Group #22550529 — subgroups

Subgroups — #22550529

Order 8128.

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

normal subgroups only

This group has 292 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×#14523249	C₂	1	2	yes	maximal, complemented
2	#16861194	C₂	1	4	yes	maximal, complemented
3	#16861185	C₂	1	2	yes	maximal, complemented
4	#16861184	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523249	—	2	8	no	—
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	4	yes	—
12	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	C₂²×D₁₂₇	C₂³	1	2	yes	—
28	D₅₀₈	—	2	8	no	—
29	D₅₀₈	—	2	2	no	—
30	D₅₀₈	D₄	1	4	yes	complemented
31	C₂×Dic₁₂₇	C₂³	1	2	yes	—
32	C₂×Dic₁₂₇	C₂³	1	2	yes	—
33	C₄×D₁₂₇	—	2	8	no	—
34	C₄×D₁₂₇	—	2	4	no	—
35	C₄×D₁₂₇	—	2	4	no	—
36	Q₈×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₂₇	—	2	2	no	—
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	2	2	no	—
45	C₂×D₁₂₇	—	2	2	no	—
46	C₂×D₁₂₇	—	2	2	no	—
47	C₂×D₁₂₇	C₂×D₄	1	4	yes	—
48	C₄×C₁₂₇	—	2	4	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	2	no	—
54	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
56	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
57	D₁₂₇	—	2	4	no	—
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	D₁₂₇	1	1	yes	characteristic, complemented
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	—	127	1	no	—
68	C₂×C₄⋊C₄	—	127	2	no	—
69	#41	—	254	8	no	—
70	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
71	C₂×D₄	—	127	1	no	—
72	C₂×D₄	—	127	1	no	—
73	C₂×D₄	—	127	4	no	—
74	C₂²×C₄	—	127	2	no	—
75	C₂²×C₄	—	127	4	no	—
76	C₂²×C₄	—	127	2	no	—
77	C₄⋊C₄	—	127	4	no	—
78	C₄⋊C₄	—	127	1	no	—
79	C₄⋊C₄	—	127	1	no	—
80	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
81	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
82	C₂²⋊C₄	—	127	2	no	—
83	C₂²⋊C₄	—	127	2	no	—
84	C₂²⋊C₄	—	127	4	no	—
85	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂³	—	127	2	no	—
87	C₂³	—	127	2	no	—
88	Q₈	—	2	4	no	—
89	D₄	—	127	4	no	—
90	D₄	—	254	8	no	—
91	D₄	—	254	2	no	—
92	C₂×C₄	—	127	2	no	—
93	C₂×C₄	—	127	2	no	—
94	C₂×C₄	—	254	8	no	—
95	C₂×C₄	—	254	4	no	—
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
99	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₂²	—	127	4	no	—
102	C₂²	—	254	2	no	—
103	C₂²	—	254	2	no	—
104	C₂²	—	254	2	no	—
105	C₂²	—	254	2	no	—
106	C₂²	—	254	2	no	—
107	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	2	1	no	—
110	C₄	—	254	2	no	—
111	C₄	—	254	2	no	—
112	C₄	D₄×D₁₂₇	1	2	yes	—
113	C₄	#14523249	1	2	yes	complemented
114	C₂	—	508	2	no	—
115	C₂	—	254	4	no	—
116	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
117	C₂	C₂×#14523249	1	1	yes	characteristic
118	C₂	#16861257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523249

C₂

yes

maximal, complemented

#16861194

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

#16861184

C₂

yes

maximal, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

Q₈×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

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

#41

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#547

yes

characteristic, complemented, Sylow

#547

—

127

maximal, Sylow

C₂×#41

—

127

—

C₄⋊Q₈

D₁₂₇

yes

characteristic, complemented

#123

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

C₂×C₄⋊C₄

—

127

—

#41

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

—

127

—

Q₈

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

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

—

127

—

102

C₂²

—

254

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

—

254

—

106

C₂²

—

254

—

107

C₂²

C₂³×D₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

254

—

111

C₄

—

254

—

112

C₄

D₄×D₁₂₇

yes

—

113

C₄

#14523249

yes

complemented

114

C₂

—

508

—

115

C₂

—

254

—

116

C₂

C₂×D₄×D₁₂₇

yes

characteristic

117

C₂

C₂×#14523249

yes

characteristic

118

C₂

#16861257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₂₇	C₂
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#547
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₂₇

C₂

hypercenter

C₄⋊Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#14523249	C₂	semidirect
2. #16861194	C₂	semidirect
3. #16861185	C₂	semidirect
4. #16861184	C₂	semidirect
5. C₄×D₅₀₈	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. C₄⋊C₄×D₁₂₇	C₂	semidirect
8. C₄⋊Q₈×C₁₂₇	C₂	semidirect
30. D₅₀₈	D₄	semidirect
51. C₄×C₁₂₇	#41	semidirect
59. C₁₂₇	#547	semidirect
62. C₄⋊Q₈	D₁₂₇	semidirect
113. C₄	#14523249	semidirect

H ≅ G/N

type

1. C₂×#14523249

C₂

semidirect

2. #16861194

C₂

semidirect

3. #16861185

C₂

semidirect

4. #16861184

C₂

semidirect

5. C₄×D₅₀₈

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. C₄⋊C₄×D₁₂₇

C₂

semidirect

8. C₄⋊Q₈×C₁₂₇

C₂

semidirect

30. D₅₀₈

D₄

semidirect

51. C₄×C₁₂₇

#41

semidirect

59. C₁₂₇

#547

semidirect

62. C₄⋊Q₈

D₁₂₇

semidirect

113. C₄

#14523249

semidirect

Subgroups — #22550529

Conjugacy classes of subgroups

Complements