Group #22550365 — subgroups

Subgroups — C₂×#16861180 #22550365

Order 8128.

426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₂²⋊D₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861180	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊Dic₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₁₂₇	—	2	8	no	—
9	C₂×C₂²⋊D₁₂₇	C₂²	1	8	yes	complemented
10	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₂₇	—	2	4	no	—
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	complemented
14	C₄⋊Dic₁₂₇	C₂²	1	4	yes	complemented
15	C₂²⋊Dic₁₂₇	C₂²	1	8	yes	complemented
16	C₂⁴×C₁₂₇	C₂²	1	2	yes	—
17	C₂×D₄×C₁₂₇	—	2	4	no	—
18	C₂×D₄×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×D₁₂₇	—	2	2	no	—
21	C₂²×D₁₂₇	—	2	2	no	—
22	C₂²×D₁₂₇	D₄	1	4	yes	complemented
23	C₂²×D₁₂₇	C₂³	1	2	yes	—
24	C₂²⋊D₁₂₇	—	2	16	no	—
25	C₂×Dic₁₂₇	—	2	2	no	—
26	C₂×Dic₁₂₇	—	2	4	no	—
27	C₂×Dic₁₂₇	C₂³	1	2	yes	—
28	C₂×Dic₁₂₇	C₂³	1	4	yes	—
29	C₄×D₁₂₇	—	2	8	no	—
30	C₂³×C₁₂₇	—	2	4	no	—
31	C₂³×C₁₂₇	—	2	4	no	—
32	C₂³×C₁₂₇	—	2	4	no	—
33	C₂³×C₁₂₇	C₂³	1	4	yes	—
34	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
35	D₄×C₁₂₇	—	2	8	no	—
36	C₂×C₄×C₁₂₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₁₂₇	—	2	4	no	—
39	C₂²×C₁₂₇	—	2	4	no	—
40	C₂²×C₁₂₇	—	2	4	no	—
41	C₂²×C₁₂₇	—	4	8	no	—
42	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂²×C₁₂₇	#41	1	2	yes	—
45	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₂₇	—	2	8	no	—
47	C₂×D₁₂₇	—	2	2	no	—
48	C₂×D₁₂₇	—	2	2	no	—
49	C₂×D₁₂₇	C₂×D₄	1	4	yes	complemented
50	C₄×C₁₂₇	C₂×D₄	1	4	yes	complemented
51	Dic₁₂₇	—	2	2	no	—
52	Dic₁₂₇	—	2	4	no	—
53	C₂×C₁₂₇	—	4	4	no	—
54	C₂×C₁₂₇	#121	1	4	yes	complemented
55	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
58	D₁₂₇	—	2	4	no	—
59	C₁₂₇	C₂×#121	1	1	yes	characteristic, complemented, Sylow
60	C₂×#121	—	127	1	no	maximal, Sylow
61	C₂²×D₄	—	127	2	no	—
62	C₂²×D₄	D₁₂₇	1	1	yes	characteristic, complemented
63	C₂³×C₄	—	127	1	no	—
64	#121	—	127	8	no	—
65	C₂×C₄⋊C₄	—	127	1	no	—
66	C₂×C₂²⋊C₄	—	127	2	no	—
67	C₂⁴	—	127	1	no	—
68	C₂⁴	C₂×D₁₂₇	1	2	yes	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	127	8	no	—
71	C₂×D₄	—	254	8	no	—
72	C₂×D₄	C₂×D₁₂₇	1	4	yes	complemented
73	C₂²×C₄	—	127	1	no	—
74	C₂²×C₄	—	127	2	no	—
75	C₂²×C₄	—	127	4	no	—
76	C₂²×C₄	—	254	4	no	—
77	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
78	C₄⋊C₄	—	127	4	no	—
79	C₂²⋊C₄	—	127	8	no	—
80	C₂³	—	2	4	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	2	4	no	—
83	C₂³	—	127	4	no	—
84	C₂³	—	127	2	no	—
85	C₂³	—	254	2	no	—
86	C₂³	—	254	2	no	—
87	C₂³	C₂²×D₁₂₇	1	4	yes	—
88	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
89	D₄	—	2	8	no	—
90	D₄	—	254	16	no	—
91	C₂×C₄	—	127	2	no	—
92	C₂×C₄	—	127	4	no	—
93	C₂×C₄	—	254	4	no	—
94	C₂×C₄	—	254	2	no	—
95	C₂×C₄	—	254	8	no	—
96	C₂×C₄	C₂²⋊D₁₂₇	1	4	yes	complemented
97	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
98	C₂²	—	2	4	no	—
99	C₂²	—	2	4	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	4	8	no	—
102	C₂²	—	127	4	no	—
103	C₂²	—	254	2	no	—
104	C₂²	—	254	2	no	—
105	C₂²	—	254	8	no	—
106	C₂²	D₄×D₁₂₇	1	2	yes	—
107	C₂²	C₂×C₂²⋊D₁₂₇	1	2	yes	—
108	C₂²	#14523247	1	2	yes	—
109	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
110	C₄	—	254	4	no	—
111	C₄	—	254	2	no	—
112	C₄	C₂×C₂²⋊D₁₂₇	1	4	yes	complemented
113	C₂	—	4	4	no	—
114	C₂	—	254	4	no	—
115	C₂	#16861180	1	4	yes	direct product factor
116	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
117	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
118	C₂	C₂×#14523247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₂₇

C₂

yes

maximal, complemented

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861180

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, complemented

C₂²×D₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

complemented

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

complemented

C₄⋊Dic₁₂₇

C₂²

yes

complemented

C₂²⋊Dic₁₂₇

C₂²

yes

complemented

C₂⁴×C₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

complemented

C₂²×D₁₂₇

C₂³

yes

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×D₄

yes

complemented

C₄×C₁₂₇

C₂×D₄

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

complemented

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

D₁₂₇

—

C₁₂₇

C₂×#121

yes

characteristic, complemented, Sylow

C₂×#121

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂²×D₄

D₁₂₇

yes

characteristic, complemented

C₂³×C₄

—

127

—

#121

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂⁴

—

127

—

C₂⁴

C₂×D₁₂₇

yes

—

C₂×D₄

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

complemented

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

127

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

—

254

—

106

C₂²

D₄×D₁₂₇

yes

—

107

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

108

C₂²

#14523247

yes

—

109

C₂²

C₂³×D₁₂₇

yes

characteristic

110

C₄

—

254

—

111

C₄

—

254

—

112

C₄

C₂×C₂²⋊D₁₂₇

yes

complemented

113

C₂

—

114

C₂

—

254

—

115

C₂

#16861180

yes

direct product factor

116

C₂

C₂×D₄×D₁₂₇

yes

characteristic

117

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

118

C₂

C₂×#14523247

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂⁴
Frattini	C₂²	C₂³×D₁₂₇
Fitting	C₂²×D₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂²×D₄	D₁₂₇
hypocenter	C₁₂₇	C₂×#121
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂⁴

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂²×D₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₁₂₇	C₂	semidirect
2. C₂²×C₄×D₁₂₇	C₂	semidirect
3. #16861180	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
5. C₂×C₂²⋊Dic₁₂₇	C₂	semidirect
6. C₂²×D₄×C₁₂₇	C₂	semidirect
9. C₂×C₂²⋊D₁₂₇	C₂²	semidirect
13. C₂×C₄×D₁₂₇	C₂²	semidirect
14. C₄⋊Dic₁₂₇	C₂²	semidirect
15. C₂²⋊Dic₁₂₇	C₂²	semidirect
18. C₂×D₄×C₁₂₇	C₂²	semidirect
22. C₂²×D₁₂₇	D₄	semidirect
36. C₂×C₄×C₁₂₇	D₄	semidirect
49. C₂×D₁₂₇	C₂×D₄	semidirect
50. C₄×C₁₂₇	C₂×D₄	semidirect
54. C₂×C₁₂₇	#121	semidirect
59. C₁₂₇	C₂×#121	semidirect
62. C₂²×D₄	D₁₂₇	semidirect
72. C₂×D₄	C₂×D₁₂₇	semidirect
96. C₂×C₄	C₂²⋊D₁₂₇	semidirect
112. C₄	C₂×C₂²⋊D₁₂₇	semidirect
115. C₂	#16861180	direct (N × H)

H ≅ G/N

type

1. C₂²×C₂²⋊D₁₂₇

C₂

semidirect

2. C₂²×C₄×D₁₂₇

C₂

semidirect

3. #16861180

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

5. C₂×C₂²⋊Dic₁₂₇

C₂

semidirect

6. C₂²×D₄×C₁₂₇

C₂

semidirect

9. C₂×C₂²⋊D₁₂₇

C₂²

semidirect

13. C₂×C₄×D₁₂₇

C₂²

semidirect

14. C₄⋊Dic₁₂₇

C₂²

semidirect

15. C₂²⋊Dic₁₂₇

C₂²

semidirect

18. C₂×D₄×C₁₂₇

C₂²

semidirect

22. C₂²×D₁₂₇

D₄

semidirect

36. C₂×C₄×C₁₂₇

D₄

semidirect

49. C₂×D₁₂₇

C₂×D₄

semidirect

50. C₄×C₁₂₇

C₂×D₄

semidirect

54. C₂×C₁₂₇

#121

semidirect

59. C₁₂₇

C₂×#121

semidirect

62. C₂²×D₄

D₁₂₇

semidirect

72. C₂×D₄

C₂×D₁₂₇

semidirect

96. C₂×C₄

C₂²⋊D₁₂₇

semidirect

112. C₄

C₂×C₂²⋊D₁₂₇

semidirect

115. C₂

#16861180

direct (N × H)

Subgroups — C₂×#16861180 #22550365

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16861180 #22550365

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16861180 #22550365