Group #22550640 — subgroups

Subgroups — C₂×#16861217 #22550640

Order 8128.

266 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14523248	C₂	1	1	yes	maximal, characteristic
2	#16861217	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523231	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#14523248	C₄	1	4	yes	—
8	#14523248	C₂²	1	4	yes	—
9	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
10	C₂×D₅₀₈	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	C₄	1	2	yes	—
12	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
13	#14523231	C₂²	1	8	yes	complemented
14	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	C₂²	1	4	yes	complemented
17	C₈×D₁₂₇	C₂²	1	8	yes	complemented
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂²×D₁₂₇	C₂×C₄	1	2	yes	—
22	Dic₂₅₄	C₂×C₄	1	4	yes	—
23	D₅₀₈	C₂×C₄	1	4	yes	—
24	C₂²⋊D₁₂₇	C₂×C₄	1	8	yes	—
25	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
26	C₄×D₁₂₇	C₂×C₄	1	4	yes	—
27	C₄×D₁₂₇	C₂³	1	4	yes	—
28	C₁₂₇⋊C₈	C₂³	1	2	yes	—
29	C₁₂₇⋊C₈	C₂³	1	2	yes	—
30	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₂₇	—	2	4	no	—
40	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
41	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
43	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
44	Dic₁₂₇	C₂²×C₄	1	4	yes	—
45	C₂×C₁₂₇	—	2	2	no	—
46	C₂×C₁₂₇	#131	1	2	yes	complemented
47	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
48	D₁₂₇	—	2	4	no	—
49	C₁₂₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	127	1	no	maximal, Sylow
51	C₂×#41	—	127	1	no	—
52	#131	—	127	8	no	—
53	C₂×M₁₆	—	127	2	no	—
54	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
55	C₂²×C₈	—	127	1	no	—
56	C₂²×C₈	—	127	2	no	—
57	#41	—	127	4	no	—
58	#41	—	127	4	no	—
59	C₂×Q₈	—	127	1	no	—
60	C₂×D₄	—	127	2	no	—
61	C₂×D₄	—	127	1	no	—
62	C₂²×C₄	—	127	2	no	—
63	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
64	M₁₆	—	127	8	no	—
65	M₁₆	C₂×D₁₂₇	1	4	yes	complemented
66	C₂×C₈	—	127	4	no	—
67	C₂×C₈	—	127	1	no	—
68	C₂×C₈	—	127	1	no	—
69	C₂×C₈	—	127	8	no	—
70	C₂×C₈	C₂×D₁₂₇	1	2	yes	—
71	C₂³	—	127	2	no	—
72	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
73	Q₈	—	127	4	no	—
74	D₄	—	127	8	no	—
75	D₄	—	127	4	no	—
76	C₂×C₄	—	127	2	no	—
77	C₂×C₄	—	127	4	no	—
78	C₂×C₄	—	127	4	no	—
79	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
81	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
82	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
83	C₈	—	127	2	no	—
84	C₈	—	127	2	no	—
85	C₈	C₂²×D₁₂₇	1	4	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	127	4	no	—
88	C₂²	—	254	4	no	—
89	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
90	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
91	C₄	—	127	4	no	—
92	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
93	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
94	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	254	4	no	—
97	C₂	#16861217	1	2	yes	direct product factor
98	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

#16861217

C₂

yes

maximal, direct product factor

C₂×#14523231

C₂

yes

maximal, complemented

C₂²×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

C₄

yes

—

#14523248

C₂²

yes

—

C₂×Dic₂₅₄

C₄

yes

characteristic

C₂×D₅₀₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

#14523231

C₂²

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

C₈×D₁₂₇

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂×C₄

yes

—

Dic₂₅₄

C₂×C₄

yes

—

D₅₀₈

C₂×C₄

yes

—

C₂²⋊D₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

C₂³

yes

—

C₁₂₇⋊C₈

C₂³

yes

—

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#131

yes

complemented

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

D₁₂₇

—

C₁₂₇

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

127

maximal, Sylow

C₂×#41

—

127

—

#131

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

D₁₂₇

yes

characteristic, complemented

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

#41

—

127

—

#41

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

127

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

—

C₂³

—

127

—

C₂³

C₄×D₁₂₇

yes

characteristic

Q₈

—

127

—

D₄

—

127

—

D₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₄×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

127

—

C₈

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

C₂×C₄×D₁₂₇

yes

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₄

—

127

—

C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₄

C₂×C₄×D₁₂₇

yes

—

C₄

C₂³×D₁₂₇

yes

characteristic

C₂

—

C₂

—

254

—

C₂

#16861217

yes

direct product factor

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Frattini	C₄	C₂³×D₁₂₇
Fitting	C₂×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₄
hypercenter	C₂×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#131
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Frattini

C₄

C₂³×D₁₂₇

Fitting

C₂×M₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×C₄

hypercenter

C₂×M₁₆

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #16861217	C₂	direct (N × H)
3. C₂×#14523231	C₂	semidirect
4. C₂²×C₁₂₇⋊C₈	C₂	semidirect
5. C₂×C₈×D₁₂₇	C₂	semidirect
6. C₂×M₁₆×C₁₂₇	C₂	semidirect
13. #14523231	C₂²	semidirect
16. C₂×C₁₂₇⋊C₈	C₂²	semidirect
17. C₈×D₁₂₇	C₂²	semidirect
19. M₁₆×C₁₂₇	C₂²	semidirect
46. C₂×C₁₂₇	#131	semidirect
49. C₁₂₇	C₂×#131	semidirect
54. C₂×M₁₆	D₁₂₇	semidirect
65. M₁₆	C₂×D₁₂₇	semidirect
97. C₂	#16861217	direct (N × H)

H ≅ G/N

type

2. #16861217

C₂

direct (N × H)

3. C₂×#14523231

C₂

semidirect

4. C₂²×C₁₂₇⋊C₈

C₂

semidirect

5. C₂×C₈×D₁₂₇

C₂

semidirect

6. C₂×M₁₆×C₁₂₇

C₂

semidirect

13. #14523231

C₂²

semidirect

16. C₂×C₁₂₇⋊C₈

C₂²

semidirect

17. C₈×D₁₂₇

C₂²

semidirect

19. M₁₆×C₁₂₇

C₂²

semidirect

46. C₂×C₁₂₇

#131

semidirect

49. C₁₂₇

C₂×#131

semidirect

54. C₂×M₁₆

D₁₂₇

semidirect

65. M₁₆

C₂×D₁₂₇

semidirect

97. C₂

#16861217

direct (N × H)

Subgroups — C₂×#16861217 #22550640

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16861217 #22550640

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16861217 #22550640