Group #22550642 — subgroups

Subgroups — #22550642

Order 8128.

258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 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	4	yes	maximal, complemented
3	#16861215	C₂	1	4	yes	maximal, complemented
4	C₂×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×D₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#14523248	C₄	1	2	yes	—
8	#14523248	C₄	1	2	yes	—
9	#14523248	C₂²	1	2	yes	—
10	#14523248	C₂²	1	2	yes	—
11	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
12	C₂×D₅₀₈	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₂₇	C₄	1	2	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	#14523232	C₂²	1	2	yes	—
16	#14523232	C₂²	1	2	yes	complemented
17	#14523231	C₂²	1	8	yes	complemented
18	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
19	C₈×D₁₂₇	C₂²	1	8	yes	complemented
20	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	M₁₆×C₁₂₇	C₂²	1	2	yes	complemented
23	C₂×C₈×C₁₂₇	C₂²	1	2	yes	complemented
24	C₂²×D₁₂₇	C₂×C₄	1	2	yes	—
25	Dic₂₅₄	C₂×C₄	1	2	yes	—
26	Dic₂₅₄	C₂×C₄	1	2	yes	—
27	D₅₀₈	C₂×C₄	1	2	yes	—
28	D₅₀₈	C₂×C₄	1	2	yes	—
29	C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
30	C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₂₇	C₂×C₄	1	4	yes	—
33	C₄×D₁₂₇	C₂³	1	4	yes	—
34	C₁₂₇⋊C₈	C₂³	1	4	yes	—
35	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₈×C₁₂₇	C₂³	1	4	yes	—
43	C₂²×C₁₂₇	—	4	1	no	—
44	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×D₁₂₇	—	4	2	no	—
48	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
49	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
50	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₂₇	C₂²×C₄	1	4	yes	—
54	C₂×C₁₂₇	—	2	1	no	—
55	C₂×C₁₂₇	—	2	1	no	—
56	C₂×C₁₂₇	—	2	1	no	—
57	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
58	D₁₂₇	—	2	4	no	—
59	C₁₂₇	#568	1	1	yes	characteristic, complemented, Sylow
60	#568	—	127	1	no	maximal, Sylow
61	C₂×#41	—	127	1	no	—
62	#131	—	127	4	no	—
63	#131	—	127	4	no	—
64	C₂×M₁₆	—	127	1	no	—
65	C₂×M₁₆	—	127	4	no	—
66	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
67	#41	—	127	2	no	—
68	#41	—	127	2	no	—
69	#41	—	127	2	no	—
70	#41	—	127	2	no	—
71	C₂×Q₈	—	127	1	no	—
72	C₂×D₄	—	127	2	no	—
73	C₂×D₄	—	127	1	no	—
74	C₂²×C₄	—	127	2	no	—
75	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
76	M₁₆	—	127	2	no	—
77	M₁₆	—	127	2	no	—
78	M₁₆	—	127	8	no	—
79	M₁₆	C₂×D₁₂₇	1	2	yes	—
80	M₁₆	C₂×D₁₂₇	1	2	yes	complemented
81	C₂×C₈	—	127	2	no	—
82	C₂×C₈	—	127	8	no	—
83	C₂×C₈	C₂×D₁₂₇	1	2	yes	complemented
84	C₂³	—	127	2	no	—
85	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
86	Q₈	—	127	2	no	—
87	Q₈	—	127	2	no	—
88	D₄	—	127	4	no	—
89	D₄	—	127	4	no	—
90	D₄	—	127	2	no	—
91	D₄	—	127	2	no	—
92	C₂×C₄	—	127	2	no	—
93	C₂×C₄	—	127	4	no	—
94	C₂×C₄	—	127	4	no	—
95	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₈	—	127	4	no	—
102	C₈	C₂²×D₁₂₇	1	4	yes	—
103	C₂²	—	4	1	no	—
104	C₂²	—	127	4	no	—
105	C₂²	—	508	2	no	—
106	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
107	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
108	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
109	C₄	—	127	4	no	—
110	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
112	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	—	2	1	no	—
117	C₂	—	254	4	no	—
118	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, complemented

#16861215

C₂

yes

maximal, complemented

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

M₁₆×D₁₂₇

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

C₄

yes

—

#14523248

C₄

yes

—

#14523248

C₂²

yes

—

#14523248

C₂²

yes

—

C₂×Dic₂₅₄

C₄

yes

characteristic

C₂×D₅₀₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

#14523232

C₂²

yes

—

#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

—

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

C₂×C₄

yes

—

Dic₂₅₄

C₂×C₄

yes

—

Dic₂₅₄

C₂×C₄

yes

—

D₅₀₈

C₂×C₄

yes

—

D₅₀₈

C₂×C₄

yes

—

C₂²⋊D₁₂₇

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

—

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#568

yes

characteristic, complemented, Sylow

#568

—

127

maximal, Sylow

C₂×#41

—

127

—

#131

—

127

—

#131

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

D₁₂₇

yes

characteristic, complemented

#41

—

127

—

#41

—

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

—

127

—

M₁₆

—

127

—

M₁₆

C₂×D₁₂₇

yes

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

127

—

C₂³

C₄×D₁₂₇

yes

characteristic

Q₈

—

127

—

Q₈

—

127

—

D₄

—

127

—

D₄

—

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

characteristic

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₈

—

127

—

102

C₈

C₂²×D₁₂₇

yes

—

103

C₂²

—

104

C₂²

—

127

—

105

C₂²

—

508

—

106

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

107

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

108

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

109

C₄

—

127

—

110

C₄

C₂×C₄×D₁₂₇

yes

characteristic

111

C₄

C₂×C₄×D₁₂₇

yes

characteristic

112

C₄

C₂×C₄×D₁₂₇

yes

characteristic

113

C₄

C₂³×D₁₂₇

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

—

117

C₂

—

254

—

118

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂³×C₄
Fitting	C₂×M₁₆×C₁₂₇	C₂
hypercenter	C₂×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#568
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₄

C₂³×D₁₂₇

commutator, socle

C₂×C₁₂₇

C₂³×C₄

Fitting

C₂×M₁₆×C₁₂₇

C₂

hypercenter

C₂×M₁₆

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #16861217	C₂	semidirect
3. #16861215	C₂	semidirect
4. C₂×#14523232	C₂	semidirect
5. M₁₆×D₁₂₇	C₂	semidirect
6. C₂×M₁₆×C₁₂₇	C₂	semidirect
16. #14523232	C₂²	semidirect
17. #14523231	C₂²	semidirect
18. C₂×C₁₂₇⋊C₈	C₂²	semidirect
19. C₈×D₁₂₇	C₂²	semidirect
22. M₁₆×C₁₂₇	C₂²	semidirect
23. C₂×C₈×C₁₂₇	C₂²	semidirect
59. C₁₂₇	#568	semidirect
66. C₂×M₁₆	D₁₂₇	semidirect
80. M₁₆	C₂×D₁₂₇	semidirect
83. C₂×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. #16861217

C₂

semidirect

3. #16861215

C₂

semidirect

4. C₂×#14523232

C₂

semidirect

5. M₁₆×D₁₂₇

C₂

semidirect

6. C₂×M₁₆×C₁₂₇

C₂

semidirect

16. #14523232

C₂²

semidirect

17. #14523231

C₂²

semidirect

18. C₂×C₁₂₇⋊C₈

C₂²

semidirect

19. C₈×D₁₂₇

C₂²

semidirect

22. M₁₆×C₁₂₇

C₂²

semidirect

23. C₂×C₈×C₁₂₇

C₂²

semidirect

59. C₁₂₇

#568

semidirect

66. C₂×M₁₆

D₁₂₇

semidirect

80. M₁₆

C₂×D₁₂₇

semidirect

83. C₂×C₈

C₂×D₁₂₇

semidirect

Subgroups — #22550642

Conjugacy classes of subgroups

Complements