Group #22550530 — subgroups

Subgroups — #22550530

Order 8128.

292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 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	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861193	C₂	1	4	yes	maximal, complemented
4	#16861189	C₂	1	2	yes	maximal, complemented
5	#16861183	C₂	1	2	yes	maximal, complemented
6	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Dic₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523247	—	2	8	no	—
10	C₂×Dic₂₅₄	C₂²	1	2	yes	—
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
13	C₂²×Dic₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
18	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
19	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	—	2	4	no	—
28	D₅₀₈	—	2	2	no	—
29	C₂²⋊D₁₂₇	—	2	8	no	—
30	C₂×Dic₁₂₇	—	2	8	no	—
31	C₂×Dic₁₂₇	C₂³	1	2	yes	—
32	C₂×Dic₁₂₇	C₂³	1	4	yes	—
33	C₄×D₁₂₇	—	2	4	no	—
34	C₂³×C₁₂₇	C₂³	1	2	yes	—
35	D₄×C₁₂₇	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	—	2	2	no	—
43	C₂²×C₁₂₇	C₂×D₄	1	4	yes	—
44	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₂₇	—	2	2	no	—
46	C₂×D₁₂₇	—	2	2	no	—
47	C₂×D₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	—	2	2	no	—
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₄×C₁₂₇	#41	1	2	yes	complemented
52	Dic₁₂₇	—	2	2	no	—
53	Dic₁₂₇	—	2	4	no	—
54	C₂×C₁₂₇	—	2	4	no	—
55	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
57	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₂₇	—	4	2	no	—
59	C₁₂₇	#547	1	1	yes	characteristic, complemented, Sylow
60	#547	—	127	1	no	maximal, Sylow
61	C₂×#41	—	127	2	no	—
62	C₄⋊Q₈	—	127	1	no	—
63	#123	—	127	4	no	—
64	C₂²⋊Q₈	—	127	2	no	—
65	#121	—	127	2	no	—
66	C₄×D₄	—	127	1	no	—
67	C₄×D₄	D₁₂₇	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	—	127	2	no	—
69	#41	—	254	8	no	—
70	C₂×Q₈	—	127	2	no	—
71	C₂×D₄	—	127	4	no	—
72	C₂×D₄	—	127	1	no	—
73	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂²×C₄	—	127	4	no	—
75	C₂²×C₄	—	127	2	no	—
76	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
77	C₄⋊C₄	—	127	1	no	—
78	C₄⋊C₄	—	127	2	no	—
79	C₄⋊C₄	—	127	4	no	—
80	C₄⋊C₄	—	127	2	no	—
81	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂²⋊C₄	—	127	2	no	—
83	C₂²⋊C₄	—	127	4	no	—
84	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
85	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂³	—	127	2	no	—
87	C₂³	C₂²×D₁₂₇	1	2	yes	—
88	Q₈	—	254	4	no	—
89	D₄	—	254	8	no	—
90	D₄	—	254	2	no	—
91	D₄	D₅₀₈	1	4	yes	complemented
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	127	2	no	—
94	C₂×C₄	—	127	4	no	—
95	C₂×C₄	—	254	8	no	—
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
99	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	254	2	no	—
104	C₂²	—	254	2	no	—
105	C₂²	—	254	2	no	—
106	C₂²	C₂×D₅₀₈	1	4	yes	—
107	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	254	2	no	—
111	C₄	—	254	4	no	—
112	C₄	C₂×D₅₀₈	1	2	yes	—
113	C₄	#14523247	1	2	yes	complemented
114	C₂	—	2	4	no	—
115	C₂	—	508	2	no	—
116	C₂	C₂²×D₅₀₈	1	1	yes	characteristic
117	C₂	C₂×#14523247	1	1	yes	characteristic
118	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523247

C₂

yes

maximal, complemented

#16861209

C₂

yes

maximal, characteristic, complemented

#16861193

C₂

yes

maximal, complemented

#16861189

C₂

yes

maximal, complemented

#16861183

C₂

yes

maximal, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, complemented

C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523247

—

C₂×Dic₂₅₄

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

—

D₄×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

#41

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#547

yes

characteristic, complemented, Sylow

#547

—

127

maximal, Sylow

C₂×#41

—

127

—

C₄⋊Q₈

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

C₄×D₄

—

127

—

C₄×D₄

D₁₂₇

yes

characteristic, complemented

C₂×C₄⋊C₄

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

—

Q₈

—

254

—

D₄

—

254

—

D₄

—

254

—

D₄

D₅₀₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

—

254

—

106

C₂²

C₂×D₅₀₈

yes

—

107

C₂²

C₂³×D₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

254

—

111

C₄

—

254

—

112

C₄

C₂×D₅₀₈

yes

—

113

C₄

#14523247

yes

complemented

114

C₂

—

115

C₂

—

508

—

116

C₂

C₂²×D₅₀₈

yes

characteristic

117

C₂

C₂×#14523247

yes

characteristic

118

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄×D₄×C₁₂₇	C₂
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#547
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄×D₄×C₁₂₇

C₂

hypercenter

C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14523247	C₂	semidirect
2. #16861209	C₂	semidirect
3. #16861193	C₂	semidirect
4. #16861189	C₂	semidirect
5. #16861183	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
8. C₄×D₄×C₁₂₇	C₂	semidirect
35. D₄×C₁₂₇	D₄	semidirect
51. C₄×C₁₂₇	#41	semidirect
59. C₁₂₇	#547	semidirect
67. C₄×D₄	D₁₂₇	semidirect
91. D₄	D₅₀₈	semidirect
113. C₄	#14523247	semidirect

H ≅ G/N

type

1. C₂×#14523247

C₂

semidirect

2. #16861209

C₂

semidirect

3. #16861193

C₂

semidirect

4. #16861189

C₂

semidirect

5. #16861183

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

8. C₄×D₄×C₁₂₇

C₂

semidirect

35. D₄×C₁₂₇

D₄

semidirect

51. C₄×C₁₂₇

#41

semidirect

59. C₁₂₇

#547

semidirect

67. C₄×D₄

D₁₂₇

semidirect

91. D₄

D₅₀₈

semidirect

113. C₄

#14523247

semidirect

Subgroups — #22550530

Conjugacy classes of subgroups

Complements