Group #22550611 — subgroups

Subgroups — #22550611

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	#16861206	C₂	1	1	yes	maximal, characteristic
2	#16861198	C₂	1	2	yes	maximal, complemented
3	#16861197	C₂	1	2	yes	maximal, complemented
4	#16861194	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₀₈	C₂	1	2	yes	maximal, complemented
7	#16861171	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	C₂×D₅₀₈	C₂²	1	2	yes	—
11	C₂×D₅₀₈	C₂²	1	2	yes	—
12	C₂×D₅₀₈	C₂²	1	2	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
15	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
16	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
17	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₂₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×C₁₂₇	C₂²	1	2	yes	—
24	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₂₇	C₂³	1	4	yes	—
26	Dic₂₅₄	—	2	2	no	—
27	D₅₀₈	—	2	4	no	—
28	D₅₀₈	—	2	4	no	—
29	D₅₀₈	—	4	2	no	—
30	C₂×Dic₁₂₇	C₂³	1	2	yes	—
31	C₂×Dic₁₂₇	C₂³	1	2	yes	—
32	C₄×D₁₂₇	—	4	4	no	—
33	Q₈×C₁₂₇	—	2	2	no	—
34	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
40	C₂×D₁₂₇	—	2	4	no	—
41	C₂×D₁₂₇	—	2	4	no	—
42	C₂×D₁₂₇	—	2	4	no	—
43	C₄×C₁₂₇	—	2	2	no	—
44	C₄×C₁₂₇	—	2	2	no	—
45	C₄×C₁₂₇	—	2	1	no	—
46	C₄×C₁₂₇	#41	1	2	yes	—
47	C₄×C₁₂₇	#41	1	2	yes	—
48	Dic₁₂₇	—	2	2	no	—
49	Dic₁₂₇	—	2	2	no	—
50	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
53	D₁₂₇	—	4	4	no	—
54	C₁₂₇	#559	1	1	yes	characteristic, complemented, Sylow
55	#559	—	127	1	no	maximal, Sylow
56	#127	—	127	1	no	—
57	#124	—	127	2	no	—
58	#124	—	127	2	no	—
59	#123	—	127	4	no	—
60	C₄×Q₈	—	127	1	no	—
61	C₄×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
62	C₄×D₄	—	127	2	no	—
63	C₄×D₄	—	127	2	no	—
64	C₂×Q₈	—	127	1	no	—
65	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂×D₄	—	127	2	no	—
67	C₂×D₄	—	127	2	no	—
68	C₂×D₄	—	127	2	no	—
69	C₂²×C₄	—	127	4	no	—
70	C₄⋊C₄	—	127	2	no	—
71	C₄⋊C₄	—	127	1	no	—
72	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
74	C₂²⋊C₄	—	127	4	no	—
75	C₂²⋊C₄	—	127	4	no	—
76	C₂²⋊C₄	—	127	4	no	—
77	C₄²	—	127	2	no	—
78	C₄²	C₂×D₁₂₇	1	2	yes	—
79	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂³	—	127	4	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	254	2	no	—
83	D₄	—	508	2	no	—
84	D₄	—	254	4	no	—
85	D₄	—	254	4	no	—
86	C₂×C₄	—	127	2	no	—
87	C₂×C₄	—	127	2	no	—
88	C₂×C₄	—	508	4	no	—
89	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
90	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
91	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₂²	—	254	4	no	—
95	C₂²	—	254	4	no	—
96	C₂²	—	254	4	no	—
97	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	254	2	no	—
102	C₄	—	254	2	no	—
103	C₄	#14523248	1	2	yes	—
104	C₄	#14523249	1	2	yes	—
105	C₂	—	508	4	no	—
106	C₂	C₂×#14523248	1	1	yes	characteristic
107	C₂	C₂×#14523249	1	1	yes	characteristic
108	C₂	#16861255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861206

C₂

yes

maximal, characteristic

#16861198

C₂

yes

maximal, complemented

#16861197

C₂

yes

maximal, complemented

#16861194

C₂

yes

maximal, complemented

C₄×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

C₄×D₅₀₈

C₂

yes

maximal, complemented

#16861171

C₂

yes

maximal, complemented

C₄×Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

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

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

D₁₂₇

—

C₁₂₇

#559

yes

characteristic, complemented, Sylow

#559

—

127

maximal, Sylow

#127

—

127

—

#124

—

127

—

#124

—

127

—

#123

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

D₁₂₇

yes

characteristic, complemented

C₄×D₄

—

127

—

C₄×D₄

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

Q₈

—

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

254

—

102

C₄

—

254

—

103

C₄

#14523248

yes

—

104

C₄

#14523249

yes

—

105

C₂

—

508

—

106

C₂

C₂×#14523248

yes

characteristic

107

C₂

C₂×#14523249

yes

characteristic

108

C₂

#16861255

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	#559
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
2. #16861198	C₂	semidirect
3. #16861197	C₂	semidirect
4. #16861194	C₂	semidirect
5. C₄×Dic₂₅₄	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. #16861171	C₂	semidirect
8. C₄×Q₈×C₁₂₇	C₂	semidirect
54. C₁₂₇	#559	semidirect
61. C₄×Q₈	D₁₂₇	semidirect

H ≅ G/N

type

2. #16861198

C₂

semidirect

3. #16861197

C₂

semidirect

4. #16861194

C₂

semidirect

5. C₄×Dic₂₅₄

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. #16861171

C₂

semidirect

8. C₄×Q₈×C₁₂₇

C₂

semidirect

54. C₁₂₇

#559

semidirect

61. C₄×Q₈

D₁₂₇

semidirect

Subgroups — #22550611

Conjugacy classes of subgroups

Complements