Group #22550337 — subgroups

Subgroups — C₂×D₄×Dic₁₂₇ #22550337

Order 8128.

426 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₂³×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₂₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₇	—	2	8	no	—
8	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
9	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₂₇	C₂²	1	8	yes	complemented
12	C₄⋊Dic₁₂₇	C₂²	1	4	yes	complemented
13	C₂²⋊Dic₁₂₇	C₂²	1	8	yes	complemented
14	C₄×Dic₁₂₇	C₂²	1	4	yes	complemented
15	C₂⁴×C₁₂₇	C₂²	1	2	yes	—
16	C₂×D₄×C₁₂₇	C₄	1	8	yes	complemented
17	C₂×D₄×C₁₂₇	C₂²	1	4	yes	—
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₂₇	—	2	4	no	—
20	C₂×Dic₁₂₇	—	2	2	no	—
21	C₂×Dic₁₂₇	—	2	16	no	—
22	C₂×Dic₁₂₇	D₄	1	4	yes	direct product factor
23	C₂×Dic₁₂₇	C₂³	1	2	yes	—
24	C₂×Dic₁₂₇	C₂³	1	4	yes	—
25	C₂×Dic₁₂₇	C₂³	1	2	yes	—
26	C₂³×C₁₂₇	—	2	4	no	—
27	C₂³×C₁₂₇	—	2	4	no	—
28	C₂³×C₁₂₇	C₂×C₄	1	8	yes	complemented
29	C₂³×C₁₂₇	C₂³	1	4	yes	—
30	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
31	D₄×C₁₂₇	C₂×C₄	1	16	yes	complemented
32	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×C₁₂₇	—	2	16	no	—
35	C₂²×C₁₂₇	—	2	4	no	—
36	C₂²×C₁₂₇	—	2	4	no	—
37	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₂₇	C₂²×C₄	1	8	yes	complemented
39	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
40	C₂²×C₁₂₇	#41	1	2	yes	—
41	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₁₂₇	C₂²×C₄	1	4	yes	complemented
43	Dic₁₂₇	—	2	4	no	—
44	Dic₁₂₇	—	2	2	no	—
45	Dic₁₂₇	C₂×D₄	1	4	yes	direct product factor
46	C₂×C₁₂₇	—	2	8	no	—
47	C₂×C₁₂₇	C₄×D₄	1	4	yes	complemented
48	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
51	C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₄×D₄	—	127	1	no	maximal, Sylow
53	C₂²×D₄	D₁₂₇	1	1	yes	characteristic
54	C₂³×C₄	—	127	2	no	—
55	C₄×D₄	—	127	8	no	—
56	C₂×C₄⋊C₄	—	127	1	no	—
57	C₂×C₂²⋊C₄	—	127	2	no	—
58	C₂×C₄²	—	127	1	no	—
59	C₂⁴	C₂×D₁₂₇	1	2	yes	—
60	C₂×D₄	Dic₁₂₇	1	8	yes	direct product factor
61	C₂×D₄	C₂×D₁₂₇	1	4	yes	—
62	C₂²×C₄	—	127	1	no	—
63	C₂²×C₄	—	127	8	no	—
64	C₂²×C₄	—	127	1	no	—
65	C₂²×C₄	—	127	2	no	—
66	C₂²×C₄	—	254	8	no	—
67	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	127	4	no	—
69	C₂²⋊C₄	—	127	8	no	—
70	C₄²	—	127	4	no	—
71	C₂³	—	2	4	no	—
72	C₂³	—	2	4	no	—
73	C₂³	C₂×Dic₁₂₇	1	8	yes	complemented
74	C₂³	C₂²×D₁₂₇	1	4	yes	—
75	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
76	D₄	C₂×Dic₁₂₇	1	16	yes	direct product factor
77	C₂×C₄	—	127	4	no	—
78	C₂×C₄	—	127	2	no	—
79	C₂×C₄	—	127	2	no	—
80	C₂×C₄	—	127	4	no	—
81	C₂×C₄	—	254	16	no	—
82	C₂×C₄	—	254	4	no	—
83	C₂×C₄	—	254	2	no	—
84	C₂×C₄	C₂×Dic₁₂₇	1	4	yes	complemented
85	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
86	C₂²	—	2	4	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	2	16	no	—
89	C₂²	C₂²×Dic₁₂₇	1	2	yes	—
90	C₂²	C₂²×Dic₁₂₇	1	8	yes	complemented
91	C₂²	D₄×D₁₂₇	1	2	yes	—
92	C₂²	#14523247	1	2	yes	—
93	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
94	C₄	—	127	4	no	—
95	C₄	—	254	4	no	—
96	C₄	—	254	2	no	—
97	C₄	C₂²×Dic₁₂₇	1	4	yes	complemented
98	C₂	—	2	8	no	—
99	C₂	D₄×Dic₁₂₇	1	4	yes	direct product factor
100	C₂	C₂³×Dic₁₂₇	1	1	yes	characteristic
101	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
102	C₂	C₂×#14523247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₂₇

C₂

yes

maximal, complemented

D₄×Dic₁₂₇

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, complemented

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

complemented

C₄⋊Dic₁₂₇

C₂²

yes

complemented

C₂²⋊Dic₁₂₇

C₂²

yes

complemented

C₄×Dic₁₂₇

C₂²

yes

complemented

C₂⁴×C₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₄

yes

complemented

C₂×D₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

direct product factor

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

complemented

C₂³×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

D₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

C₂²×C₄

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×D₄

yes

direct product factor

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

complemented

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₁₂₇

C₂×C₄×D₄

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

127

maximal, Sylow

C₂²×D₄

D₁₂₇

yes

characteristic

C₂³×C₄

—

127

—

C₄×D₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₄²

—

127

—

C₂⁴

C₂×D₁₂₇

yes

—

C₂×D₄

Dic₁₂₇

yes

direct product factor

C₂×D₄

C₂×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₂₇

yes

complemented

C₂³

C₂²×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

C₂×Dic₁₂₇

yes

direct product factor

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×Dic₁₂₇

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×Dic₁₂₇

yes

—

C₂²

C₂²×Dic₁₂₇

yes

complemented

C₂²

D₄×D₁₂₇

yes

—

C₂²

#14523247

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

127

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂²×Dic₁₂₇

yes

complemented

C₂

—

C₂

D₄×Dic₁₂₇

yes

direct product factor

100

C₂

C₂³×Dic₁₂₇

yes

characteristic

101

C₂

C₂×D₄×D₁₂₇

yes

characteristic

102

C₂

C₂×#14523247

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Frattini	C₂²	C₂³×D₁₂₇
Fitting	C₂²×D₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂²×D₄	D₁₂₇
hypocenter	C₁₂₇	C₂×C₄×D₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂²×D₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂²×D₄

D₁₂₇

hypocenter

C₁₂₇

C₂×C₄×D₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂³×Dic₁₂₇	C₂	semidirect
2. D₄×Dic₁₂₇	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
4. C₂×C₂²⋊Dic₁₂₇	C₂	semidirect
5. C₂×C₄×Dic₁₂₇	C₂	semidirect
11. C₂²×Dic₁₂₇	C₂²	semidirect
12. C₄⋊Dic₁₂₇	C₂²	semidirect
13. C₂²⋊Dic₁₂₇	C₂²	semidirect
14. C₄×Dic₁₂₇	C₂²	semidirect
16. C₂×D₄×C₁₂₇	C₄	semidirect
22. C₂×Dic₁₂₇	D₄	direct (N × H)
28. C₂³×C₁₂₇	C₂×C₄	semidirect
31. D₄×C₁₂₇	C₂×C₄	semidirect
32. C₂×C₄×C₁₂₇	C₂×C₄	semidirect
38. C₂²×C₁₂₇	C₂²×C₄	semidirect
42. C₄×C₁₂₇	C₂²×C₄	semidirect
45. Dic₁₂₇	C₂×D₄	direct (N × H)
47. C₂×C₁₂₇	C₄×D₄	semidirect
51. C₁₂₇	C₂×C₄×D₄	semidirect
60. C₂×D₄	Dic₁₂₇	direct (N × H)
73. C₂³	C₂×Dic₁₂₇	semidirect
76. D₄	C₂×Dic₁₂₇	direct (N × H)
84. C₂×C₄	C₂×Dic₁₂₇	semidirect
90. C₂²	C₂²×Dic₁₂₇	semidirect
97. C₄	C₂²×Dic₁₂₇	semidirect
99. C₂	D₄×Dic₁₂₇	direct (N × H)

H ≅ G/N

type

1. C₂³×Dic₁₂₇

C₂

semidirect

2. D₄×Dic₁₂₇

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

4. C₂×C₂²⋊Dic₁₂₇

C₂

semidirect

5. C₂×C₄×Dic₁₂₇

C₂

semidirect

11. C₂²×Dic₁₂₇

C₂²

semidirect

12. C₄⋊Dic₁₂₇

C₂²

semidirect

13. C₂²⋊Dic₁₂₇

C₂²

semidirect

14. C₄×Dic₁₂₇

C₂²

semidirect

16. C₂×D₄×C₁₂₇

C₄

semidirect

22. C₂×Dic₁₂₇

D₄

direct (N × H)

28. C₂³×C₁₂₇

C₂×C₄

semidirect

31. D₄×C₁₂₇

C₂×C₄

semidirect

32. C₂×C₄×C₁₂₇

C₂×C₄

semidirect

38. C₂²×C₁₂₇

C₂²×C₄

semidirect

42. C₄×C₁₂₇

C₂²×C₄

semidirect

45. Dic₁₂₇

C₂×D₄

direct (N × H)

47. C₂×C₁₂₇

C₄×D₄

semidirect

51. C₁₂₇

C₂×C₄×D₄

semidirect

60. C₂×D₄

Dic₁₂₇

direct (N × H)

73. C₂³

C₂×Dic₁₂₇

semidirect

76. D₄

C₂×Dic₁₂₇

direct (N × H)

84. C₂×C₄

C₂×Dic₁₂₇

semidirect

90. C₂²

C₂²×Dic₁₂₇

semidirect

97. C₄

C₂²×Dic₁₂₇

semidirect

99. C₂

D₄×Dic₁₂₇

direct (N × H)

Subgroups — C₂×D₄×Dic₁₂₇ #22550337

Conjugacy classes of subgroups

Complements

Subgroups — C2×D4×Dic127 #22550337

Conjugacy classes of subgroups

Complements

Subgroups — C₂×D₄×Dic₁₂₇ #22550337