Group #22550634 — subgroups

Subgroups — C₂×M₁₆×D₁₂₇ #22550634

Order 8128.

298 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 159 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	1	yes	maximal, characteristic
2	C₂×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523231	C₂	1	2	yes	maximal, complemented
4	M₁₆×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	C₄	1	1	yes	characteristic
8	C₂²×Dic₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₂₇	C₄	1	4	yes	—
10	C₂×C₄×D₁₂₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₁₂₇	C₄	1	1	yes	characteristic
12	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	#14523232	C₂²	1	4	yes	complemented
16	#14523231	C₂²	1	8	yes	complemented
17	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
18	C₈×D₁₂₇	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂×C₈×C₁₂₇	C₂²	1	2	yes	complemented
22	C₂²×D₁₂₇	—	2	4	no	—
23	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×D₁₂₇	C₂×C₄	1	4	yes	—
30	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
31	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₂₇	C₂×C₄	1	4	yes	—
33	C₄×D₁₂₇	C₂³	1	2	yes	—
34	C₄×D₁₂₇	C₂³	1	2	yes	—
35	C₁₂₇⋊C₈	C₂³	1	4	yes	complemented
36	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₈×C₁₂₇	C₂³	1	4	yes	complemented
42	C₂²×C₁₂₇	—	2	2	no	—
43	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×D₁₂₇	—	2	2	no	—
46	C₂×D₁₂₇	—	2	8	no	—
47	C₂×D₁₂₇	M₁₆	1	4	yes	direct product factor
48	C₂×D₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×D₁₂₇	C₂²×C₄	1	2	yes	—
50	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₂₇	C₂²×C₄	1	2	yes	—
54	Dic₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂×C₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	C₂×M₁₆	1	2	yes	complemented
57	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
58	D₁₂₇	—	2	2	no	—
59	D₁₂₇	C₂×M₁₆	1	4	yes	direct product factor
60	C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
61	C₂²×M₁₆	—	127	1	no	maximal, Sylow
62	C₂³×C₄	—	127	1	no	—
63	C₂×M₁₆	—	127	1	no	—
64	C₂×M₁₆	—	127	8	no	—
65	C₂×M₁₆	—	127	2	no	—
66	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic, direct product factor
67	C₂²×C₈	—	127	2	no	—
68	C₂⁴	—	127	1	no	—
69	C₂²×C₄	—	127	1	no	—
70	C₂²×C₄	—	127	1	no	—
71	C₂²×C₄	—	127	4	no	—
72	C₂²×C₄	—	127	1	no	—
73	C₂²×C₄	—	127	4	no	—
74	C₂²×C₄	—	127	1	no	—
75	C₂²×C₄	—	127	1	no	—
76	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
77	M₁₆	—	127	4	no	—
78	M₁₆	—	127	8	no	—
79	M₁₆	C₂×D₁₂₇	1	4	yes	direct product factor
80	C₂×C₈	—	127	2	no	—
81	C₂×C₈	—	127	8	no	—
82	C₂×C₈	C₂×D₁₂₇	1	2	yes	complemented
83	C₂³	—	127	1	no	—
84	C₂³	—	127	4	no	—
85	C₂³	—	127	1	no	—
86	C₂³	—	254	4	no	—
87	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	—	127	4	no	—
89	C₂×C₄	—	127	1	no	—
90	C₂×C₄	—	127	1	no	—
91	C₂×C₄	—	127	4	no	—
92	C₂×C₄	—	127	4	no	—
93	C₂×C₄	—	127	2	no	—
94	C₂×C₄	—	127	2	no	—
95	C₂×C₄	—	127	2	no	—
96	C₂×C₄	—	127	2	no	—
97	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
99	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₈	—	127	4	no	—
102	C₈	C₂²×D₁₂₇	1	4	yes	complemented
103	C₂²	—	2	2	no	—
104	C₂²	—	127	4	no	—
105	C₂²	—	127	2	no	—
106	C₂²	—	127	2	no	—
107	C₂²	—	254	8	no	—
108	C₂²	—	254	2	no	—
109	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
110	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₄	—	127	2	no	—
112	C₄	—	127	2	no	—
113	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
114	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
115	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	—	127	4	no	—
118	C₂	—	254	2	no	—
119	C₂	M₁₆×D₁₂₇	1	2	yes	direct product factor
120	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×#14523231

C₂

yes

maximal, complemented

M₁₆×D₁₂₇

C₂

yes

maximal, direct product factor

C₂×C₈×D₁₂₇

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₄

yes

characteristic

C₂²×Dic₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₄

yes

—

C₂×C₄×D₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523232

C₂²

yes

complemented

#14523231

C₂²

yes

complemented

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

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

—

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂³

yes

—

C₄×D₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

C₂³

yes

complemented

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

M₁₆

yes

direct product factor

C₂×D₁₂₇

C₂²×C₄

yes

—

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

C₂²×C₄

yes

—

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×M₁₆

yes

complemented

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

D₁₂₇

—

D₁₂₇

C₂×M₁₆

yes

direct product factor

C₁₂₇

C₂²×M₁₆

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

127

maximal, Sylow

C₂³×C₄

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

D₁₂₇

yes

characteristic, direct product factor

C₂²×C₈

—

127

—

C₂⁴

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

C₂×D₁₂₇

yes

direct product factor

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

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

—

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₈

—

127

—

102

C₈

C₂²×D₁₂₇

yes

complemented

103

C₂²

—

104

C₂²

—

127

—

105

C₂²

—

127

—

106

C₂²

—

127

—

107

C₂²

—

254

—

108

C₂²

—

254

—

109

C₂²

C₂×C₄×D₁₂₇

yes

—

110

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

111

C₄

—

127

—

112

C₄

—

127

—

113

C₄

C₂×C₄×D₁₂₇

yes

—

114

C₄

C₂×C₄×D₁₂₇

yes

characteristic

115

C₄

C₂³×D₁₂₇

yes

characteristic

116

C₂

—

117

C₂

—

127

—

118

C₂

—

254

—

119

C₂

M₁₆×D₁₂₇

yes

direct product factor

120

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	C₂²×M₁₆
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

hypocenter

C₁₂₇

C₂²×M₁₆

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. C₂×#14523232	C₂	semidirect
3. C₂×#14523231	C₂	semidirect
4. M₁₆×D₁₂₇	C₂	direct (N × H)
5. C₂×C₈×D₁₂₇	C₂	semidirect
6. C₂×M₁₆×C₁₂₇	C₂	semidirect
15. #14523232	C₂²	semidirect
16. #14523231	C₂²	semidirect
17. C₂×C₁₂₇⋊C₈	C₂²	semidirect
18. C₈×D₁₂₇	C₂²	semidirect
20. M₁₆×C₁₂₇	C₂²	semidirect
21. C₂×C₈×C₁₂₇	C₂²	semidirect
35. C₁₂₇⋊C₈	C₂³	semidirect
41. C₈×C₁₂₇	C₂³	semidirect
47. C₂×D₁₂₇	M₁₆	direct (N × H)
56. C₂×C₁₂₇	C₂×M₁₆	semidirect
59. D₁₂₇	C₂×M₁₆	direct (N × H)
60. C₁₂₇	C₂²×M₁₆	semidirect
66. C₂×M₁₆	D₁₂₇	direct (N × H)
79. M₁₆	C₂×D₁₂₇	direct (N × H)
82. C₂×C₈	C₂×D₁₂₇	semidirect
102. C₈	C₂²×D₁₂₇	semidirect
119. C₂	M₁₆×D₁₂₇	direct (N × H)

H ≅ G/N

type

2. C₂×#14523232

C₂

semidirect

3. C₂×#14523231

C₂

semidirect

4. M₁₆×D₁₂₇

C₂

direct (N × H)

5. C₂×C₈×D₁₂₇

C₂

semidirect

6. C₂×M₁₆×C₁₂₇

C₂

semidirect

15. #14523232

C₂²

semidirect

16. #14523231

C₂²

semidirect

17. C₂×C₁₂₇⋊C₈

C₂²

semidirect

18. C₈×D₁₂₇

C₂²

semidirect

20. M₁₆×C₁₂₇

C₂²

semidirect

21. C₂×C₈×C₁₂₇

C₂²

semidirect

35. C₁₂₇⋊C₈

C₂³

semidirect

41. C₈×C₁₂₇

C₂³

semidirect

47. C₂×D₁₂₇

M₁₆

direct (N × H)

56. C₂×C₁₂₇

C₂×M₁₆

semidirect

59. D₁₂₇

C₂×M₁₆

direct (N × H)

60. C₁₂₇

C₂²×M₁₆

semidirect

66. C₂×M₁₆

D₁₂₇

direct (N × H)

79. M₁₆

C₂×D₁₂₇

direct (N × H)

82. C₂×C₈

C₂×D₁₂₇

semidirect

102. C₈

C₂²×D₁₂₇

semidirect

119. C₂

M₁₆×D₁₂₇

direct (N × H)

Subgroups — C₂×M₁₆×D₁₂₇ #22550634

Conjugacy classes of subgroups

Complements

Subgroups — C2×M16×D127 #22550634

Conjugacy classes of subgroups

Complements

Subgroups — C₂×M₁₆×D₁₂₇ #22550634