Group #22550322 — subgroups

Subgroups — C₂×C₂²⋊C₄×D₁₂₇ #22550322

Order 8128.

674 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 207 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	C₂⁴×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	—	2	8	no	—
8	C₂³×D₁₂₇	C₄	1	8	yes	complemented
9	C₂³×D₁₂₇	C₂²	1	4	yes	—
10	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₂₇	C₂²	1	2	yes	complemented
13	C₂×C₄×D₁₂₇	—	2	8	no	—
14	C₂×C₄×D₁₂₇	C₂²	1	8	yes	complemented
15	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	complemented
16	(C₂×D₁₂₇)⋊C₄	C₂²	1	8	yes	complemented
17	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	complemented
19	C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂²×D₁₂₇	—	2	4	no	—
21	C₂²×D₁₂₇	—	2	32	no	—
22	C₂²×D₁₂₇	—	2	16	no	—
23	C₂²×D₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×D₁₂₇	C₂×C₄	1	16	yes	complemented
25	C₂²×D₁₂₇	C₂×C₄	1	4	yes	complemented
26	C₂²×D₁₂₇	D₄	1	8	yes	—
27	C₂²×D₁₂₇	C₂³	1	2	yes	—
28	C₂²×D₁₂₇	C₂³	1	2	yes	—
29	C₂×Dic₁₂₇	—	2	4	no	—
30	C₂×Dic₁₂₇	C₂³	1	4	yes	complemented
31	C₄×D₁₂₇	—	2	16	no	—
32	C₂³×C₁₂₇	—	2	4	no	—
33	C₂³×C₁₂₇	C₂×C₄	1	4	yes	complemented
34	C₂³×C₁₂₇	C₂³	1	2	yes	—
35	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	4	yes	complemented
38	C₂²×C₁₂₇	—	2	8	no	—
39	C₂²×C₁₂₇	—	2	4	no	—
40	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₁₂₇	C₂²×C₄	1	4	yes	complemented
42	C₂²×C₁₂₇	C₂×D₄	1	4	yes	—
43	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
44	C₂×D₁₂₇	—	2	8	no	—
45	C₂×D₁₂₇	—	2	4	no	—
46	C₂×D₁₂₇	—	2	32	no	—
47	C₂×D₁₂₇	C₂²⋊C₄	1	16	yes	direct product factor
48	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
49	C₂×D₁₂₇	C₂²×C₄	1	4	yes	complemented
50	C₂×D₁₂₇	C₂×D₄	1	8	yes	—
51	C₄×C₁₂₇	—	2	4	no	—
52	Dic₁₂₇	—	2	4	no	—
53	C₂×C₁₂₇	—	2	4	no	—
54	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	complemented
55	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂²×D₄	1	2	yes	—
57	D₁₂₇	—	2	4	no	—
58	D₁₂₇	C₂×C₂²⋊C₄	1	8	yes	direct product factor
59	C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₂²⋊C₄	—	127	1	no	maximal, Sylow
61	C₂⁵	—	127	1	no	—
62	C₂³×C₄	—	127	2	no	—
63	C₂×C₂²⋊C₄	—	127	1	no	—
64	C₂×C₂²⋊C₄	—	127	2	no	—
65	C₂×C₂²⋊C₄	—	127	8	no	—
66	C₂×C₂²⋊C₄	D₁₂₇	1	1	yes	characteristic, direct product factor
67	C₂⁴	—	127	1	no	—
68	C₂⁴	—	127	8	no	—
69	C₂⁴	—	127	4	no	—
70	C₂⁴	—	127	1	no	—
71	C₂⁴	—	254	8	no	—
72	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
73	C₂²×C₄	—	127	2	no	—
74	C₂²×C₄	—	127	8	no	—
75	C₂²×C₄	—	254	8	no	—
76	C₂²×C₄	C₂×D₁₂₇	1	2	yes	complemented
77	C₂²⋊C₄	—	127	4	no	—
78	C₂²⋊C₄	—	127	8	no	—
79	C₂²⋊C₄	C₂×D₁₂₇	1	4	yes	direct product factor
80	C₂³	—	2	4	no	—
81	C₂³	—	127	4	no	—
82	C₂³	—	127	8	no	—
83	C₂³	—	127	2	no	—
84	C₂³	—	127	4	no	—
85	C₂³	—	127	16	no	—
86	C₂³	—	127	2	no	—
87	C₂³	—	254	16	no	—
88	C₂³	—	254	32	no	—
89	C₂³	—	254	4	no	—
90	C₂³	C₄×D₁₂₇	1	4	yes	complemented
91	C₂³	C₂²×D₁₂₇	1	2	yes	—
92	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	127	4	no	—
95	C₂×C₄	—	254	16	no	—
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	C₂²×D₁₂₇	1	4	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	2	8	no	—
100	C₂²	—	127	4	no	—
101	C₂²	—	127	8	no	—
102	C₂²	—	127	16	no	—
103	C₂²	—	127	4	no	—
104	C₂²	—	254	32	no	—
105	C₂²	—	254	4	no	—
106	C₂²	—	254	8	no	—
107	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
108	C₂²	C₂×C₄×D₁₂₇	1	4	yes	complemented
109	C₂²	D₄×D₁₂₇	1	4	yes	—
110	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
111	C₄	—	2	4	no	—
112	C₄	—	254	4	no	—
113	C₂	—	2	4	no	—
114	C₂	—	127	8	no	—
115	C₂	—	254	4	no	—
116	C₂	C₂²⋊C₄×D₁₂₇	1	4	yes	direct product factor
117	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
118	C₂	C₂×D₄×D₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×D₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₄×D₁₂₇

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

C₂²⋊C₄×D₁₂₇

C₂

yes

maximal, direct product factor

C₂×C₂²⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂³×D₁₂₇

C₄

yes

complemented

C₂³×D₁₂₇

C₂²

yes

—

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

complemented

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

complemented

C₂²⋊Dic₁₂₇

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

—

C₂²×D₁₂₇

C₂×C₄

yes

complemented

C₂²×D₁₂₇

C₂×C₄

yes

complemented

C₂²×D₁₂₇

D₄

yes

—

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

complemented

C₄×D₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

complemented

C₂³×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

direct product factor

C₂×D₁₂₇

C₂²×C₄

yes

—

C₂×D₁₂₇

C₂²×C₄

yes

complemented

C₂×D₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

complemented

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×C₁₂₇

C₂²×D₄

yes

—

D₁₂₇

—

D₁₂₇

C₂×C₂²⋊C₄

yes

direct product factor

C₁₂₇

C₂²×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₂²⋊C₄

—

127

maximal, Sylow

C₂⁵

—

127

—

C₂³×C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

D₁₂₇

yes

characteristic, direct product factor

C₂⁴

—

127

—

C₂⁴

—

127

—

C₂⁴

—

127

—

C₂⁴

—

127

—

C₂⁴

—

254

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

complemented

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

direct product factor

C₂³

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₄×D₁₂₇

yes

complemented

C₂³

C₂²×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

127

—

101

C₂²

—

127

—

102

C₂²

—

127

—

103

C₂²

—

127

—

104

C₂²

—

254

—

105

C₂²

—

254

—

106

C₂²

—

254

—

107

C₂²

C₂×C₄×D₁₂₇

yes

—

108

C₂²

C₂×C₄×D₁₂₇

yes

complemented

109

C₂²

D₄×D₁₂₇

yes

—

110

C₂²

C₂³×D₁₂₇

yes

characteristic

111

C₄

—

112

C₄

—

254

—

113

C₂

—

114

C₂

—

127

—

115

C₂

—

254

—

116

C₂

C₂²⋊C₄×D₁₂₇

yes

direct product factor

117

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

118

C₂

C₂×D₄×D₁₂₇

yes

—

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

subgroup

G/H

center

C₂³

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂×C₂²⋊C₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₂₇

hypocenter

C₁₂₇

C₂²×C₂²⋊C₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂²×C₄×D₁₂₇	C₂	semidirect
3. C₂×C₂²⋊Dic₁₂₇	C₂	semidirect
4. C₂×(C₂×D₁₂₇)⋊C₄	C₂	semidirect
5. C₂²⋊C₄×D₁₂₇	C₂	direct (N × H)
6. C₂×C₂²⋊C₄×C₁₂₇	C₂	semidirect
8. C₂³×D₁₂₇	C₄	semidirect
12. C₂²×Dic₁₂₇	C₂²	semidirect
14. C₂×C₄×D₁₂₇	C₂²	semidirect
15. C₂²⋊Dic₁₂₇	C₂²	semidirect
16. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
18. C₂²×C₄×C₁₂₇	C₂²	semidirect
19. C₂²⋊C₄×C₁₂₇	C₂²	semidirect
24. C₂²×D₁₂₇	C₂×C₄	semidirect
25. C₂²×D₁₂₇	C₂×C₄	semidirect
30. C₂×Dic₁₂₇	C₂³	semidirect
33. C₂³×C₁₂₇	C₂×C₄	semidirect
37. C₂×C₄×C₁₂₇	C₂³	semidirect
41. C₂²×C₁₂₇	C₂²×C₄	semidirect
47. C₂×D₁₂₇	C₂²⋊C₄	direct (N × H)
49. C₂×D₁₂₇	C₂²×C₄	semidirect
54. C₂×C₁₂₇	C₂×C₂²⋊C₄	semidirect
58. D₁₂₇	C₂×C₂²⋊C₄	direct (N × H)
59. C₁₂₇	C₂²×C₂²⋊C₄	semidirect
66. C₂×C₂²⋊C₄	D₁₂₇	direct (N × H)
76. C₂²×C₄	C₂×D₁₂₇	semidirect
79. C₂²⋊C₄	C₂×D₁₂₇	direct (N × H)
90. C₂³	C₄×D₁₂₇	semidirect
97. C₂×C₄	C₂²×D₁₂₇	semidirect
108. C₂²	C₂×C₄×D₁₂₇	semidirect
116. C₂	C₂²⋊C₄×D₁₂₇	direct (N × H)

H ≅ G/N

type

2. C₂²×C₄×D₁₂₇

C₂

semidirect

3. C₂×C₂²⋊Dic₁₂₇

C₂

semidirect

4. C₂×(C₂×D₁₂₇)⋊C₄

C₂

semidirect

5. C₂²⋊C₄×D₁₂₇

C₂

direct (N × H)

6. C₂×C₂²⋊C₄×C₁₂₇

C₂

semidirect

8. C₂³×D₁₂₇

C₄

semidirect

12. C₂²×Dic₁₂₇

C₂²

semidirect

14. C₂×C₄×D₁₂₇

C₂²

semidirect

15. C₂²⋊Dic₁₂₇

C₂²

semidirect

16. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

18. C₂²×C₄×C₁₂₇

C₂²

semidirect

19. C₂²⋊C₄×C₁₂₇

C₂²

semidirect

24. C₂²×D₁₂₇

C₂×C₄

semidirect

25. C₂²×D₁₂₇

C₂×C₄

semidirect

30. C₂×Dic₁₂₇

C₂³

semidirect

33. C₂³×C₁₂₇

C₂×C₄

semidirect

37. C₂×C₄×C₁₂₇

C₂³

semidirect

41. C₂²×C₁₂₇

C₂²×C₄

semidirect

47. C₂×D₁₂₇

C₂²⋊C₄

direct (N × H)

49. C₂×D₁₂₇

C₂²×C₄

semidirect

54. C₂×C₁₂₇

C₂×C₂²⋊C₄

semidirect

58. D₁₂₇

C₂×C₂²⋊C₄

direct (N × H)

59. C₁₂₇

C₂²×C₂²⋊C₄

semidirect

66. C₂×C₂²⋊C₄

D₁₂₇

direct (N × H)

76. C₂²×C₄

C₂×D₁₂₇

semidirect

79. C₂²⋊C₄

C₂×D₁₂₇

direct (N × H)

90. C₂³

C₄×D₁₂₇

semidirect

97. C₂×C₄

C₂²×D₁₂₇

semidirect

108. C₂²

C₂×C₄×D₁₂₇

semidirect

116. C₂

C₂²⋊C₄×D₁₂₇

direct (N × H)

Subgroups — C₂×C₂²⋊C₄×D₁₂₇ #22550322

Conjugacy classes of subgroups

Complements

Subgroups — C2×C22⋊C4×D127 #22550322

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₂²⋊C₄×D₁₂₇ #22550322