Group #22550103 — subgroups

Subgroups — #22550103

Order 8128.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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×#14523236	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861188	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861140	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	#14523236	—	2	2	no	—
10	C₂×C₁₂₇⋊C₈	—	2	2	no	—
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
14	Dic₁₂₇⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₂₇	—	2	1	no	—
16	C₂×D₄×C₁₂₇	—	2	1	no	—
17	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
21	Dic₂₅₄	—	2	1	no	—
22	Dic₂₅₄	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₁₂₇⋊C₈	—	2	1	no	—
26	C₁₂₇⋊C₈	D₄	1	2	yes	complemented
27	C₂³×C₁₂₇	—	2	1	no	—
28	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
29	D₄×C₁₂₇	—	2	1	no	—
30	D₄×C₁₂₇	—	2	1	no	—
31	D₄×C₁₂₇	—	4	2	no	—
32	C₂×C₄×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₁₂₇	—	2	1	no	—
38	C₂²×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	4	1	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	SD₁₆	1	2	yes	complemented
43	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₂₇	—	2	1	no	—
45	C₄×C₁₂₇	—	4	1	no	—
46	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₂₇	#41	1	1	yes	characteristic
48	Dic₁₂₇	—	4	1	no	—
49	Dic₁₂₇	—	4	1	no	—
50	C₂×C₁₂₇	—	2	2	no	—
51	C₂×C₁₂₇	—	8	1	no	—
52	C₂×C₁₂₇	#121	1	1	yes	characteristic
53	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₂₇	#135	1	1	yes	characteristic
55	C₁₂₇	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	127	1	no	maximal, Sylow
57	C₂×SD₁₆	—	127	1	no	—
58	C₂²×C₈	—	127	1	no	—
59	C₂²⋊Q₈	—	127	1	no	—
60	#121	D₁₂₇	1	1	yes	characteristic
61	#106	—	127	1	no	—
62	#103	—	127	1	no	—
63	#102	—	127	1	no	—
64	C₂×Q₈	—	127	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	SD₁₆	—	254	2	no	—
69	C₂×C₈	—	127	1	no	—
70	C₂×C₈	—	127	1	no	—
71	C₂×C₈	—	254	2	no	—
72	C₄⋊C₄	—	127	1	no	—
73	C₄⋊C₄	—	254	1	no	—
74	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	2	1	no	—
76	C₂²⋊C₄	—	254	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
79	Q₈	—	254	1	no	—
80	Q₈	—	254	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	254	1	no	—
88	C₂×C₄	—	254	1	no	—
89	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₈	—	127	2	no	—
92	C₈	—	254	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	#14523236	1	2	yes	complemented
99	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	508	1	no	—
103	C₄	—	508	1	no	—
104	C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₄	#14523247	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	8	1	no	—
108	C₂	#16861181	1	1	yes	characteristic
109	C₂	C₂×#14523236	1	1	yes	characteristic
110	C₂	#16861229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523236

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, characteristic, complemented

#16861140

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

#121×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

#14523236

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

D₄

yes

complemented

C₂³×C₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#465

—

127

maximal, Sylow

C₂×SD₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²⋊Q₈

—

127

—

#121

D₁₂₇

yes

characteristic

#106

—

127

—

#103

—

127

—

#102

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

254

—

C₂³

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

254

—

Q₈

—

254

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523236

yes

complemented

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

508

—

103

C₄

—

508

—

104

C₄

D₄×D₁₂₇

yes

characteristic

105

C₄

#14523247

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#16861181

yes

characteristic

109

C₂

C₂×#14523236

yes

characteristic

110

C₂

#16861229

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#121×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#121	D₁₂₇
hypocenter	C₁₂₇	#465
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#121×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂×#14523236	C₂	semidirect
2. C₂²×C₁₂₇⋊C₈	C₂	semidirect
3. #16861188	C₂	semidirect
4. #16861140	C₂	semidirect
5. #16861132	C₂	semidirect
6. #16861130	C₂	semidirect
26. C₁₂₇⋊C₈	D₄	semidirect
42. C₂²×C₁₂₇	SD₁₆	semidirect
55. C₁₂₇	#465	semidirect
98. C₂²	#14523236	semidirect

H ≅ G/N

type

1. C₂×#14523236

C₂

semidirect

2. C₂²×C₁₂₇⋊C₈

C₂

semidirect

3. #16861188

C₂

semidirect

4. #16861140

C₂

semidirect

5. #16861132

C₂

semidirect

6. #16861130

C₂

semidirect

26. C₁₂₇⋊C₈

D₄

semidirect

42. C₂²×C₁₂₇

SD₁₆

semidirect

55. C₁₂₇

#465

semidirect

98. C₂²

#14523236

semidirect

Subgroups — #22550103

Conjugacy classes of subgroups

Complements