Group #22550325 — subgroups

Subgroups — C₂×C₄⋊C₄×D₁₂₇ #22550325

Order 8128.

418 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 207 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×C₄×D₁₂₇	C₂	1	2	yes	maximal
2	C₂²×C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
9	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₂₇	—	2	8	no	—
11	C₂×C₄×D₁₂₇	C₄	1	8	yes	complemented
12	C₂×C₄×D₁₂₇	C₂²	1	8	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₂₇	C₂²	1	4	yes	complemented
15	Dic₁₂₇⋊C₄	C₂²	1	8	yes	complemented
16	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂²×D₁₂₇	C₂×C₄	1	4	yes	—
20	C₂²×D₁₂₇	D₄	1	4	yes	—
21	C₂²×D₁₂₇	Q₈	1	4	yes	—
22	C₂²×D₁₂₇	C₂³	1	2	yes	—
23	C₂×Dic₁₂₇	—	2	4	no	—
24	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	complemented
25	C₂×Dic₁₂₇	C₂³	1	4	yes	—
26	C₂×Dic₁₂₇	C₂³	1	2	yes	—
27	C₄×D₁₂₇	—	2	16	no	—
28	C₄×D₁₂₇	C₂×C₄	1	16	yes	complemented
29	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	2	4	no	—
31	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
32	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
36	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
37	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₂₇	C₄⋊C₄	1	16	yes	direct product factor
39	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
40	C₂×D₁₂₇	C₂×D₄	1	4	yes	—
41	C₂×D₁₂₇	C₂×Q₈	1	4	yes	—
42	C₄×C₁₂₇	—	2	4	no	—
43	C₄×C₁₂₇	C₂²×C₄	1	4	yes	complemented
44	Dic₁₂₇	—	2	4	no	—
45	Dic₁₂₇	C₂²×C₄	1	4	yes	complemented
46	C₂×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	complemented
47	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
50	D₁₂₇	C₂×C₄⋊C₄	1	8	yes	direct product factor
51	C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₄⋊C₄	—	127	1	no	maximal, Sylow
53	C₂³×C₄	—	127	1	no	—
54	C₂³×C₄	—	127	2	no	—
55	C₂×C₄⋊C₄	—	127	2	no	—
56	C₂×C₄⋊C₄	—	127	1	no	—
57	C₂×C₄⋊C₄	—	127	8	no	—
58	C₂×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic, direct product factor
59	C₂⁴	—	127	1	no	—
60	C₂²×C₄	—	127	1	no	—
61	C₂²×C₄	—	127	8	no	—
62	C₂²×C₄	—	127	4	no	—
63	C₂²×C₄	—	127	2	no	—
64	C₂²×C₄	—	127	8	no	—
65	C₂²×C₄	—	254	8	no	—
66	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
67	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	127	8	no	—
69	C₄⋊C₄	—	127	4	no	—
70	C₄⋊C₄	C₂×D₁₂₇	1	4	yes	direct product factor
71	C₂³	—	127	4	no	—
72	C₂³	—	127	4	no	—
73	C₂³	—	127	4	no	—
74	C₂³	—	127	2	no	—
75	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	127	4	no	—
78	C₂×C₄	—	127	2	no	—
79	C₂×C₄	—	127	16	no	—
80	C₂×C₄	—	127	4	no	—
81	C₂×C₄	—	254	16	no	—
82	C₂×C₄	—	254	4	no	—
83	C₂×C₄	C₄×D₁₂₇	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
85	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
86	C₂²	—	127	4	no	—
87	C₂²	—	127	4	no	—
88	C₂²	—	127	4	no	—
89	C₂²	—	127	16	no	—
90	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
91	C₂²	D₄×D₁₂₇	1	2	yes	—
92	C₂²	Q₈×D₁₂₇	1	2	yes	—
93	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	127	4	no	—
96	C₄	—	254	4	no	—
97	C₄	C₂×C₄×D₁₂₇	1	4	yes	complemented
98	C₂	—	127	8	no	—
99	C₂	C₄⋊C₄×D₁₂₇	1	4	yes	direct product factor
100	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
101	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
102	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₂₇

C₂

yes

maximal

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₄

yes

complemented

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

complemented

Dic₁₂₇⋊C₄

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

C₂×C₄

yes

—

C₂²×D₁₂₇

D₄

yes

—

C₂²×D₁₂₇

Q₈

yes

—

C₂²×D₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×C₄

yes

complemented

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

C₄⋊C₄

yes

direct product factor

C₂×D₁₂₇

C₂²×C₄

yes

—

C₂×D₁₂₇

C₂×D₄

yes

—

C₂×D₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

C₂²×C₄

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊C₄

yes

complemented

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

D₁₂₇

C₂×C₄⋊C₄

yes

direct product factor

C₁₂₇

C₂²×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

127

maximal, Sylow

C₂³×C₄

—

127

—

C₂³×C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

D₁₂₇

yes

characteristic, direct product factor

C₂⁴

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

direct product factor

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

C₂×C₄×D₁₂₇

yes

—

C₂²

D₄×D₁₂₇

yes

—

C₂²

Q₈×D₁₂₇

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

127

—

C₄

—

254

—

C₄

C₂×C₄×D₁₂₇

yes

complemented

C₂

—

127

—

C₂

C₄⋊C₄×D₁₂₇

yes

direct product factor

100

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

101

C₂

C₂×D₄×D₁₂₇

yes

characteristic

102

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

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

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂×C₄⋊C₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₂₇

hypocenter

C₁₂₇

C₂²×C₄⋊C₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
4. C₂×Dic₁₂₇⋊C₄	C₂	semidirect
5. C₄⋊C₄×D₁₂₇	C₂	direct (N × H)
6. C₂×C₄⋊C₄×C₁₂₇	C₂	semidirect
11. C₂×C₄×D₁₂₇	C₄	semidirect
14. C₄⋊Dic₁₂₇	C₂²	semidirect
15. Dic₁₂₇⋊C₄	C₂²	semidirect
18. C₄⋊C₄×C₁₂₇	C₂²	semidirect
24. C₂×Dic₁₂₇	C₂×C₄	semidirect
28. C₄×D₁₂₇	C₂×C₄	semidirect
31. C₂×C₄×C₁₂₇	C₂×C₄	semidirect
38. C₂×D₁₂₇	C₄⋊C₄	direct (N × H)
43. C₄×C₁₂₇	C₂²×C₄	semidirect
45. Dic₁₂₇	C₂²×C₄	semidirect
46. C₂×C₁₂₇	C₂×C₄⋊C₄	semidirect
50. D₁₂₇	C₂×C₄⋊C₄	direct (N × H)
51. C₁₂₇	C₂²×C₄⋊C₄	semidirect
58. C₂×C₄⋊C₄	D₁₂₇	direct (N × H)
70. C₄⋊C₄	C₂×D₁₂₇	direct (N × H)
83. C₂×C₄	C₄×D₁₂₇	semidirect
97. C₄	C₂×C₄×D₁₂₇	semidirect
99. C₂	C₄⋊C₄×D₁₂₇	direct (N × H)

H ≅ G/N

type

3. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

4. C₂×Dic₁₂₇⋊C₄

C₂

semidirect

5. C₄⋊C₄×D₁₂₇

C₂

direct (N × H)

6. C₂×C₄⋊C₄×C₁₂₇

C₂

semidirect

11. C₂×C₄×D₁₂₇

C₄

semidirect

14. C₄⋊Dic₁₂₇

C₂²

semidirect

15. Dic₁₂₇⋊C₄

C₂²

semidirect

18. C₄⋊C₄×C₁₂₇

C₂²

semidirect

24. C₂×Dic₁₂₇

C₂×C₄

semidirect

28. C₄×D₁₂₇

C₂×C₄

semidirect

31. C₂×C₄×C₁₂₇

C₂×C₄

semidirect

38. C₂×D₁₂₇

C₄⋊C₄

direct (N × H)

43. C₄×C₁₂₇

C₂²×C₄

semidirect

45. Dic₁₂₇

C₂²×C₄

semidirect

46. C₂×C₁₂₇

C₂×C₄⋊C₄

semidirect

50. D₁₂₇

C₂×C₄⋊C₄

direct (N × H)

51. C₁₂₇

C₂²×C₄⋊C₄

semidirect

58. C₂×C₄⋊C₄

D₁₂₇

direct (N × H)

70. C₄⋊C₄

C₂×D₁₂₇

direct (N × H)

83. C₂×C₄

C₄×D₁₂₇

semidirect

97. C₄

C₂×C₄×D₁₂₇

semidirect

99. C₂

C₄⋊C₄×D₁₂₇

direct (N × H)

Subgroups — C₂×C₄⋊C₄×D₁₂₇ #22550325

Conjugacy classes of subgroups

Complements

Subgroups — C2×C4⋊C4×D127 #22550325

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄⋊C₄×D₁₂₇ #22550325