Group #22549793 — subgroups

Subgroups — #22549793

Order 8128.

238 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 63 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×D₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	C₂²	1	2	yes	—
7	C₂×D₅₀₈	—	2	4	no	—
8	C₂×D₅₀₈	—	2	2	no	—
9	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
10	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
11	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
12	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₂₇	—	2	2	no	—
16	C₂²×D₁₂₇	—	2	2	no	—
17	C₂²×D₁₂₇	—	2	2	no	—
18	C₂²×D₁₂₇	—	2	2	no	—
19	C₂²×D₁₂₇	—	2	2	no	—
20	C₂²×D₁₂₇	—	2	2	no	—
21	C₂²×D₁₂₇	—	2	2	no	—
22	C₂²×D₁₂₇	D₄	1	4	yes	—
23	D₅₀₈	—	2	8	no	—
24	C₂×Dic₁₂₇	—	2	2	no	—
25	C₂×Dic₁₂₇	—	2	2	no	—
26	C₂×Dic₁₂₇	—	2	2	no	—
27	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
34	C₂×C₄×C₁₂₇	D₄	1	2	yes	complemented
35	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
36	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₂₇	#41	1	1	yes	characteristic
39	C₂²×C₁₂₇	#41	1	2	yes	—
40	C₂×D₁₂₇	—	2	4	no	—
41	C₂×D₁₂₇	—	2	4	no	—
42	C₂×D₁₂₇	—	2	4	no	—
43	C₂×D₁₂₇	—	4	2	no	—
44	C₂×D₁₂₇	—	4	2	no	—
45	C₂×D₁₂₇	—	4	2	no	—
46	C₂×D₁₂₇	—	4	2	no	—
47	C₄×C₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	—	2	2	no	—
49	C₄×C₁₂₇	—	4	1	no	—
50	Dic₁₂₇	—	4	2	no	—
51	C₂×C₁₂₇	#120	1	1	yes	characteristic
52	C₂×C₁₂₇	#121	1	2	yes	—
53	C₂×C₁₂₇	#121	1	1	yes	characteristic
54	C₂×C₁₂₇	#123	1	2	yes	—
55	C₂×C₁₂₇	#124	1	1	yes	characteristic
56	D₁₂₇	—	4	4	no	—
57	C₁₂₇	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	127	1	no	maximal, Sylow
59	C₂²×D₄	—	127	1	no	—
60	C₂×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic, complemented
61	C₂×C₂²⋊C₄	—	127	2	no	—
62	C₂×C₂²⋊C₄	—	127	2	no	—
63	#95	—	127	1	no	—
64	C₂⁴	—	127	2	no	—
65	C₂×D₄	—	254	4	no	—
66	C₂×D₄	—	254	2	no	—
67	C₂²×C₄	—	127	2	no	—
68	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
71	C₄⋊C₄	—	2	2	no	—
72	C₂²⋊C₄	—	254	4	no	—
73	C₂²⋊C₄	—	254	4	no	—
74	C₂³	—	127	4	no	—
75	C₂³	—	254	2	no	—
76	C₂³	—	254	2	no	—
77	C₂³	—	254	2	no	—
78	C₂³	—	254	2	no	—
79	C₂³	—	254	2	no	—
80	C₂³	—	254	2	no	—
81	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
82	D₄	—	254	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	254	2	no	—
89	C₂×C₄	—	254	2	no	—
90	C₂×C₄	—	254	2	no	—
91	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
92	C₂×C₄	D₅₀₈	1	2	yes	complemented
93	C₂²	—	508	2	no	—
94	C₂²	—	508	2	no	—
95	C₂²	—	508	2	no	—
96	C₂²	—	508	2	no	—
97	C₂²	—	254	4	no	—
98	C₂²	—	254	4	no	—
99	C₂²	—	254	4	no	—
100	C₂²	D₄×D₁₂₇	1	2	yes	—
101	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
103	C₂²	#14523248	1	1	yes	characteristic
104	C₂²	#14523249	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	1	no	—
108	C₄	—	508	2	no	—
109	C₂	—	508	4	no	—
110	C₂	#16861177	1	1	yes	characteristic
111	C₂	#16861183	1	1	yes	characteristic
112	C₂	#16861184	1	2	yes	—
113	C₂	#16861194	1	2	yes	—
114	C₂	#16861198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₀₈

C₂

yes

maximal, characteristic

C₂×(C₂×D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

C₂×(C₂×D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

C₁₂₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂²×Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#120

yes

characteristic

C₂×C₁₂₇

#121

yes

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

#123

yes

—

C₂×C₁₂₇

#124

yes

characteristic

D₁₂₇

—

C₁₂₇

#394

yes

characteristic, complemented, Sylow

#394

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×C₄⋊C₄

D₁₂₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

#95

—

127

—

C₂⁴

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

D₅₀₈

yes

complemented

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

100

C₂²

D₄×D₁₂₇

yes

—

101

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

C₂²

C₂×D₅₀₈

yes

characteristic

103

C₂²

#14523248

yes

characteristic

104

C₂²

#14523249

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

508

—

109

C₂

—

508

—

110

C₂

#16861177

yes

characteristic

111

C₂

#16861183

yes

characteristic

112

C₂

#16861184

yes

—

113

C₂

#16861194

yes

—

114

C₂

#16861198

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator, socle	C₂³×C₁₂₇	C₂³
Fitting	C₂×C₄⋊C₄×C₁₂₇	C₂
hypercenter	C₂×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#394
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator, socle

C₂³×C₁₂₇

C₂³

Fitting

C₂×C₄⋊C₄×C₁₂₇

C₂

hypercenter

C₂×C₄⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×(C₂×D₁₂₇)⋊C₄	C₂	semidirect
3. C₂×(C₂×D₁₂₇)⋊C₄	C₂	semidirect
4. C₁₂₇⋊#95	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₂₇	C₂	semidirect
34. C₂×C₄×C₁₂₇	D₄	semidirect
57. C₁₂₇	#394	semidirect
60. C₂×C₄⋊C₄	D₁₂₇	semidirect
92. C₂×C₄	D₅₀₈	semidirect

H ≅ G/N

type

2. C₂×(C₂×D₁₂₇)⋊C₄

C₂

semidirect

3. C₂×(C₂×D₁₂₇)⋊C₄

C₂

semidirect

4. C₁₂₇⋊#95

C₂

semidirect

5. C₂×C₄⋊C₄×C₁₂₇

C₂

semidirect

34. C₂×C₄×C₁₂₇

D₄

semidirect

57. C₁₂₇

#394

semidirect

60. C₂×C₄⋊C₄

D₁₂₇

semidirect

92. C₂×C₄

D₅₀₈

semidirect

Subgroups — #22549793

Conjugacy classes of subgroups

Complements