Group #22549894 — subgroups

Subgroups — #22549894

Order 8128.

154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 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×#14523248	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861137	C₂	1	4	yes	maximal, complemented
4	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#14523248	—	2	1	no	—
6	#14523248	—	2	1	no	—
7	#14523248	C₄	1	2	yes	complemented
8	#14523248	C₂²	1	2	yes	—
9	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic, complemented
10	C₂×D₅₀₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
12	C₂×C₄×D₁₂₇	—	2	1	no	—
13	#14523232	—	2	1	no	—
14	#14523232	C₂²	1	2	yes	complemented
15	C₂×C₁₂₇⋊C₈	—	2	1	no	—
16	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₂₇	—	2	1	no	—
18	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
19	C₄²×C₁₂₇	C₂²	1	2	yes	complemented
20	C₂²×D₁₂₇	—	2	1	no	—
21	Dic₂₅₄	—	2	1	no	—
22	Dic₂₅₄	C₂×C₄	1	2	yes	—
23	D₅₀₈	—	2	1	no	—
24	D₅₀₈	C₂×C₄	1	2	yes	—
25	C₂²⋊D₁₂₇	—	2	2	no	—
26	C₂²⋊D₁₂₇	—	2	2	no	—
27	C₂×Dic₁₂₇	—	2	1	no	—
28	C₄×D₁₂₇	—	2	2	no	—
29	C₄×D₁₂₇	—	2	2	no	—
30	C₁₂₇⋊C₈	—	2	2	no	—
31	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	4	1	no	—
40	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	4	2	no	—
45	C₄×C₁₂₇	—	4	2	no	—
46	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	Dic₁₂₇	—	2	2	no	—
51	C₂×C₁₂₇	—	2	1	no	—
52	C₂×C₁₂₇	—	2	1	no	—
53	C₂×C₁₂₇	—	2	1	no	—
54	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	D₁₂₇	—	4	2	no	—
56	C₁₂₇	#421	1	1	yes	characteristic, complemented, Sylow
57	#421	—	127	1	no	maximal, Sylow
58	C₂×#41	—	127	1	no	—
59	C₂×M₁₆	—	127	1	no	—
60	#117	D₁₂₇	1	1	yes	characteristic, complemented
61	#104	—	127	4	no	—
62	#41	—	127	2	no	—
63	#41	—	127	2	no	—
64	#41	—	254	1	no	—
65	#41	—	254	1	no	—
66	C₂×Q₈	—	127	1	no	—
67	C₂×D₄	—	127	1	no	—
68	C₂×D₄	—	254	1	no	—
69	C₂²×C₄	—	254	1	no	—
70	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
71	M₁₆	—	127	2	no	—
72	M₁₆	—	254	1	no	—
73	C₂×C₈	—	254	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₂²⋊C₄	—	2	1	no	—
76	C₄²	C₂×D₁₂₇	1	2	yes	complemented
77	C₂³	—	254	1	no	—
78	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
79	Q₈	—	127	2	no	—
80	Q₈	—	254	1	no	—
81	D₄	—	127	2	no	—
82	D₄	—	254	2	no	—
83	D₄	—	254	2	no	—
84	D₄	—	254	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	254	1	no	—
87	C₂×C₄	—	254	2	no	—
88	C₂×C₄	—	254	2	no	—
89	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
92	C₂×C₄	D₅₀₈	1	1	yes	characteristic
93	C₂×C₄	D₅₀₈	1	1	yes	characteristic
94	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₈	—	254	2	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	508	2	no	—
98	C₂²	—	254	2	no	—
99	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
100	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
101	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	254	2	no	—
104	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
105	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
106	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
107	C₄	C₂×D₅₀₈	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	508	2	no	—
112	C₂	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

#16861137

C₂

yes

maximal, complemented

#117×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

—

#14523248

C₄

yes

complemented

#14523248

C₂²

yes

—

C₂×Dic₂₅₄

C₄

yes

characteristic, complemented

C₂×D₅₀₈

C₄

yes

characteristic, complemented

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523232

—

#14523232

C₂²

yes

complemented

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

C₂×C₄

yes

—

D₅₀₈

—

D₅₀₈

C₂×C₄

yes

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#421

yes

characteristic, complemented, Sylow

#421

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

#117

D₁₂₇

yes

characteristic, complemented

#104

—

127

—

#41

—

127

—

#41

—

127

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

127

—

M₁₆

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₁₂₇

yes

complemented

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

127

—

Q₈

—

254

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

D₅₀₈

yes

characteristic

C₂×C₄

D₅₀₈

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₂²

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

(C₂×D₁₂₇)⋊C₄

yes

characteristic

100

C₂²

(C₂×D₁₂₇)⋊C₄

yes

characteristic

101

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

C₄

—

103

C₄

—

254

—

104

C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

105

C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

106

C₄

C₂×C₄×D₁₂₇

yes

characteristic

107

C₄

C₂×D₅₀₈

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

111

C₂

—

508

—

112

C₂

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#117×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₂²⋊C₄
hypercenter	#117	D₁₂₇
hypocenter	C₁₂₇	#421
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₄

C₂×D₅₀₈

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#117×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂×C₂²⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
2. C₂×#14523232	C₂	semidirect
3. #16861137	C₂	semidirect
4. #117×C₁₂₇	C₂	semidirect
7. #14523248	C₄	semidirect
9. C₂×Dic₂₅₄	C₄	semidirect
10. C₂×D₅₀₈	C₄	semidirect
14. #14523232	C₂²	semidirect
19. C₄²×C₁₂₇	C₂²	semidirect
56. C₁₂₇	#421	semidirect
60. #117	D₁₂₇	semidirect
76. C₄²	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523232

C₂

semidirect

3. #16861137

C₂

semidirect

4. #117×C₁₂₇

C₂

semidirect

7. #14523248

C₄

semidirect

9. C₂×Dic₂₅₄

C₄

semidirect

10. C₂×D₅₀₈

C₄

semidirect

14. #14523232

C₂²

semidirect

19. C₄²×C₁₂₇

C₂²

semidirect

56. C₁₂₇

#421

semidirect

60. #117

D₁₂₇

semidirect

76. C₄²

C₂×D₁₂₇

semidirect

Subgroups — #22549894

Conjugacy classes of subgroups

Complements