Group #22550591 — subgroups

Subgroups — #22550591

Order 8128.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861195	C₂	1	4	yes	maximal, complemented
3	#16861183	C₂	1	2	yes	maximal, complemented
4	#16861180	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861168	C₂	1	2	yes	maximal, complemented
7	#16861167	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₅₄	C₂²	1	2	yes	—
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
17	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₂₇	C₂²	1	4	yes	—
20	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₂₇	C₂³	1	2	yes	—
26	Dic₂₅₄	—	4	2	no	—
27	D₅₀₈	—	2	2	no	—
28	C₂²⋊D₁₂₇	—	4	4	no	—
29	C₂×Dic₁₂₇	C₂³	1	2	yes	—
30	C₂×Dic₁₂₇	C₂³	1	4	yes	—
31	C₄×D₁₂₇	—	2	4	no	—
32	C₂³×C₁₂₇	C₂³	1	2	yes	—
33	D₄×C₁₂₇	—	2	2	no	—
34	C₂×C₄×C₁₂₇	—	2	4	no	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	—	2	2	no	—
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	2	2	no	—
45	C₂×D₁₂₇	—	2	2	no	—
46	C₄×C₁₂₇	—	2	2	no	—
47	C₄×C₁₂₇	—	2	1	no	—
48	C₄×C₁₂₇	#41	1	2	yes	—
49	C₄×C₁₂₇	#41	1	2	yes	—
50	Dic₁₂₇	—	2	2	no	—
51	Dic₁₂₇	—	2	4	no	—
52	C₂×C₁₂₇	—	4	2	no	—
53	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₂₇	—	4	2	no	—
57	C₁₂₇	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	127	1	no	maximal, Sylow
59	C₄⋊Q₈	—	127	1	no	—
60	#124	—	127	4	no	—
61	#121	—	127	2	no	—
62	#121	—	127	2	no	—
63	C₄×D₄	—	127	1	no	—
64	C₄×D₄	D₁₂₇	1	1	yes	characteristic, complemented
65	#117	—	127	2	no	—
66	#117	—	127	2	no	—
67	C₂×Q₈	—	127	2	no	—
68	C₂×D₄	—	127	4	no	—
69	C₂×D₄	—	127	1	no	—
70	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
71	C₂²×C₄	—	127	2	no	—
72	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
73	C₄⋊C₄	—	127	1	no	—
74	C₄⋊C₄	—	127	2	no	—
75	C₄⋊C₄	—	127	2	no	—
76	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂²⋊C₄	—	127	4	no	—
78	C₂²⋊C₄	—	127	4	no	—
79	C₂²⋊C₄	—	127	2	no	—
80	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
81	C₄²	—	127	4	no	—
82	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
83	C₂³	—	127	2	no	—
84	C₂³	C₂²×D₁₂₇	1	2	yes	—
85	Q₈	—	508	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	508	4	no	—
88	D₄	—	254	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	127	2	no	—
91	C₂×C₄	—	127	4	no	—
92	C₂×C₄	—	254	4	no	—
93	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
94	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	254	2	no	—
101	C₂²	—	254	2	no	—
102	C₂²	—	254	2	no	—
103	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	254	2	no	—
107	C₄	—	254	4	no	—
108	C₄	#14523247	1	2	yes	—
109	C₄	#14523248	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	508	2	no	—
112	C₂	C₂×#14523247	1	1	yes	characteristic
113	C₂	C₂×#14523248	1	1	yes	characteristic
114	C₂	#16861255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861209

C₂

yes

maximal, characteristic, complemented

#16861195

C₂

yes

maximal, complemented

#16861183

C₂

yes

maximal, complemented

#16861180

C₂

yes

maximal, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

#16861168

C₂

yes

maximal, complemented

#16861167

C₂

yes

maximal, complemented

C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂²⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

—

D₄×C₁₂₇

—

C₂×C₄×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

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

D₁₂₇

—

C₁₂₇

#555

yes

characteristic, complemented, Sylow

#555

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

#124

—

127

—

#121

—

127

—

#121

—

127

—

C₄×D₄

—

127

—

C₄×D₄

D₁₂₇

yes

characteristic, complemented

#117

—

127

—

#117

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

—

Q₈

—

508

—

D₄

—

D₄

—

508

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

254

—

101

C₂²

—

254

—

102

C₂²

—

254

—

103

C₂²

C₂³×D₁₂₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

254

—

107

C₄

—

254

—

108

C₄

#14523247

yes

—

109

C₄

#14523248

yes

—

110

C₂

—

111

C₂

—

508

—

112

C₂

C₂×#14523247

yes

characteristic

113

C₂

C₂×#14523248

yes

characteristic

114

C₂

#16861255

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄×D₄×C₁₂₇	C₂
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#555
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄×D₄×C₁₂₇

C₂

hypercenter

C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #16861209	C₂	semidirect
2. #16861195	C₂	semidirect
3. #16861183	C₂	semidirect
4. #16861180	C₂	semidirect
5. C₄×D₅₀₈	C₂	semidirect
6. #16861168	C₂	semidirect
7. #16861167	C₂	semidirect
8. C₄×D₄×C₁₂₇	C₂	semidirect
57. C₁₂₇	#555	semidirect
64. C₄×D₄	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861209

C₂

semidirect

2. #16861195

C₂

semidirect

3. #16861183

C₂

semidirect

4. #16861180

C₂

semidirect

5. C₄×D₅₀₈

C₂

semidirect

6. #16861168

C₂

semidirect

7. #16861167

C₂

semidirect

8. C₄×D₄×C₁₂₇

C₂

semidirect

57. C₁₂₇

#555

semidirect

64. C₄×D₄

D₁₂₇

semidirect

Subgroups — #22550591

Conjugacy classes of subgroups

Complements