Group #22549876 — subgroups

Subgroups — #22549876

Order 8128.

162 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14523249	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861133	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861131	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	—	2	2	no	—
9	#14523249	C₄	1	4	yes	complemented
10	C₂×D₅₀₈	—	2	1	no	—
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₂₇	—	2	1	no	—
13	C₂×C₄×D₁₂₇	—	2	1	no	—
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	#14523231	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	—	2	1	no	—
19	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²×D₁₂₇	—	2	1	no	—
23	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
24	D₅₀₈	—	2	4	no	—
25	D₅₀₈	—	2	1	no	—
26	D₅₀₈	C₂×C₄	1	2	yes	complemented
27	C₂×Dic₁₂₇	—	2	1	no	—
28	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
29	C₄×D₁₂₇	—	2	4	no	—
30	C₄×D₁₂₇	—	4	2	no	—
31	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₂₇	D₄	1	2	yes	—
33	C₁₂₇⋊C₈	—	2	1	no	—
34	Q₈×C₁₂₇	—	2	1	no	—
35	Q₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	—	2	1	no	—
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₈×C₁₂₇	—	2	1	no	—
40	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₂₇	—	2	1	no	—
42	C₂×D₁₂₇	—	2	1	no	—
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	4	1	no	—
45	C₂×D₁₂₇	—	4	1	no	—
46	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
47	C₄×C₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	—	4	1	no	—
49	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
50	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	Dic₁₂₇	—	4	1	no	—
52	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₁₂₇	#137	1	1	yes	characteristic
56	D₁₂₇	—	2	2	no	—
57	D₁₂₇	—	4	2	no	—
58	C₁₂₇	#417	1	1	yes	characteristic, complemented, Sylow
59	#417	—	127	1	no	maximal, Sylow
60	C₂×#41	—	127	1	no	—
61	C₂×M₁₆	—	127	1	no	—
62	C₂×C₄⋊C₄	—	127	1	no	—
63	#103	—	127	1	no	—
64	#103	D₁₂₇	1	1	yes	characteristic, complemented
65	#102	—	127	1	no	—
66	#102	—	127	1	no	—
67	#41	—	127	4	no	—
68	#41	—	254	2	no	—
69	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂×D₄	—	127	1	no	—
71	C₂×D₄	—	254	1	no	—
72	C₂²×C₄	—	127	1	no	—
73	C₂²×C₄	—	254	1	no	—
74	C₂²×C₄	—	254	1	no	—
75	M₁₆	—	254	2	no	—
76	C₂×C₈	—	127	1	no	—
77	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
78	C₄⋊C₄	—	127	1	no	—
79	C₄⋊C₄	—	254	1	no	—
80	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂³	—	127	1	no	—
82	C₂³	—	254	1	no	—
83	Q₈	—	2	1	no	—
84	Q₈	C₄×D₁₂₇	1	2	yes	complemented
85	D₄	—	127	2	no	—
86	D₄	—	254	4	no	—
87	D₄	—	254	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	127	1	no	—
91	C₂×C₄	—	127	2	no	—
92	C₂×C₄	—	127	2	no	—
93	C₂×C₄	—	508	2	no	—
94	C₂×C₄	—	254	4	no	—
95	C₂×C₄	—	254	1	no	—
96	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	254	1	no	—
99	C₂²	—	127	2	no	—
100	C₂²	—	508	1	no	—
101	C₂²	—	508	1	no	—
102	C₂²	—	254	2	no	—
103	C₂²	—	254	1	no	—
104	C₂²	—	254	1	no	—
105	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	4	1	no	—
108	C₄	—	127	2	no	—
109	C₄	—	508	1	no	—
110	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₄	D₄×D₁₂₇	1	1	yes	characteristic
112	C₂	—	508	2	no	—
113	C₂	—	254	2	no	—
114	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
115	C₂	#16861236	1	1	yes	characteristic
116	C₂	#16861241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523249

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861133

C₂

yes

maximal, characteristic, complemented

#16861131

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

#103×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

#14523249

C₄

yes

complemented

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

characteristic

D₅₀₈

—

D₅₀₈

—

D₅₀₈

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#417

yes

characteristic, complemented, Sylow

#417

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂×C₄⋊C₄

—

127

—

#103

—

127

—

#103

D₁₂₇

yes

characteristic, complemented

#102

—

127

—

#102

—

127

—

#41

—

127

—

#41

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

—

254

—

Q₈

—

Q₈

C₄×D₁₂₇

yes

complemented

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

127

—

100

C₂²

—

508

—

101

C₂²

—

508

—

102

C₂²

—

254

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

D₄×D₁₂₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

127

—

109

C₄

—

508

—

110

C₄

C₂×C₄×D₁₂₇

yes

characteristic

111

C₄

D₄×D₁₂₇

yes

characteristic

112

C₂

—

508

—

113

C₂

—

254

—

114

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

115

C₂

#16861236

yes

characteristic

116

C₂

#16861241

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#103×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#103	D₁₂₇
hypocenter	C₁₂₇	#417
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#103×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
2. C₂×#14523231	C₂	semidirect
3. C₄⋊C₄×D₁₂₇	C₂	semidirect
4. #16861133	C₂	semidirect
5. #16861131	C₂	semidirect
6. #16861129	C₂	semidirect
7. #103×C₁₂₇	C₂	semidirect
9. #14523249	C₄	semidirect
26. D₅₀₈	C₂×C₄	semidirect
35. Q₈×C₁₂₇	C₂×C₄	semidirect
58. C₁₂₇	#417	semidirect
64. #103	D₁₂₇	semidirect
84. Q₈	C₄×D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523231

C₂

semidirect

3. C₄⋊C₄×D₁₂₇

C₂

semidirect

4. #16861133

C₂

semidirect

5. #16861131

C₂

semidirect

6. #16861129

C₂

semidirect

7. #103×C₁₂₇

C₂

semidirect

9. #14523249

C₄

semidirect

26. D₅₀₈

C₂×C₄

semidirect

35. Q₈×C₁₂₇

C₂×C₄

semidirect

58. C₁₂₇

#417

semidirect

64. #103

D₁₂₇

semidirect

84. Q₈

C₄×D₁₂₇

semidirect

Subgroups — #22549876

Conjugacy classes of subgroups

Complements