Group #27164786 — subgroups

Subgroups — C₂×D₈×D₁₉₉ #27164786

Order 12736.

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

normal subgroups only

This group has 338 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×D₄×D₁₉₉	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₅₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15691871	C₂	1	2	yes	maximal, complemented
4	D₈×D₁₉₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₉₉	—	2	2	no	—
8	C₂×D₇₉₆	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₁₉₉	—	2	2	no	—
10	D₄×D₁₉₉	—	2	4	no	—
11	D₄×D₁₉₉	C₂²	1	8	yes	complemented
12	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
13	D₁₅₉₂	C₂²	1	4	yes	complemented
14	#15691871	C₂²	1	8	yes	complemented
15	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
16	C₈×D₁₉₉	C₂²	1	4	yes	complemented
17	C₂×D₄×C₁₉₉	C₂²	1	2	yes	complemented
18	D₈×C₁₉₉	C₂²	1	4	yes	complemented
19	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₁₉₉	—	2	2	no	—
21	C₂²×D₁₉₉	—	2	8	no	—
22	C₂²×D₁₉₉	—	4	8	no	—
23	C₂²×D₁₉₉	D₄	1	1	yes	characteristic
24	D₇₉₆	—	2	2	no	—
25	D₇₉₆	C₂³	1	4	yes	complemented
26	C₂²⋊D₁₉₉	—	2	8	no	—
27	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
28	C₄×D₁₉₉	D₄	1	2	yes	—
29	C₄×D₁₉₉	C₂³	1	2	yes	—
30	C₁₉₉⋊C₈	C₂³	1	2	yes	complemented
31	C₂³×C₁₉₉	—	2	2	no	—
32	D₄×C₁₉₉	—	2	2	no	—
33	D₄×C₁₉₉	C₂³	1	4	yes	complemented
34	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
35	C₈×C₁₉₉	C₂³	1	2	yes	complemented
36	C₂²×C₁₉₉	—	2	4	no	—
37	C₂²×C₁₉₉	—	4	4	no	—
38	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₉₉	—	2	4	no	—
40	C₂×D₁₉₉	—	4	4	no	—
41	C₂×D₁₉₉	—	4	16	no	—
42	C₂×D₁₉₉	D₈	1	4	yes	direct product factor
43	C₂×D₁₉₉	C₂×D₄	1	2	yes	—
44	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
45	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
46	Dic₁₉₉	C₂×D₄	1	2	yes	—
47	C₂×C₁₉₉	—	4	4	no	—
48	C₂×C₁₉₉	C₂×D₈	1	2	yes	complemented
49	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
50	D₁₉₉	—	4	4	no	—
51	D₁₉₉	C₂×D₈	1	4	yes	direct product factor
52	C₁₉₉	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
53	C₂²×D₈	—	199	1	no	maximal, Sylow
54	C₂²×D₄	—	199	2	no	—
55	C₂×D₈	—	199	2	no	—
56	C₂×D₈	—	199	8	no	—
57	C₂×D₈	—	199	1	no	—
58	C₂×D₈	D₁₉₉	1	1	yes	characteristic, direct product factor
59	C₂²×C₈	—	199	1	no	—
60	C₂⁴	—	398	2	no	—
61	C₂×D₄	—	398	4	no	—
62	C₂×D₄	—	398	2	no	—
63	C₂×D₄	—	199	8	no	—
64	C₂×D₄	—	199	2	no	—
65	C₂×D₄	C₂×D₁₉₉	1	2	yes	complemented
66	C₂²×C₄	—	199	1	no	—
67	D₈	—	199	8	no	—
68	D₈	—	199	4	no	—
69	D₈	C₂×D₁₉₉	1	4	yes	direct product factor
70	C₂×C₈	—	199	1	no	—
71	C₂×C₈	—	199	4	no	—
72	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic, complemented
73	C₂³	—	2	2	no	—
74	C₂³	—	796	8	no	—
75	C₂³	—	398	8	no	—
76	C₂³	—	398	2	no	—
77	C₂³	—	199	1	no	—
78	D₄	—	2	2	no	—
79	D₄	—	398	8	no	—
80	D₄	—	398	2	no	—
81	D₄	—	199	4	no	—
82	D₄	C₂²×D₁₉₉	1	4	yes	complemented
83	C₂×C₄	—	199	1	no	—
84	C₂×C₄	—	199	2	no	—
85	C₂×C₄	—	199	2	no	—
86	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
87	C₈	—	199	2	no	—
88	C₈	C₂²×D₁₉₉	1	2	yes	complemented
89	C₂²	—	2	4	no	—
90	C₂²	—	4	4	no	—
91	C₂²	—	796	16	no	—
92	C₂²	—	796	4	no	—
93	C₂²	—	398	4	no	—
94	C₂²	—	199	4	no	—
95	C₂²	—	199	2	no	—
96	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
97	C₄	—	199	2	no	—
98	C₄	D₄×D₁₉₉	1	1	yes	characteristic
99	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
100	C₂	—	4	4	no	—
101	C₂	—	796	4	no	—
102	C₂	—	199	4	no	—
103	C₂	D₈×D₁₉₉	1	2	yes	direct product factor
104	C₂	C₂×D₄×D₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₉₉

C₂

yes

maximal, complemented

C₂×D₁₅₉₂

C₂

yes

maximal, characteristic, complemented

C₂×#15691871

C₂

yes

maximal, complemented

D₈×D₁₉₉

C₂

yes

maximal, direct product factor

C₂×C₈×D₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₉₉

—

C₂×D₇₉₆

C₂²

yes

complemented

C₂×C₂²⋊D₁₉₉

—

D₄×D₁₉₉

—

D₄×D₁₉₉

C₂²

yes

complemented

C₂×C₄×D₁₉₉

C₂²

yes

characteristic

D₁₅₉₂

C₂²

yes

complemented

#15691871

C₂²

yes

complemented

C₂×C₁₉₉⋊C₈

C₂²

yes

characteristic, complemented

C₈×D₁₉₉

C₂²

yes

complemented

C₂×D₄×C₁₉₉

C₂²

yes

complemented

D₈×C₁₉₉

C₂²

yes

complemented

C₂×C₈×C₁₉₉

C₂²

yes

characteristic, complemented

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

D₄

yes

characteristic

D₇₉₆

—

D₇₉₆

C₂³

yes

complemented

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

D₄

yes

characteristic

C₄×D₁₉₉

D₄

yes

—

C₄×D₁₉₉

C₂³

yes

—

C₁₉₉⋊C₈

C₂³

yes

complemented

C₂³×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

C₂³

yes

complemented

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₈×C₁₉₉

C₂³

yes

complemented

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×D₄

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

D₈

yes

direct product factor

C₂×D₁₉₉

C₂×D₄

yes

—

C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₄×C₁₉₉

C₂⁴

yes

characteristic

Dic₁₉₉

C₂×D₄

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂×D₈

yes

complemented

C₂×C₁₉₉

C₂²×D₄

yes

characteristic

D₁₉₉

—

D₁₉₉

C₂×D₈

yes

direct product factor

C₁₉₉

C₂²×D₈

yes

characteristic, complemented, Sylow

C₂²×D₈

—

199

maximal, Sylow

C₂²×D₄

—

199

—

C₂×D₈

—

199

—

C₂×D₈

—

199

—

C₂×D₈

—

199

—

C₂×D₈

D₁₉₉

yes

characteristic, direct product factor

C₂²×C₈

—

199

—

C₂⁴

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂×D₄

—

199

—

C₂×D₄

C₂×D₁₉₉

yes

complemented

C₂²×C₄

—

199

—

D₈

—

199

—

D₈

—

199

—

D₈

C₂×D₁₉₉

yes

direct product factor

C₂×C₈

—

199

—

C₂×C₈

—

199

—

C₂×C₈

C₂×D₁₉₉

yes

characteristic, complemented

C₂³

—

C₂³

—

796

—

C₂³

—

398

—

C₂³

—

398

—

C₂³

—

199

—

D₄

—

D₄

—

398

—

D₄

—

398

—

D₄

—

199

—

D₄

C₂²×D₁₉₉

yes

complemented

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

C₈

—

199

—

C₈

C₂²×D₁₉₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

796

—

C₂²

—

796

—

C₂²

—

398

—

C₂²

—

199

—

C₂²

—

199

—

C₂²

D₄×D₁₉₉

yes

characteristic

C₄

—

199

—

C₄

D₄×D₁₉₉

yes

characteristic

C₄

C₂³×D₁₉₉

yes

characteristic

100

C₂

—

101

C₂

—

796

—

102

C₂

—

199

—

103

C₂

D₈×D₁₉₉

yes

direct product factor

104

C₂

C₂×D₄×D₁₉₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₄×C₁₉₉	C₂⁴
Frattini	C₄	C₂³×D₁₉₉
Fitting	C₂×D₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₂×D₈	D₁₉₉
hypocenter	C₁₉₉	C₂²×D₈
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×D₁₉₉

commutator, cosocle

C₄×C₁₉₉

C₂⁴

Frattini

C₄

C₂³×D₁₉₉

Fitting

C₂×D₈×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×D₄

hypercenter

C₂×D₈

D₁₉₉

hypocenter

C₁₉₉

C₂²×D₈

LCS

G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. C₂×D₄×D₁₉₉	C₂	semidirect
2. C₂×D₁₅₉₂	C₂	semidirect
3. C₂×#15691871	C₂	semidirect
4. D₈×D₁₉₉	C₂	direct (N × H)
5. C₂×C₈×D₁₉₉	C₂	semidirect
6. C₂×D₈×C₁₉₉	C₂	semidirect
8. C₂×D₇₉₆	C₂²	semidirect
11. D₄×D₁₉₉	C₂²	semidirect
13. D₁₅₉₂	C₂²	semidirect
14. #15691871	C₂²	semidirect
15. C₂×C₁₉₉⋊C₈	C₂²	semidirect
16. C₈×D₁₉₉	C₂²	semidirect
17. C₂×D₄×C₁₉₉	C₂²	semidirect
18. D₈×C₁₉₉	C₂²	semidirect
19. C₂×C₈×C₁₉₉	C₂²	semidirect
25. D₇₉₆	C₂³	semidirect
30. C₁₉₉⋊C₈	C₂³	semidirect
33. D₄×C₁₉₉	C₂³	semidirect
35. C₈×C₁₉₉	C₂³	semidirect
42. C₂×D₁₉₉	D₈	direct (N × H)
48. C₂×C₁₉₉	C₂×D₈	semidirect
51. D₁₉₉	C₂×D₈	direct (N × H)
52. C₁₉₉	C₂²×D₈	semidirect
58. C₂×D₈	D₁₉₉	direct (N × H)
65. C₂×D₄	C₂×D₁₉₉	semidirect
69. D₈	C₂×D₁₉₉	direct (N × H)
72. C₂×C₈	C₂×D₁₉₉	semidirect
82. D₄	C₂²×D₁₉₉	semidirect
88. C₈	C₂²×D₁₉₉	semidirect
103. C₂	D₈×D₁₉₉	direct (N × H)

H ≅ G/N

type

1. C₂×D₄×D₁₉₉

C₂

semidirect

2. C₂×D₁₅₉₂

C₂

semidirect

3. C₂×#15691871

C₂

semidirect

4. D₈×D₁₉₉

C₂

direct (N × H)

5. C₂×C₈×D₁₉₉

C₂

semidirect

6. C₂×D₈×C₁₉₉

C₂

semidirect

8. C₂×D₇₉₆

C₂²

semidirect

11. D₄×D₁₉₉

C₂²

semidirect

13. D₁₅₉₂

C₂²

semidirect

14. #15691871

C₂²

semidirect

15. C₂×C₁₉₉⋊C₈

C₂²

semidirect

16. C₈×D₁₉₉

C₂²

semidirect

17. C₂×D₄×C₁₉₉

C₂²

semidirect

18. D₈×C₁₉₉

C₂²

semidirect

19. C₂×C₈×C₁₉₉

C₂²

semidirect

25. D₇₉₆

C₂³

semidirect

30. C₁₉₉⋊C₈

C₂³

semidirect

33. D₄×C₁₉₉

C₂³

semidirect

35. C₈×C₁₉₉

C₂³

semidirect

42. C₂×D₁₉₉

D₈

direct (N × H)

48. C₂×C₁₉₉

C₂×D₈

semidirect

51. D₁₉₉

C₂×D₈

direct (N × H)

52. C₁₉₉

C₂²×D₈

semidirect

58. C₂×D₈

D₁₉₉

direct (N × H)

65. C₂×D₄

C₂×D₁₉₉

semidirect

69. D₈

C₂×D₁₉₉

direct (N × H)

72. C₂×C₈

C₂×D₁₉₉

semidirect

82. D₄

C₂²×D₁₉₉

semidirect

88. C₈

C₂²×D₁₉₉

semidirect

103. C₂

D₈×D₁₉₉

direct (N × H)

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

subgroup	index
C₂×D₄×D₁₉₉	2
C₂×D₁₅₉₂	2
C₂×#15691871	2
D₈×D₁₉₉	2
C₂×C₈×D₁₉₉	2
C₂×D₈×C₁₉₉	2
C₂²×D₈	199

subgroup

index

C₂×D₄×D₁₉₉

C₂×D₁₅₉₂

C₂×#15691871

D₈×D₁₉₉

C₂×C₈×D₁₉₉

C₂×D₈×C₁₉₉

C₂²×D₈

199

Subgroups — C₂×D₈×D₁₉₉ #27164786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D8×D199 #27164786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₈×D₁₉₉ #27164786