₂×D₄×C₃₃₇

—

C₂×D₄×C₃₃₇

C₂²

yes

—

C₂×D₄×C₃₃₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₃₃₇

C₂²

yes

characteristic

D₈×C₃₃₇

—

C₂×C₈×C₃₃₇

C₂²

yes

complemented

C₄⋊C₄×C₃₃₇

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₃₃₇

—

C₂³×C₃₃₇

—

C₂³×C₃₃₇

—

C₂³×C₃₃₇

—

C₂³×C₃₃₇

—

C₂³×C₃₃₇

—

C₂³×C₃₃₇

D₄

yes

characteristic

D₄×C₃₃₇

—

D₄×C₃₃₇

—

D₄×C₃₃₇

—

D₄×C₃₃₇

—

D₄×C₃₃₇

D₄

yes

complemented

C₂×C₄×C₃₃₇

—

C₂×C₄×C₃₃₇

—

C₂×C₄×C₃₃₇

D₄

yes

characteristic

C₂×C₄×C₃₃₇

C₂³

yes

characteristic

C₈×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

—

C₂²×C₃₃₇

D₈

yes

complemented

C₂²×C₃₃₇

C₂×D₄

yes

characteristic

C₄×C₃₃₇

—

C₄×C₃₃₇

—

C₄×C₃₃₇

C₂×D₄

yes

—

C₂×C₃₃₇

—

C₂×C₃₃₇

—

C₂×C₃₃₇

—

C₂×C₃₃₇

#120

yes

characteristic

C₂×C₃₃₇

C₂×D₈

yes

C₂×C₃₃₇

yes

C₃₃₇

yes

characteristic, direct product factor, Sylow

#447

C₃₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×D₄

C₂×C₃₃₇

yes

characteristic, complemented

C₂×D₈

C₂×C₃₃₇

yes

complemented

#121

C₂×C₃₃₇

yes

characteristic, complemented

#102

C₂×C₃₃₇

yes

complemented

C₂²⋊C₈

C₂×C₃₃₇

yes

characteristic, complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₃₃₇

yes

—

C₂×D₄

C₂²×C₃₃₇

yes

characteristic, complemented

C₂²×C₄

C₂²×C₃₃₇

yes

characteristic

D₈

—

C₂×C₈

C₂²×C₃₃₇

yes

complemented

C₄⋊C₄

C₂²×C₃₃₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

D₄×C₃₃₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

D₄×C₃₃₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×C₃₃₇

yes

characteristic

C₂×C₄

C₂³×C₃₃₇

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₈×C₃₃₇

yes

complemented

C₂²

C₂×D₄×C₃₃₇

yes

characteristic

C₄

—

C₄

—

100

C₄

C₂×D₄×C₃₃₇

yes

—

101

C₂

—

102

C₂

—

103

C₂

—

104

C₂

#120×C₃₃₇

yes

characteristic

105

C₂

C₂×D₈×C₃₃₇

yes

characteristic

106

C₂

AGL(1,ℤ/8)×C₃₃₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₃₇ ◁ C₂²×C₄×C₃₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₃₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂³×C₃₃₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₃₇ ◁ C₂²×C₄×C₃₃₇ ◁ G

N	H ≅ G/N	type
1. C₂²×D₄×C₃₃₇	C₂	semidirect
2. C₂×D₈×C₃₃₇	C₂	semidirect
3. #121×C₃₃₇	C₂	semidirect
4. #102×C₃₃₇	C₂	semidirect
5. C₂²⋊C₈×C₃₃₇	C₂	semidirect
11. C₂×D₄×C₃₃₇	C₂²	semidirect
14. C₂×C₈×C₃₃₇	C₂²	semidirect
15. C₄⋊C₄×C₃₃₇	C₂²	semidirect
27. D₄×C₃₃₇	D₄	semidirect
42. C₂²×C₃₃₇	D₈	semidirect
53. C₃₃₇	#447	direct (N × H)
54. #447	C₃₃₇	direct (N × H)
55. C₂²×D₄	C₂×C₃₃₇	semidirect
56. C₂×D₈	C₂×C₃₃₇	semidirect
57. #121	C₂×C₃₃₇	semidirect
58. #102	C₂×C₃₃₇	semidirect
59. C₂²⋊C₈	C₂×C₃₃₇	semidirect
65. C₂×D₄	C₂²×C₃₃₇	semidirect
68. C₂×C₈	C₂²×C₃₃₇	semidirect
69. C₄⋊C₄	C₂²×C₃₃₇	semidirect
81. D₄	D₄×C₃₃₇	semidirect
96. C₂²	D₈×C₃₃₇	semidirect

H ≅ G/N

type

1. C₂²×D₄×C₃₃₇

C₂

semidirect

2. C₂×D₈×C₃₃₇

C₂

semidirect

3. #121×C₃₃₇

C₂

semidirect

4. #102×C₃₃₇

C₂

semidirect

5. C₂²⋊C₈×C₃₃₇

C₂

semidirect

11. C₂×D₄×C₃₃₇

C₂²

semidirect

14. C₂×C₈×C₃₃₇

C₂²

semidirect

15. C₄⋊C₄×C₃₃₇

C₂²

semidirect

27. D₄×C₃₃₇

D₄

semidirect

42. C₂²×C₃₃₇

D₈

semidirect

53. C₃₃₇

#447

direct (N × H)

54. #447

C₃₃₇

direct (N × H)

55. C₂²×D₄

C₂×C₃₃₇

semidirect

56. C₂×D₈

C₂×C₃₃₇

semidirect

57. #121

C₂×C₃₃₇

semidirect

58. #102

C₂×C₃₃₇

semidirect

59. C₂²⋊C₈

C₂×C₃₃₇

semidirect

65. C₂×D₄

C₂²×C₃₃₇

semidirect

68. C₂×C₈

C₂²×C₃₃₇

semidirect

69. C₄⋊C₄

C₂²×C₃₃₇

semidirect

81. D₄

D₄×C₃₃₇

semidirect

96. C₂²

D₈×C₃₃₇

semidirect

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

subgroup	index
C₂²×D₄×C₃₃₇	2
C₂×D₈×C₃₃₇	2
#121×C₃₃₇	2
#102×C₃₃₇	2
C₂²⋊C₈×C₃₃₇	2
#447	337

subgroup

index

C₂²×D₄×C₃₃₇

C₂×D₈×C₃₃₇

#121×C₃₃₇

#102×C₃₃₇

C₂²⋊C₈×C₃₃₇

#447

337

Subgroups — #447×C₃₃₇ #36340884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #447×C337 #36340884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #447×C₃₃₇ #36340884