Group #22550546 — subgroups

Subgroups — #22550546

Order 8128.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14523247	C₂	1	2	yes	maximal, complemented
2	#16861205	C₂	1	2	yes	maximal, complemented
3	#16861181	C₂	1	4	yes	maximal, complemented
4	#16861180	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
8	#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523247	—	2	8	no	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
14	C₂²×Dic₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
21	C₄×Dic₁₂₇	C₂²	1	2	yes	—
22	C₂×D₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂×D₄×C₁₂₇	C₂²	1	2	yes	—
24	C₂×D₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	D₄	1	4	yes	complemented
28	D₅₀₈	—	2	2	no	—
29	C₂²⋊D₁₂₇	—	2	8	no	—
30	C₂²⋊D₁₂₇	—	4	4	no	—
31	C₂×Dic₁₂₇	—	2	8	no	—
32	C₂×Dic₁₂₇	C₂³	1	2	yes	—
33	C₂×Dic₁₂₇	C₂³	1	2	yes	—
34	C₂×Dic₁₂₇	C₂³	1	2	yes	—
35	C₄×D₁₂₇	—	2	4	no	—
36	C₂³×C₁₂₇	C₂³	1	4	yes	—
37	D₄×C₁₂₇	—	2	4	no	—
38	D₄×C₁₂₇	—	2	4	no	—
39	D₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₁₂₇	—	2	4	no	—
44	C₂²×C₁₂₇	—	2	4	no	—
45	C₂²×C₁₂₇	—	2	4	no	—
46	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₁₂₇	—	2	2	no	—
48	C₂×D₁₂₇	—	2	2	no	—
49	C₂×D₁₂₇	—	2	2	no	—
50	C₄×C₁₂₇	—	2	1	no	—
51	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
52	C₄×C₁₂₇	#41	1	2	yes	complemented
53	Dic₁₂₇	—	2	2	no	—
54	Dic₁₂₇	—	2	2	no	—
55	Dic₁₂₇	C₂×D₄	1	4	yes	—
56	C₂×C₁₂₇	—	4	4	no	—
57	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
60	D₁₂₇	—	4	2	no	—
61	C₁₂₇	#550	1	1	yes	characteristic, complemented, Sylow
62	#550	—	127	1	no	maximal, Sylow
63	C₂×#41	—	127	2	no	—
64	#127	—	127	2	no	—
65	#127	D₁₂₇	1	1	yes	characteristic, complemented
66	#121	—	127	2	no	—
67	#121	—	127	4	no	—
68	C₄×Q₈	—	127	1	no	—
69	C₄×D₄	—	127	2	no	—
70	C₄×D₄	—	127	1	no	—
71	#41	—	254	8	no	—
72	C₂×Q₈	—	127	1	no	—
73	C₂×D₄	—	127	4	no	—
74	C₂×D₄	—	127	4	no	—
75	C₂×D₄	—	127	1	no	—
76	C₂×D₄	C₂×D₁₂₇	1	2	yes	—
77	C₂×D₄	C₂×D₁₂₇	1	2	yes	—
78	C₂×D₄	C₂×D₁₂₇	1	2	yes	—
79	C₂²×C₄	—	127	4	no	—
80	C₂²×C₄	—	127	2	no	—
81	C₄⋊C₄	—	127	2	no	—
82	C₄⋊C₄	—	127	1	no	—
83	C₄⋊C₄	—	127	1	no	—
84	C₂²⋊C₄	—	127	4	no	—
85	C₂²⋊C₄	—	127	2	no	—
86	C₄²	—	127	2	no	—
87	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
88	C₂³	—	127	2	no	—
89	C₂³	C₂²×D₁₂₇	1	4	yes	—
90	Q₈	—	127	4	no	—
91	D₄	—	2	4	no	—
92	D₄	—	2	4	no	—
93	D₄	—	4	2	no	—
94	D₄	—	508	4	no	—
95	D₄	—	254	8	no	—
96	D₄	—	254	2	no	—
97	C₂×C₄	—	127	2	no	—
98	C₂×C₄	—	127	2	no	—
99	C₂×C₄	—	127	2	no	—
100	C₂×C₄	—	254	8	no	—
101	C₂×C₄	—	254	4	no	—
102	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
105	C₂²	—	2	4	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	254	2	no	—
109	C₂²	—	254	2	no	—
110	C₂²	—	254	2	no	—
111	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	127	4	no	—
114	C₄	—	254	2	no	—
115	C₄	—	254	2	no	—
116	C₄	D₄×D₁₂₇	1	2	yes	—
117	C₄	#14523247	1	2	yes	complemented
118	C₂	—	4	4	no	—
119	C₂	—	508	2	no	—
120	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
121	C₂	C₂×#14523247	1	1	yes	characteristic
122	C₂	#16861254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523247

C₂

yes

maximal, complemented

#16861205

C₂

yes

maximal, complemented

#16861181

C₂

yes

maximal, complemented

#16861180

C₂

yes

maximal, complemented

C₄×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₂₇

C₂

yes

maximal, complemented

#127×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523247

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

D₄

yes

complemented

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

#41

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

D₁₂₇

—

C₁₂₇

#550

yes

characteristic, complemented, Sylow

#550

—

127

maximal, Sylow

C₂×#41

—

127

—

#127

—

127

—

#127

D₁₂₇

yes

characteristic, complemented

#121

—

127

—

#121

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

—

C₂×D₄

C₂×D₁₂₇

yes

—

C₂×D₄

C₂×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

—

Q₈

—

127

—

D₄

—

D₄

—

D₄

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

100

C₂×C₄

—

254

—

101

C₂×C₄

—

254

—

102

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

103

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

104

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

254

—

109

C₂²

—

254

—

110

C₂²

—

254

—

111

C₂²

C₂³×D₁₂₇

yes

characteristic

112

C₄

—

113

C₄

—

127

—

114

C₄

—

254

—

115

C₄

—

254

—

116

C₄

D₄×D₁₂₇

yes

—

117

C₄

#14523247

yes

complemented

118

C₂

—

119

C₂

—

508

—

120

C₂

C₂×D₄×D₁₂₇

yes

characteristic

121

C₂

C₂×#14523247

yes

characteristic

122

C₂

#16861254

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#127×C₁₂₇	C₂
hypercenter	#127	D₁₂₇
hypocenter	C₁₂₇	#550
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#127×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂×#14523247	C₂	semidirect
2. #16861205	C₂	semidirect
3. #16861181	C₂	semidirect
4. #16861180	C₂	semidirect
5. C₄×Dic₂₅₄	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. D₄×Dic₁₂₇	C₂	semidirect
8. #127×C₁₂₇	C₂	semidirect
27. Dic₂₅₄	D₄	semidirect
52. C₄×C₁₂₇	#41	semidirect
61. C₁₂₇	#550	semidirect
65. #127	D₁₂₇	semidirect
117. C₄	#14523247	semidirect

H ≅ G/N

type

1. C₂×#14523247

C₂

semidirect

2. #16861205

C₂

semidirect

3. #16861181

C₂

semidirect

4. #16861180

C₂

semidirect

5. C₄×Dic₂₅₄

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. D₄×Dic₁₂₇

C₂

semidirect

8. #127×C₁₂₇

C₂

semidirect

27. Dic₂₅₄

D₄

semidirect

52. C₄×C₁₂₇

#41

semidirect

61. C₁₂₇

#550

semidirect

65. #127

D₁₂₇

semidirect

117. C₄

#14523247

semidirect

Subgroups — #22550546

Conjugacy classes of subgroups

Complements