Group #22550669 — subgroups

Subgroups — C₂×#16861240 #22550669

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	C₂×#14523248	C₂	1	1	yes	maximal, characteristic
2	#16861240	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523238	C₂	1	2	yes	maximal, complemented
4	C₂×#14523235	C₂	1	2	yes	maximal, complemented
5	C₂×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#14523248	—	2	2	no	—
8	#14523248	C₂²	1	4	yes	—
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
12	C₂×C₄×D₁₂₇	—	2	1	no	—
13	#14523238	C₂²	1	8	yes	complemented
14	#14523235	C₂²	1	8	yes	complemented
15	#14523232	C₂²	1	4	yes	complemented
16	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
17	C₂×Q₈×C₁₂₇	—	2	1	no	—
18	C₂×Q₈×C₁₂₇	—	2	2	no	—
19	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂²×C₄×C₁₂₇	—	2	1	no	—
22	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₂₇	—	2	1	no	—
24	Dic₂₅₄	—	2	1	no	—
25	Dic₂₅₄	C₂³	1	2	yes	—
26	D₅₀₈	—	2	1	no	—
27	D₅₀₈	C₂³	1	2	yes	—
28	C₂²⋊D₁₂₇	—	2	4	no	—
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₄×D₁₂₇	—	2	4	no	—
31	C₁₂₇⋊C₈	C₂³	1	4	yes	—
32	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
33	Q₈×C₁₂₇	—	2	2	no	—
34	Q₈×C₁₂₇	—	2	4	no	—
35	Q₈×C₁₂₇	C₂³	1	4	yes	—
36	C₂×C₄×C₁₂₇	—	2	2	no	—
37	C₂×C₄×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₁₂₇	—	2	2	no	—
43	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×D₁₂₇	—	2	2	no	—
46	C₂×D₁₂₇	—	4	2	no	—
47	C₄×C₁₂₇	—	2	4	no	—
48	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	Dic₁₂₇	—	2	2	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	#137	1	2	yes	complemented
54	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
55	D₁₂₇	—	4	2	no	—
56	C₁₂₇	C₂×#137	1	1	yes	characteristic, complemented, Sylow
57	C₂×#137	—	127	1	no	maximal, Sylow
58	C₂×#41	—	127	1	no	—
59	C₂²×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
60	#137	—	127	8	no	—
61	C₂×Q₁₆	—	127	2	no	—
62	C₂×SD₁₆	—	127	2	no	—
63	C₂×M₁₆	—	127	1	no	—
64	#41	—	127	4	no	—
65	#41	—	254	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	127	1	no	—
69	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
70	C₂×Q₈	C₂×D₁₂₇	1	4	yes	complemented
71	C₂×D₄	—	127	1	no	—
72	C₂×D₄	—	254	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	254	1	no	—
75	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
76	Q₁₆	—	127	8	no	—
77	SD₁₆	—	127	8	no	—
78	M₁₆	—	127	4	no	—
79	C₂×C₈	—	127	2	no	—
80	C₂³	—	254	1	no	—
81	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	127	2	no	—
85	Q₈	—	254	1	no	—
86	Q₈	C₂²×D₁₂₇	1	4	yes	—
87	D₄	—	127	2	no	—
88	D₄	—	254	4	no	—
89	D₄	—	254	1	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	254	1	no	—
93	C₂×C₄	—	254	4	no	—
94	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
95	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₈	—	127	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	508	2	no	—
101	C₂²	—	254	2	no	—
102	C₂²	C₂×C₂²⋊D₁₂₇	1	2	yes	—
103	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
104	C₄	—	2	4	no	—
105	C₄	—	254	2	no	—
106	C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
107	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
108	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	—	508	2	no	—
111	C₂	#16861240	1	2	yes	direct product factor
112	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

#16861240

C₂

yes

maximal, direct product factor

C₂×#14523238

C₂

yes

maximal, complemented

C₂×#14523235

C₂

yes

maximal, complemented

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523238

C₂²

yes

complemented

#14523235

C₂²

yes

complemented

#14523232

C₂²

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

C₂³

yes

—

D₅₀₈

—

D₅₀₈

C₂³

yes

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

C₂³

yes

—

C₂³×C₁₂₇

D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#137

yes

complemented

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

C₁₂₇

C₂×#137

yes

characteristic, complemented, Sylow

C₂×#137

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×Q₈

D₁₂₇

yes

characteristic, complemented

#137

—

127

—

C₂×Q₁₆

—

127

—

C₂×SD₁₆

—

127

—

C₂×M₁₆

—

127

—

#41

—

127

—

#41

—

254

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂×Q₈

C₂×D₁₂₇

yes

complemented

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

127

—

SD₁₆

—

127

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

C₂²×D₁₂₇

yes

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₂²

—

100

C₂²

—

508

—

101

C₂²

—

254

—

102

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

103

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

104

C₄

—

105

C₄

—

254

—

106

C₄

C₂×C₂²⋊D₁₂₇

yes

—

107

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

108

C₄

C₂³×D₁₂₇

yes

characteristic

109

C₂

—

110

C₂

—

508

—

111

C₂

#16861240

yes

direct product factor

112

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	C₂²×Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂²×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#137
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂⁴

Frattini

C₄

C₂³×D₁₂₇

Fitting

C₂²×Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂²×Q₈

D₁₂₇

hypocenter

C₁₂₇

C₂×#137

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #16861240	C₂	direct (N × H)
3. C₂×#14523238	C₂	semidirect
4. C₂×#14523235	C₂	semidirect
5. C₂×#14523232	C₂	semidirect
6. C₂²×Q₈×C₁₂₇	C₂	semidirect
13. #14523238	C₂²	semidirect
14. #14523235	C₂²	semidirect
15. #14523232	C₂²	semidirect
20. C₂×Q₈×C₁₂₇	C₂²	semidirect
53. C₂×C₁₂₇	#137	semidirect
56. C₁₂₇	C₂×#137	semidirect
59. C₂²×Q₈	D₁₂₇	semidirect
70. C₂×Q₈	C₂×D₁₂₇	semidirect
111. C₂	#16861240	direct (N × H)

H ≅ G/N

type

2. #16861240

C₂

direct (N × H)

3. C₂×#14523238

C₂

semidirect

4. C₂×#14523235

C₂

semidirect

5. C₂×#14523232

C₂

semidirect

6. C₂²×Q₈×C₁₂₇

C₂

semidirect

13. #14523238

C₂²

semidirect

14. #14523235

C₂²

semidirect

15. #14523232

C₂²

semidirect

20. C₂×Q₈×C₁₂₇

C₂²

semidirect

53. C₂×C₁₂₇

#137

semidirect

56. C₁₂₇

C₂×#137

semidirect

59. C₂²×Q₈

D₁₂₇

semidirect

70. C₂×Q₈

C₂×D₁₂₇

semidirect

111. C₂

#16861240

direct (N × H)

Subgroups — C₂×#16861240 #22550669

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16861240 #22550669

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16861240 #22550669