Group #22550349 — subgroups

Subgroups — #22550349

Order 8128.

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

normal subgroups only

This group has 338 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	C₂²×C₂²⋊D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊D₁₂₇	C₂	1	4	yes	maximal, complemented
3	(C₂²⋊D₁₂₇)⋊C₄	C₂	1	4	yes	maximal, complemented
4	#16861165	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₄×D₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₂₇	C₄	1	8	yes	complemented
10	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
12	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₂₇	C₂²	1	2	yes	complemented
14	C₂×C₄×D₁₂₇	C₂²	1	4	yes	complemented
15	Dic₁₂₇⋊C₄	C₂²	1	4	yes	complemented
16	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
17	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	complemented
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
19	C₄×Dic₁₂₇	C₂²	1	4	yes	complemented
20	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	complemented
22	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	complemented
24	C₂²×D₁₂₇	—	4	2	no	—
25	C₂²×D₁₂₇	C₂×C₄	1	4	yes	complemented
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	C₂²⋊D₁₂₇	C₂×C₄	1	8	yes	complemented
28	C₂²⋊D₁₂₇	C₂×C₄	1	8	yes	complemented
29	C₂×Dic₁₂₇	—	4	2	no	—
30	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	complemented
31	C₂×Dic₁₂₇	C₂³	1	4	yes	—
32	C₂×Dic₁₂₇	C₂³	1	2	yes	—
33	C₄×D₁₂₇	—	4	4	no	—
34	C₂³×C₁₂₇	—	4	2	no	—
35	C₂³×C₁₂₇	C₂×C₄	1	4	yes	complemented
36	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	—	2	2	no	—
43	C₂²×C₁₂₇	—	2	2	no	—
44	C₂²×C₁₂₇	—	4	4	no	—
45	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
49	C₂×D₁₂₇	—	2	4	no	—
50	C₂×D₁₂₇	—	4	4	no	—
51	C₂×D₁₂₇	C₂²×C₄	1	4	yes	—
52	C₄×C₁₂₇	—	2	4	no	—
53	Dic₁₂₇	—	2	4	no	—
54	Dic₁₂₇	C₂²×C₄	1	4	yes	—
55	C₂×C₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	—	2	2	no	—
57	C₂×C₁₂₇	—	2	2	no	—
58	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
59	C₂×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
60	D₁₂₇	—	2	4	no	—
61	C₁₂₇	#518	1	1	yes	characteristic, complemented, Sylow
62	#518	—	127	1	no	maximal, Sylow
63	C₂²×D₄	—	127	1	no	—
64	C₄×D₄	—	127	4	no	—
65	C₄×D₄	—	127	4	no	—
66	#117	—	127	2	no	—
67	C₂×C₂²⋊C₄	—	127	1	no	—
68	C₂×C₂²⋊C₄	—	127	2	no	—
69	C₂×C₂²⋊C₄	D₁₂₇	1	1	yes	characteristic, complemented
70	C₂⁴	—	127	1	no	—
71	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂×D₄	—	127	8	no	—
73	C₂×D₄	—	127	2	no	—
74	C₂×D₄	—	127	2	no	—
75	C₂²×C₄	—	127	1	no	—
76	C₂²×C₄	—	127	2	no	—
77	C₂²×C₄	—	127	4	no	—
78	C₂²×C₄	C₂×D₁₂₇	1	2	yes	complemented
79	C₄⋊C₄	—	127	4	no	—
80	C₂²⋊C₄	—	127	2	no	—
81	C₂²⋊C₄	—	127	2	no	—
82	C₂²⋊C₄	—	127	4	no	—
83	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
84	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	complemented
85	C₄²	—	127	4	no	—
86	C₂³	—	4	2	no	—
87	C₂³	—	127	4	no	—
88	C₂³	—	127	2	no	—
89	C₂³	—	508	2	no	—
90	C₂³	C₄×D₁₂₇	1	4	yes	complemented
91	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
94	D₄	—	127	8	no	—
95	D₄	—	127	8	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	127	4	no	—
98	C₂×C₄	—	127	2	no	—
99	C₂×C₄	—	127	4	no	—
100	C₂×C₄	—	508	4	no	—
101	C₂×C₄	—	508	2	no	—
102	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	—	127	4	no	—
108	C₂²	—	508	4	no	—
109	C₂²	—	254	4	no	—
110	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
111	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
112	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
113	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	127	4	no	—
116	C₄	—	254	4	no	—
117	C₂	—	2	2	no	—
118	C₂	—	2	2	no	—
119	C₂	—	2	2	no	—
120	C₂	—	254	4	no	—
121	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
122	C₂	#16861254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₂₇

C₂

yes

maximal, characteristic

C₄×C₂²⋊D₁₂₇

C₂

yes

maximal, complemented

(C₂²⋊D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

#16861165

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₄×D₁₂₇

C₂

yes

maximal, complemented

C₂×C₂²⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

complemented

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

complemented

C₂×C₄×D₁₂₇

C₂²

yes

complemented

Dic₁₂₇⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

C₄×Dic₁₂₇

C₂²

yes

complemented

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₂²⋊C₄×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

complemented

C₂²×D₁₂₇

C₂³

yes

—

C₂²⋊D₁₂₇

C₂×C₄

yes

complemented

C₂²⋊D₁₂₇

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

complemented

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

—

D₁₂₇

—

C₁₂₇

#518

yes

characteristic, complemented, Sylow

#518

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

127

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

complemented

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

C₂×D₁₂₇

yes

complemented

C₄²

—

127

—

C₂³

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

508

—

C₂³

C₄×D₁₂₇

yes

complemented

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

—

127

—

D₄

—

127

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

100

C₂×C₄

—

508

—

101

C₂×C₄

—

508

—

102

C₂×C₄

C₂²×D₁₂₇

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

127

—

108

C₂²

—

508

—

109

C₂²

—

254

—

110

C₂²

C₂×C₄×D₁₂₇

yes

—

111

C₂²

C₂×C₄×D₁₂₇

yes

—

112

C₂²

C₂×C₄×D₁₂₇

yes

—

113

C₂²

C₂³×D₁₂₇

yes

characteristic

114

C₄

—

115

C₄

—

127

—

116

C₄

—

254

—

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

—

254

—

121

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

122

C₂

#16861254

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₄×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂⁴
hypercenter	C₂×C₂²⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#518
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator

C₂×C₁₂₇

C₂³×C₄

Fitting

C₂×C₂²⋊C₄×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂⁴

hypercenter

C₂×C₂²⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
2. C₄×C₂²⋊D₁₂₇	C₂	semidirect
3. (C₂²⋊D₁₂₇)⋊C₄	C₂	semidirect
4. #16861165	C₂	semidirect
5. C₂×C₂²⋊Dic₁₂₇	C₂	semidirect
6. C₂²⋊C₄×D₁₂₇	C₂	semidirect
7. C₂×C₂²⋊C₄×C₁₂₇	C₂	semidirect
9. C₂×C₂²⋊D₁₂₇	C₄	semidirect
13. C₂²×Dic₁₂₇	C₂²	semidirect
14. C₂×C₄×D₁₂₇	C₂²	semidirect
15. Dic₁₂₇⋊C₄	C₂²	semidirect
17. C₂²⋊Dic₁₂₇	C₂²	semidirect
18. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
19. C₄×Dic₁₂₇	C₂²	semidirect
21. C₂²×C₄×C₁₂₇	C₂²	semidirect
23. C₂²⋊C₄×C₁₂₇	C₂²	semidirect
25. C₂²×D₁₂₇	C₂×C₄	semidirect
27. C₂²⋊D₁₂₇	C₂×C₄	semidirect
28. C₂²⋊D₁₂₇	C₂×C₄	semidirect
30. C₂×Dic₁₂₇	C₂×C₄	semidirect
35. C₂³×C₁₂₇	C₂×C₄	semidirect
61. C₁₂₇	#518	semidirect
69. C₂×C₂²⋊C₄	D₁₂₇	semidirect
78. C₂²×C₄	C₂×D₁₂₇	semidirect
84. C₂²⋊C₄	C₂×D₁₂₇	semidirect
90. C₂³	C₄×D₁₂₇	semidirect

H ≅ G/N

type

2. C₄×C₂²⋊D₁₂₇

C₂

semidirect

3. (C₂²⋊D₁₂₇)⋊C₄

C₂

semidirect

4. #16861165

C₂

semidirect

5. C₂×C₂²⋊Dic₁₂₇

C₂

semidirect

6. C₂²⋊C₄×D₁₂₇

C₂

semidirect

7. C₂×C₂²⋊C₄×C₁₂₇

C₂

semidirect

9. C₂×C₂²⋊D₁₂₇

C₄

semidirect

13. C₂²×Dic₁₂₇

C₂²

semidirect

14. C₂×C₄×D₁₂₇

C₂²

semidirect

15. Dic₁₂₇⋊C₄

C₂²

semidirect

17. C₂²⋊Dic₁₂₇

C₂²

semidirect

18. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

19. C₄×Dic₁₂₇

C₂²

semidirect

21. C₂²×C₄×C₁₂₇

C₂²

semidirect

23. C₂²⋊C₄×C₁₂₇

C₂²

semidirect

25. C₂²×D₁₂₇

C₂×C₄

semidirect

27. C₂²⋊D₁₂₇

C₂×C₄

semidirect

28. C₂²⋊D₁₂₇

C₂×C₄

semidirect

30. C₂×Dic₁₂₇

C₂×C₄

semidirect

35. C₂³×C₁₂₇

C₂×C₄

semidirect

61. C₁₂₇

#518

semidirect

69. C₂×C₂²⋊C₄

D₁₂₇

semidirect

78. C₂²×C₄

C₂×D₁₂₇

semidirect

84. C₂²⋊C₄

C₂×D₁₂₇

semidirect

90. C₂³

C₄×D₁₂₇

semidirect

Subgroups — #22550349

Conjugacy classes of subgroups

Complements