Group #44543674 — subgroups

Subgroups — #44543674

Order 31936.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×#22547811	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₉₉₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#30630351	C₂	1	2	yes	maximal, complemented
4	#30630339	C₂	1	2	yes	maximal, complemented
5	#30630335	C₂	1	4	yes	maximal, complemented
6	#30630331	C₂	1	4	yes	maximal, complemented
7	#117×C₄₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#22547811	—	2	4	no	—
9	C₂×Dic₉₉₈	—	2	4	no	—
10	C₂×Dic₉₉₈	C₂²	1	2	yes	—
11	C₂×Dic₉₉₈	C₂²	1	2	yes	—
12	C₂×Dic₉₉₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₉₉₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₉₉	C₂²	1	2	yes	—
15	C₂²×Dic₄₉₉	C₂²	1	2	yes	—
16	C₂×C₄×D₄₉₉	C₂²	1	2	yes	—
17	C₄⋊Dic₄₉₉	C₂²	1	8	yes	—
18	(C₂×D₄₉₉)⋊C₄	C₂²	1	8	yes	—
19	C₂²×C₄×C₄₉₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₄₉₉	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₄₉₉	C₂²	1	2	yes	—
22	C₄²×C₄₉₉	C₂²	1	2	yes	—
23	C₂²×D₄₉₉	C₂³	1	2	yes	—
24	Dic₉₉₈	—	2	4	no	—
25	Dic₉₉₈	—	2	4	no	—
26	Dic₉₉₈	—	2	2	no	—
27	D₁₉₉₆	—	2	2	no	—
28	C₂²⋊D₄₉₉	—	2	4	no	—
29	C₂×Dic₄₉₉	—	2	4	no	—
30	C₂×Dic₄₉₉	C₂³	1	4	yes	—
31	C₂×Dic₄₉₉	C₂³	1	2	yes	—
32	C₄×D₄₉₉	—	2	4	no	—
33	C₂³×C₄₉₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₄₉₉	D₄	1	4	yes	—
35	C₂×C₄×C₄₉₉	C₂³	1	4	yes	—
36	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄₉₉	—	2	2	no	—
39	C₂²×C₄₉₉	C₂×D₄	1	2	yes	—
40	C₂²×C₄₉₉	C₂⁴	1	1	yes	characteristic
41	C₂×D₄₉₉	—	2	2	no	—
42	C₂×D₄₉₉	—	2	4	no	—
43	C₄×C₄₉₉	—	2	4	no	—
44	C₄×C₄₉₉	C₂×D₄	1	2	yes	—
45	C₄×C₄₉₉	C₂×D₄	1	2	yes	—
46	Dic₄₉₉	—	2	2	no	—
47	Dic₄₉₉	—	2	4	no	—
48	C₂×C₄₉₉	—	2	2	no	—
49	C₂×C₄₉₉	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₄₉₉	2₋¹⁺⁴	1	2	yes	—
51	D₄₉₉	—	4	2	no	—
52	C₄₉₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	499	1	no	maximal, Sylow
54	C₂×#41	—	499	1	no	—
55	C₂²×Q₈	—	499	1	no	—
56	C₄⋊Q₈	—	499	2	no	—
57	#124	—	499	2	no	—
58	#123	—	499	4	no	—
59	C₂²⋊Q₈	—	499	4	no	—
60	#117	D₄₉₉	1	1	yes	characteristic, complemented
61	#41	—	998	4	no	—
62	C₂×Q₈	—	998	4	no	—
63	C₂×Q₈	—	499	2	no	—
64	C₂×Q₈	—	499	2	no	—
65	C₂×Q₈	—	499	1	no	—
66	C₂×D₄	—	499	2	no	—
67	C₂×D₄	—	499	1	no	—
68	C₂²×C₄	—	499	2	no	—
69	C₂²×C₄	—	499	2	no	—
70	C₂²×C₄	C₂×D₄₉₉	1	1	yes	characteristic
71	C₄⋊C₄	—	499	8	no	—
72	C₄⋊C₄	C₂×D₄₉₉	1	2	yes	—
73	C₂²⋊C₄	—	499	8	no	—
74	C₂²⋊C₄	C₂×D₄₉₉	1	2	yes	—
75	C₄²	C₂×D₄₉₉	1	2	yes	—
76	C₂³	—	499	2	no	—
77	C₂³	C₂²×D₄₉₉	1	1	yes	characteristic
78	Q₈	—	998	4	no	—
79	Q₈	—	998	4	no	—
80	Q₈	—	998	2	no	—
81	D₄	—	998	4	no	—
82	D₄	—	998	2	no	—
83	C₂×C₄	—	998	4	no	—
84	C₂×C₄	—	998	4	no	—
85	C₂×C₄	—	499	4	no	—
86	C₂×C₄	—	499	2	no	—
87	C₂×C₄	D₁₉₉₆	1	4	yes	—
88	C₂×C₄	C₂²×D₄₉₉	1	4	yes	—
89	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	998	2	no	—
93	C₂²	—	998	4	no	—
94	C₂²	C₂×D₁₉₉₆	1	2	yes	—
95	C₂²	C₂³×D₄₉₉	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	998	2	no	—
98	C₄	—	998	4	no	—
99	C₄	C₂×D₁₉₉₆	1	2	yes	—
100	C₄	C₂×D₁₉₉₆	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	1996	2	no	—
103	C₂	C₂²×D₁₉₉₆	1	1	yes	characteristic
104	C₂	#30630398	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22547811

C₂

yes

maximal, characteristic

C₂²×Dic₉₉₈

C₂

yes

maximal, characteristic, complemented

#30630351

C₂

yes

maximal, complemented

#30630339

C₂

yes

maximal, complemented

#30630335

C₂

yes

maximal, complemented

#30630331

C₂

yes

maximal, complemented

#117×C₄₉₉

C₂

yes

maximal, characteristic, complemented

#22547811

—

C₂×Dic₉₉₈

—

C₂×Dic₉₉₈

C₂²

yes

—

C₂×Dic₉₉₈

C₂²

yes

—

C₂×Dic₉₉₈

C₂²

yes

characteristic

C₂×D₁₉₉₆

C₂²

yes

characteristic

C₂×C₂²⋊D₄₉₉

C₂²

yes

—

C₂²×Dic₄₉₉

C₂²

yes

—

C₂×C₄×D₄₉₉

C₂²

yes

—

C₄⋊Dic₄₉₉

C₂²

yes

—

(C₂×D₄₉₉)⋊C₄

C₂²

yes

—

C₂²×C₄×C₄₉₉

C₂²

yes

characteristic

C₄⋊C₄×C₄₉₉

C₂²

yes

—

C₂²⋊C₄×C₄₉₉

C₂²

yes

—

C₄²×C₄₉₉

C₂²

yes

—

C₂²×D₄₉₉

C₂³

yes

—

Dic₉₉₈

—

Dic₉₉₈

—

Dic₉₉₈

—

D₁₉₉₆

—

C₂²⋊D₄₉₉

—

C₂×Dic₄₉₉

—

C₂×Dic₄₉₉

C₂³

yes

—

C₂×Dic₄₉₉

C₂³

yes

—

C₄×D₄₉₉

—

C₂³×C₄₉₉

C₂³

yes

characteristic

C₂×C₄×C₄₉₉

D₄

yes

—

C₂×C₄×C₄₉₉

C₂³

yes

—

C₂×C₄×C₄₉₉

C₂³

yes

characteristic

C₂×C₄×C₄₉₉

C₂³

yes

characteristic

C₂²×C₄₉₉

—

C₂²×C₄₉₉

C₂×D₄

yes

—

C₂²×C₄₉₉

C₂⁴

yes

characteristic

C₂×D₄₉₉

—

C₂×D₄₉₉

—

C₄×C₄₉₉

—

C₄×C₄₉₉

C₂×D₄

yes

—

C₄×C₄₉₉

C₂×D₄

yes

—

Dic₄₉₉

—

Dic₄₉₉

—

C₂×C₄₉₉

—

C₂×C₄₉₉

C₂²×D₄

yes

characteristic

C₂×C₄₉₉

2₋¹⁺⁴

yes

—

D₄₉₉

—

C₄₉₉

#536

yes

characteristic, complemented, Sylow

#536

—

499

maximal, Sylow

C₂×#41

—

499

—

C₂²×Q₈

—

499

—

C₄⋊Q₈

—

499

—

#124

—

499

—

#123

—

499

—

C₂²⋊Q₈

—

499

—

#117

D₄₉₉

yes

characteristic, complemented

#41

—

998

—

C₂×Q₈

—

998

—

C₂×Q₈

—

499

—

C₂×Q₈

—

499

—

C₂×Q₈

—

499

—

C₂×D₄

—

499

—

C₂×D₄

—

499

—

C₂²×C₄

—

499

—

C₂²×C₄

—

499

—

C₂²×C₄

C₂×D₄₉₉

yes

characteristic

C₄⋊C₄

—

499

—

C₄⋊C₄

C₂×D₄₉₉

yes

—

C₂²⋊C₄

—

499

—

C₂²⋊C₄

C₂×D₄₉₉

yes

—

C₄²

C₂×D₄₉₉

yes

—

C₂³

—

499

—

C₂³

C₂²×D₄₉₉

yes

characteristic

Q₈

—

998

—

Q₈

—

998

—

Q₈

—

998

—

D₄

—

998

—

D₄

—

998

—

C₂×C₄

—

998

—

C₂×C₄

—

998

—

C₂×C₄

—

499

—

C₂×C₄

—

499

—

C₂×C₄

D₁₉₉₆

yes

—

C₂×C₄

C₂²×D₄₉₉

yes

—

C₂×C₄

C₂²×D₄₉₉

yes

characteristic

C₂×C₄

C₂²×D₄₉₉

yes

characteristic

C₂²

—

C₂²

—

998

—

C₂²

—

998

—

C₂²

C₂×D₁₉₉₆

yes

—

C₂²

C₂³×D₄₉₉

yes

characteristic

C₄

—

C₄

—

998

—

C₄

—

998

—

C₄

C₂×D₁₉₉₆

yes

—

100

C₄

C₂×D₁₉₉₆

yes

—

101

C₂

—

102

C₂

—

1996

—

103

C₂

C₂²×D₁₉₉₆

yes

characteristic

104

C₂

#30630398

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₉₉
commutator, socle, cosocle	C₂²×C₄₉₉	C₂⁴
Fitting	#117×C₄₉₉	C₂
prime radical	C₂×#22547811	C₂
hypercenter	#117	D₄₉₉
hypocenter	C₄₉₉	#536
LCS	G ▷ C₂²×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₉₉

commutator, socle, cosocle

C₂²×C₄₉₉

C₂⁴

Fitting

#117×C₄₉₉

C₂

C₂

D₄₉₉

C₄₉₉

G ▷ C₂²×C₄₉₉ ▷ C₄₉₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₂²×Dic₉₉₈	C₂	semidirect
3. #30630351	C₂	semidirect
4. #30630339	C₂	semidirect
5. #30630335	C₂	semidirect
6. #30630331	C₂	semidirect
7. #117×C₄₉₉	C₂	semidirect
52. C₄₉₉	#536	semidirect
60. #117	D₄₉₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₉₉₈

C₂

semidirect

3. #30630351

C₂

semidirect

4. #30630339

C₂

semidirect

5. #30630335

C₂

semidirect

6. #30630331

C₂

semidirect

7. #117×C₄₉₉

C₂

semidirect

52. C₄₉₉

#536

semidirect

60. #117

D₄₉₉

semidirect

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

subgroup	index
C₂×#22547811	2
C₂²×Dic₉₉₈	2
#30630351	2
#30630339	2
#30630335	2
#30630331	2
#117×C₄₉₉	2
#536	499

subgroup

index

C₂²×Dic₉₉₈

#117×C₄₉₉

499

Subgroups — #44543674

Conjugacy classes of subgroups

Complements

Maximal subgroups