Group #22550357 — subgroups

Subgroups — #22550357

Order 8128.

294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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	#16861174	C₂	1	2	yes	maximal, complemented
3	#16861171	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₁₂₇	C₂	1	4	yes	maximal, complemented
5	#16861168	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861167	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523248	C₄	1	8	yes	complemented
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
19	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₁₂₇	C₂²	1	4	yes	—
21	C₄×Dic₁₂₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	C₂×C₄	1	4	yes	complemented
28	D₅₀₈	C₂×C₄	1	4	yes	complemented
29	C₂²⋊D₁₂₇	C₂×C₄	1	8	yes	complemented
30	C₂×Dic₁₂₇	C₂³	1	2	yes	—
31	C₂×Dic₁₂₇	C₂³	1	2	yes	—
32	C₂×Dic₁₂₇	C₂³	1	2	yes	—
33	C₄×D₁₂₇	—	4	4	no	—
34	C₄×D₁₂₇	C₂×C₄	1	8	yes	complemented
35	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	4	2	no	—
37	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
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	—	2	1	no	—
42	C₂²×C₁₂₇	—	2	1	no	—
43	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₂₇	—	2	2	no	—
46	C₂×D₁₂₇	—	2	2	no	—
47	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
48	C₄×C₁₂₇	—	2	4	no	—
49	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
51	Dic₁₂₇	—	2	2	no	—
52	Dic₁₂₇	—	2	2	no	—
53	Dic₁₂₇	C₂²×C₄	1	4	yes	—
54	C₂×C₁₂₇	—	2	2	no	—
55	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₂₇	—	2	4	no	—
59	C₁₂₇	#520	1	1	yes	characteristic, complemented, Sylow
60	#520	—	127	1	no	maximal, Sylow
61	C₂×#41	—	127	1	no	—
62	C₄×Q₈	—	127	2	no	—
63	C₄×D₄	—	127	4	no	—
64	C₄×D₄	—	127	2	no	—
65	#117	—	127	1	no	—
66	#117	—	127	2	no	—
67	C₂×C₄⋊C₄	—	127	2	no	—
68	C₂×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic, complemented
69	#41	—	127	8	no	—
70	C₂×Q₈	—	127	1	no	—
71	C₂×D₄	—	127	2	no	—
72	C₂×D₄	—	127	1	no	—
73	C₂²×C₄	—	127	2	no	—
74	C₂²×C₄	—	127	4	no	—
75	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
76	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
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	4	no	—
84	C₄²	—	127	4	no	—
85	C₄²	—	127	2	no	—
86	C₂³	—	127	2	no	—
87	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
88	Q₈	—	127	4	no	—
89	D₄	—	127	8	no	—
90	D₄	—	127	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	127	2	no	—
93	C₂×C₄	—	127	8	no	—
94	C₂×C₄	—	127	2	no	—
95	C₂×C₄	—	127	2	no	—
96	C₂×C₄	—	508	4	no	—
97	C₂×C₄	C₄×D₁₂₇	1	4	yes	complemented
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	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	127	4	no	—
104	C₂²	—	254	2	no	—
105	C₂²	—	254	2	no	—
106	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
107	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	127	4	no	—
110	C₄	—	254	2	no	—
111	C₄	—	254	2	no	—
112	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
113	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	—	254	4	no	—
116	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
117	C₂	#16861254	1	1	yes	characteristic
118	C₂	#16861257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

#16861174

C₂

yes

maximal, complemented

#16861171

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₂₇

C₂

yes

maximal, complemented

#16861168

C₂

yes

maximal, characteristic, complemented

#16861167

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

C₄

yes

complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊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

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

C₂×C₄

yes

complemented

D₅₀₈

C₂×C₄

yes

complemented

C₂²⋊D₁₂₇

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×C₄

yes

complemented

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#520

yes

characteristic, complemented, Sylow

#520

—

127

maximal, Sylow

C₂×#41

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

#117

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

D₁₂₇

yes

characteristic, complemented

#41

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

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

—

C₄²

—

127

—

C₄²

—

127

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

127

—

D₄

—

127

—

D₄

—

127

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

C₄×D₁₂₇

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

127

—

104

C₂²

—

254

—

105

C₂²

—

254

—

106

C₂²

C₂×C₄×D₁₂₇

yes

—

107

C₂²

C₂³×D₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

127

—

110

C₄

—

254

—

111

C₄

—

254

—

112

C₄

C₂×C₄×D₁₂₇

yes

—

113

C₄

C₂×C₄×D₁₂₇

yes

—

114

C₂

—

115

C₂

—

254

—

116

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

117

C₂

#16861254

yes

characteristic

118

C₂

#16861257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂⁴
hypercenter	C₂×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#520
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Fitting

C₂×C₄⋊C₄×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂⁴

hypercenter

C₂×C₄⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #16861174	C₂	semidirect
3. #16861171	C₂	semidirect
4. C₄×C₂²⋊D₁₂₇	C₂	semidirect
5. #16861168	C₂	semidirect
6. #16861167	C₂	semidirect
7. C₄⋊C₄×D₁₂₇	C₂	semidirect
8. C₂×C₄⋊C₄×C₁₂₇	C₂	semidirect
9. #14523248	C₄	semidirect
27. Dic₂₅₄	C₂×C₄	semidirect
28. D₅₀₈	C₂×C₄	semidirect
29. C₂²⋊D₁₂₇	C₂×C₄	semidirect
34. C₄×D₁₂₇	C₂×C₄	semidirect
37. C₂×C₄×C₁₂₇	C₂×C₄	semidirect
59. C₁₂₇	#520	semidirect
68. C₂×C₄⋊C₄	D₁₂₇	semidirect
97. C₂×C₄	C₄×D₁₂₇	semidirect

H ≅ G/N

type

2. #16861174

C₂

semidirect

3. #16861171

C₂

semidirect

4. C₄×C₂²⋊D₁₂₇

C₂

semidirect

5. #16861168

C₂

semidirect

6. #16861167

C₂

semidirect

7. C₄⋊C₄×D₁₂₇

C₂

semidirect

8. C₂×C₄⋊C₄×C₁₂₇

C₂

semidirect

9. #14523248

C₄

semidirect

27. Dic₂₅₄

C₂×C₄

semidirect

28. D₅₀₈

C₂×C₄

semidirect

29. C₂²⋊D₁₂₇

C₂×C₄

semidirect

34. C₄×D₁₂₇

C₂×C₄

semidirect

37. C₂×C₄×C₁₂₇

C₂×C₄

semidirect

59. C₁₂₇

#520

semidirect

68. C₂×C₄⋊C₄

D₁₂₇

semidirect

97. C₂×C₄

C₄×D₁₂₇

semidirect

Subgroups — #22550357

Conjugacy classes of subgroups

Complements