Group #22550696 — subgroups

Subgroups — C₂²×D₄×D₁₂₇ #22550696

Order 8128.

1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.

normal subgroups only

This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 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	C₂⁴×D₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂²⋊D₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×D₁₂₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	—	2	16	no	—
8	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
9	C₂³×D₁₂₇	C₂²	1	2	yes	complemented
10	C₂³×D₁₂₇	C₂²	1	24	yes	complemented
11	C₂×D₅₀₈	C₂²	1	12	yes	complemented
12	C₂×C₂²⋊D₁₂₇	C₂²	1	24	yes	complemented
13	D₄×D₁₂₇	C₂²	1	64	yes	direct product factor
14	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄×D₁₂₇	C₂²	1	12	yes	complemented
16	C₂⁴×C₁₂₇	C₂²	1	2	yes	complemented
17	C₂×D₄×C₁₂₇	C₂²	1	12	yes	complemented
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₁₂₇	—	2	8	no	—
20	C₂²×D₁₂₇	—	2	96	no	—
21	C₂²×D₁₂₇	D₄	1	8	yes	direct product factor
22	C₂²×D₁₂₇	C₂³	1	6	yes	—
23	C₂²×D₁₂₇	C₂³	1	32	yes	complemented
24	C₂²×D₁₂₇	C₂³	1	12	yes	complemented
25	D₅₀₈	C₂³	1	16	yes	complemented
26	C₂²⋊D₁₂₇	C₂³	1	32	yes	complemented
27	C₂×Dic₁₂₇	C₂³	1	6	yes	complemented
28	C₄×D₁₂₇	C₂³	1	16	yes	complemented
29	C₂³×C₁₂₇	—	2	8	no	—
30	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂³×C₁₂₇	C₂³	1	12	yes	complemented
32	D₄×C₁₂₇	C₂³	1	16	yes	complemented
33	C₂×C₄×C₁₂₇	C₂³	1	6	yes	complemented
34	C₂²×C₁₂₇	—	2	24	no	—
35	C₂²×C₁₂₇	C₂×D₄	1	4	yes	complemented
36	C₂²×C₁₂₇	C₂⁴	1	3	yes	—
37	C₂²×C₁₂₇	C₂⁴	1	8	yes	complemented
38	C₂×D₁₂₇	—	2	24	no	—
39	C₂×D₁₂₇	—	2	64	no	—
40	C₂×D₁₂₇	C₂×D₄	1	24	yes	direct product factor
41	C₂×D₁₂₇	C₂⁴	1	4	yes	—
42	C₂×D₁₂₇	C₂⁴	1	8	yes	complemented
43	C₄×C₁₂₇	C₂⁴	1	4	yes	complemented
44	Dic₁₂₇	C₂⁴	1	4	yes	complemented
45	C₂×C₁₂₇	—	2	8	no	—
46	C₂×C₁₂₇	C₂²×D₄	1	6	yes	complemented
47	C₂×C₁₂₇	C₂⁵	1	1	yes	characteristic
48	D₁₂₇	—	2	8	no	—
49	D₁₂₇	C₂²×D₄	1	8	yes	direct product factor
50	C₁₂₇	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
51	C₂³×D₄	—	127	1	no	maximal, Sylow
52	C₂⁵	—	127	2	no	—
53	C₂²×D₄	—	127	2	no	—
54	C₂²×D₄	—	127	24	no	—
55	C₂²×D₄	—	127	1	no	—
56	C₂²×D₄	D₁₂₇	1	1	yes	characteristic, direct product factor
57	C₂³×C₄	—	127	1	no	—
58	C₂⁴	—	127	1	no	—
59	C₂⁴	—	127	2	no	—
60	C₂⁴	—	127	24	no	—
61	C₂⁴	—	254	16	no	—
62	C₂⁴	C₂×D₁₂₇	1	2	yes	complemented
63	C₂×D₄	—	127	24	no	—
64	C₂×D₄	—	127	64	no	—
65	C₂×D₄	—	127	12	no	—
66	C₂×D₄	C₂×D₁₂₇	1	12	yes	direct product factor
67	C₂²×C₄	—	127	1	no	—
68	C₂²×C₄	—	127	12	no	—
69	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
70	C₂³	—	2	8	no	—
71	C₂³	—	127	8	no	—
72	C₂³	—	127	6	no	—
73	C₂³	—	127	12	no	—
74	C₂³	—	127	32	no	—
75	C₂³	—	254	96	no	—
76	C₂³	—	254	8	no	—
77	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
78	C₂³	C₂²×D₁₂₇	1	12	yes	complemented
79	D₄	—	127	32	no	—
80	D₄	—	127	16	no	—
81	D₄	C₂²×D₁₂₇	1	16	yes	direct product factor
82	C₂×C₄	—	127	6	no	—
83	C₂×C₄	—	127	16	no	—
84	C₂×C₄	C₂²×D₁₂₇	1	6	yes	complemented
85	C₂²	—	2	24	no	—
86	C₂²	—	127	24	no	—
87	C₂²	—	127	4	no	—
88	C₂²	—	127	8	no	—
89	C₂²	—	254	64	no	—
90	C₂²	—	254	24	no	—
91	C₂²	D₄×D₁₂₇	1	4	yes	direct product factor
92	C₂²	C₂³×D₁₂₇	1	3	yes	—
93	C₂²	C₂³×D₁₂₇	1	8	yes	complemented
94	C₄	—	127	4	no	—
95	C₄	C₂³×D₁₂₇	1	4	yes	complemented
96	C₂	—	2	8	no	—
97	C₂	—	127	8	no	—
98	C₂	—	254	8	no	—
99	C₂	C₂×D₄×D₁₂₇	1	6	yes	direct product factor
100	C₂	C₂⁴×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×D₁₂₇

C₂

yes

maximal, complemented

C₂²×D₅₀₈

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₂₇

C₂

yes

maximal, complemented

C₂×D₄×D₁₂₇

C₂

yes

maximal, direct product factor

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂³×D₁₂₇

C₂²

yes

complemented

C₂³×D₁₂₇

C₂²

yes

complemented

C₂×D₅₀₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₂₇

C₂²

yes

complemented

D₄×D₁₂₇

C₂²

yes

direct product factor

C₂²×Dic₁₂₇

C₂²

yes

characteristic, complemented

C₂×C₄×D₁₂₇

C₂²

yes

complemented

C₂⁴×C₁₂₇

C₂²

yes

complemented

C₂×D₄×C₁₂₇

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

direct product factor

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

complemented

C₂²×D₁₂₇

C₂³

yes

complemented

D₅₀₈

C₂³

yes

complemented

C₂²⋊D₁₂₇

C₂³

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

complemented

C₄×D₁₂₇

C₂³

yes

complemented

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂³×C₁₂₇

C₂³

yes

complemented

D₄×C₁₂₇

C₂³

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

complemented

C₂²×C₁₂₇

C₂⁴

yes

—

C₂²×C₁₂₇

C₂⁴

yes

complemented

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×D₄

yes

direct product factor

C₂×D₁₂₇

C₂⁴

yes

—

C₂×D₁₂₇

C₂⁴

yes

complemented

C₄×C₁₂₇

C₂⁴

yes

complemented

Dic₁₂₇

C₂⁴

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

complemented

C₂×C₁₂₇

C₂⁵

yes

characteristic

D₁₂₇

—

D₁₂₇

C₂²×D₄

yes

direct product factor

C₁₂₇

C₂³×D₄

yes

characteristic, complemented, Sylow

C₂³×D₄

—

127

maximal, Sylow

C₂⁵

—

127

—

C₂²×D₄

—

127

—

C₂²×D₄

—

127

—

C₂²×D₄

—

127

—

C₂²×D₄

D₁₂₇

yes

characteristic, direct product factor

C₂³×C₄

—

127

—

C₂⁴

—

127

—

C₂⁴

—

127

—

C₂⁴

—

127

—

C₂⁴

—

254

—

C₂⁴

C₂×D₁₂₇

yes

complemented

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

direct product factor

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂³

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂³

C₂²×D₁₂₇

yes

complemented

D₄

—

127

—

D₄

—

127

—

D₄

C₂²×D₁₂₇

yes

direct product factor

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

C₂²×D₁₂₇

yes

complemented

C₂²

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

D₄×D₁₂₇

yes

direct product factor

C₂²

C₂³×D₁₂₇

yes

—

C₂²

C₂³×D₁₂₇

yes

complemented

C₄

—

127

—

C₄

C₂³×D₁₂₇

yes

complemented

C₂

—

C₂

—

127

—

C₂

—

254

—

C₂

C₂×D₄×D₁₂₇

yes

direct product factor

100

C₂

C₂⁴×D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁵
Frattini	C₂	C₂⁴×D₁₂₇
Fitting	C₂²×D₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂²×D₄	D₁₂₇
hypocenter	C₁₂₇	C₂³×D₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂⁵

Frattini

C₂

C₂⁴×D₁₂₇

Fitting

C₂²×D₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂²×D₄

D₁₂₇

hypocenter

C₁₂₇

C₂³×D₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂⁴×D₁₂₇	C₂	semidirect
2. C₂²×D₅₀₈	C₂	semidirect
3. C₂²×C₂²⋊D₁₂₇	C₂	semidirect
4. C₂×D₄×D₁₂₇	C₂	direct (N × H)
5. C₂²×C₄×D₁₂₇	C₂	semidirect
6. C₂²×D₄×C₁₂₇	C₂	semidirect
9. C₂³×D₁₂₇	C₂²	semidirect
10. C₂³×D₁₂₇	C₂²	semidirect
11. C₂×D₅₀₈	C₂²	semidirect
12. C₂×C₂²⋊D₁₂₇	C₂²	semidirect
13. D₄×D₁₂₇	C₂²	direct (N × H)
14. C₂²×Dic₁₂₇	C₂²	semidirect
15. C₂×C₄×D₁₂₇	C₂²	semidirect
16. C₂⁴×C₁₂₇	C₂²	semidirect
17. C₂×D₄×C₁₂₇	C₂²	semidirect
18. C₂²×C₄×C₁₂₇	C₂²	semidirect
21. C₂²×D₁₂₇	D₄	direct (N × H)
23. C₂²×D₁₂₇	C₂³	semidirect
24. C₂²×D₁₂₇	C₂³	semidirect
25. D₅₀₈	C₂³	semidirect
26. C₂²⋊D₁₂₇	C₂³	semidirect
27. C₂×Dic₁₂₇	C₂³	semidirect
28. C₄×D₁₂₇	C₂³	semidirect
31. C₂³×C₁₂₇	C₂³	semidirect
32. D₄×C₁₂₇	C₂³	semidirect
33. C₂×C₄×C₁₂₇	C₂³	semidirect
35. C₂²×C₁₂₇	C₂×D₄	semidirect
37. C₂²×C₁₂₇	C₂⁴	semidirect
40. C₂×D₁₂₇	C₂×D₄	direct (N × H)
42. C₂×D₁₂₇	C₂⁴	semidirect
43. C₄×C₁₂₇	C₂⁴	semidirect
44. Dic₁₂₇	C₂⁴	semidirect
46. C₂×C₁₂₇	C₂²×D₄	semidirect
49. D₁₂₇	C₂²×D₄	direct (N × H)
50. C₁₂₇	C₂³×D₄	semidirect
56. C₂²×D₄	D₁₂₇	direct (N × H)
62. C₂⁴	C₂×D₁₂₇	semidirect
66. C₂×D₄	C₂×D₁₂₇	direct (N × H)
69. C₂²×C₄	C₂×D₁₂₇	semidirect
78. C₂³	C₂²×D₁₂₇	semidirect
81. D₄	C₂²×D₁₂₇	direct (N × H)
84. C₂×C₄	C₂²×D₁₂₇	semidirect
91. C₂²	D₄×D₁₂₇	direct (N × H)
93. C₂²	C₂³×D₁₂₇	semidirect
95. C₄	C₂³×D₁₂₇	semidirect
99. C₂	C₂×D₄×D₁₂₇	direct (N × H)

H ≅ G/N

type

1. C₂⁴×D₁₂₇

C₂

semidirect

2. C₂²×D₅₀₈

C₂

semidirect

3. C₂²×C₂²⋊D₁₂₇

C₂

semidirect

4. C₂×D₄×D₁₂₇

C₂

direct (N × H)

5. C₂²×C₄×D₁₂₇

C₂

semidirect

6. C₂²×D₄×C₁₂₇

C₂

semidirect

9. C₂³×D₁₂₇

C₂²

semidirect

10. C₂³×D₁₂₇

C₂²

semidirect

11. C₂×D₅₀₈

C₂²

semidirect

12. C₂×C₂²⋊D₁₂₇

C₂²

semidirect

13. D₄×D₁₂₇

C₂²

direct (N × H)

14. C₂²×Dic₁₂₇

C₂²

semidirect

15. C₂×C₄×D₁₂₇

C₂²

semidirect

16. C₂⁴×C₁₂₇

C₂²

semidirect

17. C₂×D₄×C₁₂₇

C₂²

semidirect

18. C₂²×C₄×C₁₂₇

C₂²

semidirect

21. C₂²×D₁₂₇

D₄

direct (N × H)

23. C₂²×D₁₂₇

C₂³

semidirect

24. C₂²×D₁₂₇

C₂³

semidirect

25. D₅₀₈

C₂³

semidirect

26. C₂²⋊D₁₂₇

C₂³

semidirect

27. C₂×Dic₁₂₇

C₂³

semidirect

28. C₄×D₁₂₇

C₂³

semidirect

31. C₂³×C₁₂₇

C₂³

semidirect

32. D₄×C₁₂₇

C₂³

semidirect

33. C₂×C₄×C₁₂₇

C₂³

semidirect

35. C₂²×C₁₂₇

C₂×D₄

semidirect

37. C₂²×C₁₂₇

C₂⁴

semidirect

40. C₂×D₁₂₇

C₂×D₄

direct (N × H)

42. C₂×D₁₂₇

C₂⁴

semidirect

43. C₄×C₁₂₇

C₂⁴

semidirect

44. Dic₁₂₇

C₂⁴

semidirect

46. C₂×C₁₂₇

C₂²×D₄

semidirect

49. D₁₂₇

C₂²×D₄

direct (N × H)

50. C₁₂₇

C₂³×D₄

semidirect

56. C₂²×D₄

D₁₂₇

direct (N × H)

62. C₂⁴

C₂×D₁₂₇

semidirect

66. C₂×D₄

C₂×D₁₂₇

direct (N × H)

69. C₂²×C₄

C₂×D₁₂₇

semidirect

78. C₂³

C₂²×D₁₂₇

semidirect

81. D₄

C₂²×D₁₂₇

direct (N × H)

84. C₂×C₄

C₂²×D₁₂₇

semidirect

91. C₂²

D₄×D₁₂₇

direct (N × H)

93. C₂²

C₂³×D₁₂₇

semidirect

95. C₄

C₂³×D₁₂₇

semidirect

99. C₂

C₂×D₄×D₁₂₇

direct (N × H)

Subgroups — C₂²×D₄×D₁₂₇ #22550696

Conjugacy classes of subgroups

Complements

Subgroups — C22×D4×D127 #22550696

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×D₄×D₁₂₇ #22550696