Group #22549726 — subgroups

Subgroups — C₂²⋊C₄×Dic₁₂₇ #22549726

Order 8128.

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

normal subgroups only

This group has 258 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	C₂³×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₁₂₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₇	—	2	4	no	—
7	C₂²×Dic₁₂₇	C₄	1	4	yes	complemented
8	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
10	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₁₂₇	C₄	1	4	yes	complemented
12	C₄×Dic₁₂₇	—	2	4	no	—
13	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	complemented
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	—	2	2	no	—
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₂×Dic₁₂₇	—	2	8	no	—
20	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	complemented
23	C₂×Dic₁₂₇	D₄	1	4	yes	—
24	C₂³×C₁₂₇	—	2	2	no	—
25	C₂³×C₁₂₇	—	2	2	no	—
26	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	2	2	no	—
31	C₂×C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
33	C₂²×C₁₂₇	—	2	2	no	—
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	—	2	4	no	—
36	C₂²×C₁₂₇	—	2	4	no	—
37	C₂²×C₁₂₇	C₄²	1	4	yes	complemented
38	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
42	C₂²×C₁₂₇	#41	1	2	yes	—
43	C₄×C₁₂₇	—	2	4	no	—
44	Dic₁₂₇	—	2	2	no	—
45	Dic₁₂₇	—	2	4	no	—
46	Dic₁₂₇	C₂²⋊C₄	1	4	yes	direct product factor
47	C₂×C₁₂₇	—	2	4	no	—
48	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₁₂₇	#117	1	1	yes	characteristic
51	C₂×C₁₂₇	C₄×D₄	1	2	yes	—
52	C₂×C₁₂₇	C₄×D₄	1	2	yes	—
53	C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
54	C₄×C₂²⋊C₄	—	127	1	no	maximal, Sylow
55	C₂³×C₄	—	127	1	no	—
56	C₂×C₂²⋊C₄	—	127	1	no	—
57	C₂×C₂²⋊C₄	D₁₂₇	1	1	yes	characteristic
58	C₂×C₄²	—	127	2	no	—
59	#95	—	127	2	no	—
60	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂²×C₄	—	127	1	no	—
62	C₂²×C₄	—	127	4	no	—
63	C₂²×C₄	—	127	1	no	—
64	C₂²×C₄	—	127	2	no	—
65	C₂²×C₄	—	254	4	no	—
66	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
67	C₂²⋊C₄	—	127	4	no	—
68	C₂²⋊C₄	Dic₁₂₇	1	4	yes	direct product factor
69	C₄²	—	254	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	2	no	—
72	C₂³	C₄×D₁₂₇	1	2	yes	—
73	C₂³	C₄×D₁₂₇	1	2	yes	—
74	C₂³	C₂×Dic₁₂₇	1	2	yes	—
75	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	127	2	no	—
79	C₂×C₄	—	127	4	no	—
80	C₂×C₄	—	127	4	no	—
81	C₂×C₄	—	127	2	no	—
82	C₂×C₄	—	254	8	no	—
83	C₂×C₄	—	254	2	no	—
84	C₂×C₄	—	254	2	no	—
85	C₂×C₄	—	254	2	no	—
86	C₂×C₄	C₂×Dic₁₂₇	1	4	yes	complemented
87	C₂²	—	2	2	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	2	4	no	—
91	C₂²	C₄×Dic₁₂₇	1	4	yes	complemented
92	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
93	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
94	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
95	C₂²	D₄×D₁₂₇	1	2	yes	—
96	C₂²	#14523247	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	127	4	no	—
99	C₄	—	254	2	no	—
100	C₄	—	254	4	no	—
101	C₂	—	2	4	no	—
102	C₂	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
103	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
104	C₂	#16861165	1	1	yes	characteristic
105	C₂	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
106	C₂	D₄×Dic₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, complemented

C₁₂₇⋊#95

C₂

yes

maximal, complemented

C₂×C₂²⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

C₄

yes

complemented

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²⋊Dic₁₂₇

C₄

yes

complemented

C₄×Dic₁₂₇

—

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²⋊C₄×C₁₂₇

C₄

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

D₄

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄²

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

#41

yes

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

direct product factor

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄²

yes

characteristic

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

#117

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

—

C₂×C₁₂₇

C₄×D₄

yes

—

C₁₂₇

C₄×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₄×C₂²⋊C₄

—

127

maximal, Sylow

C₂³×C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

D₁₂₇

yes

characteristic

C₂×C₄²

—

127

—

#95

—

127

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

Dic₁₂₇

yes

direct product factor

C₄²

—

254

—

C₂³

—

C₂³

—

C₂³

C₄×D₁₂₇

yes

—

C₂³

C₄×D₁₂₇

yes

—

C₂³

C₂×Dic₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×Dic₁₂₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₄×Dic₁₂₇

yes

complemented

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₂²

C₂²×Dic₁₂₇

yes

characteristic

C₂²

D₄×D₁₂₇

yes

—

C₂²

#14523247

yes

—

C₄

—

C₄

—

127

—

C₄

—

254

—

100

C₄

—

254

—

101

C₂

—

102

C₂

C₂×C₄×Dic₁₂₇

yes

characteristic

103

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

104

C₂

#16861165

yes

characteristic

105

C₂

(C₂²⋊D₁₂₇)⋊C₄

yes

—

106

C₂

D₄×Dic₁₂₇

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄²
Fitting	C₂×C₂²⋊C₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₂₇
hypocenter	C₁₂₇	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂×C₄²

Fitting

C₂×C₂²⋊C₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₂₇

hypocenter

C₁₂₇

C₄×C₂²⋊C₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
3. C₂×C₄×Dic₁₂₇	C₂	semidirect
4. C₁₂₇⋊#95	C₂	semidirect
7. C₂²×Dic₁₂₇	C₄	semidirect
11. C₂²⋊Dic₁₂₇	C₄	semidirect
15. C₂²⋊C₄×C₁₂₇	C₄	semidirect
22. C₂×Dic₁₂₇	C₂×C₄	semidirect
32. C₂×C₄×C₁₂₇	C₂×C₄	semidirect
37. C₂²×C₁₂₇	C₄²	semidirect
46. Dic₁₂₇	C₂²⋊C₄	direct (N × H)
53. C₁₂₇	C₄×C₂²⋊C₄	semidirect
68. C₂²⋊C₄	Dic₁₂₇	direct (N × H)
86. C₂×C₄	C₂×Dic₁₂₇	semidirect
91. C₂²	C₄×Dic₁₂₇	semidirect

H ≅ G/N

type

3. C₂×C₄×Dic₁₂₇

C₂

semidirect

4. C₁₂₇⋊#95

C₂

semidirect

7. C₂²×Dic₁₂₇

C₄

semidirect

11. C₂²⋊Dic₁₂₇

C₄

semidirect

15. C₂²⋊C₄×C₁₂₇

C₄

semidirect

22. C₂×Dic₁₂₇

C₂×C₄

semidirect

32. C₂×C₄×C₁₂₇

C₂×C₄

semidirect

37. C₂²×C₁₂₇

C₄²

semidirect

46. Dic₁₂₇

C₂²⋊C₄

direct (N × H)

53. C₁₂₇

C₄×C₂²⋊C₄

semidirect

68. C₂²⋊C₄

Dic₁₂₇

direct (N × H)

86. C₂×C₄

C₂×Dic₁₂₇

semidirect

91. C₂²

C₄×Dic₁₂₇

semidirect

Subgroups — C₂²⋊C₄×Dic₁₂₇ #22549726

Conjugacy classes of subgroups

Complements

Subgroups — C22⋊C4×Dic127 #22549726

Conjugacy classes of subgroups

Complements

Subgroups — C₂²⋊C₄×Dic₁₂₇ #22549726