Group #30640449 — subgroups

Subgroups — #30640449

Order 16256.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861149	C₂	1	1	yes	maximal, characteristic
3	#22549679	C₂	1	4	yes	maximal, complemented
4	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	1	no	—
6	C₂×Dic₅₀₈	—	2	1	no	—
7	C₂×Dic₅₀₈	—	2	2	no	—
8	C₂×Dic₅₀₈	C₄	1	4	yes	complemented
9	C₂×Dic₅₀₈	C₂²	1	2	yes	—
10	#16861149	—	2	2	no	—
11	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	#117×C₁₂₇	—	2	1	no	—
14	#107×C₁₂₇	C₂²	1	2	yes	—
15	#106×C₁₂₇	—	2	1	no	—
16	C₂×Dic₂₅₄	—	2	4	no	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	—	4	1	no	—
19	C₂×Dic₂₅₄	—	4	2	no	—
20	C₂²×Dic₁₂₇	—	4	1	no	—
21	Dic₅₀₈	—	2	4	no	—
22	Dic₅₀₈	—	2	2	no	—
23	Dic₅₀₈	C₂×C₄	1	4	yes	—
24	C₁₂₇⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
32	C₄²×C₁₂₇	—	4	1	no	—
33	Dic₂₅₄	—	2	4	no	—
34	Dic₂₅₄	—	4	4	no	—
35	Dic₂₅₄	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	4	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂³×C₁₂₇	D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
45	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	#102	1	2	yes	—
49	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₂₇	—	8	2	no	—
51	C₄×C₁₂₇	#102	1	2	yes	—
52	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₂₇	—	4	4	no	—
55	C₂×C₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	C₂×#102	1	1	yes	characteristic
57	C₂×C₁₂₇	#510	1	2	yes	—
58	#2141	—	127	1	no	maximal, Sylow
59	C₁₂₇	#2141	1	1	yes	characteristic, complemented, Sylow
60	C₂²×Q₁₆	—	127	1	no	—
61	C₂×M₃₂	—	127	1	no	—
62	#427	D₁₂₇	1	1	yes	characteristic
63	#358	—	127	4	no	—
64	C₂²×Q₈	—	254	1	no	—
65	C₂×Q₁₆	—	127	4	no	—
66	C₂×Q₁₆	—	127	2	no	—
67	C₂×Q₁₆	—	254	1	no	—
68	C₂×Q₁₆	—	254	2	no	—
69	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	254	2	no	—
72	C₂×C₁₆	—	127	2	no	—
73	#107	C₂×D₁₂₇	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	508	1	no	—
76	C₂×Q₈	—	508	2	no	—
77	C₂×Q₈	—	254	4	no	—
78	C₂×Q₈	—	254	2	no	—
79	C₂²×C₄	—	508	1	no	—
80	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
81	Q₁₆	—	127	4	no	—
82	Q₁₆	—	254	4	no	—
83	Q₁₆	—	254	2	no	—
84	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
85	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
86	C₂×C₈	D₅₀₈	1	2	yes	—
87	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	254	2	no	—
92	C₂³	#14523233	1	1	yes	characteristic
93	Q₈	—	508	2	no	—
94	Q₈	—	508	4	no	—
95	Q₈	—	254	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	508	2	no	—
98	C₂×C₄	—	508	4	no	—
99	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
100	C₂×C₄	#14523233	1	1	yes	characteristic
101	C₂×C₄	#14523235	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
103	C₈	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
105	C₈	C₂×D₅₀₈	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#16861129	1	2	yes	—
108	C₂²	C₂×#14523233	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	508	4	no	—
111	C₄	#16861129	1	2	yes	—
112	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#14523235	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#16861129	1	1	yes	characteristic
116	C₂	#22550316	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

C₂×#16861149

C₂

yes

maximal, characteristic

#22549679

C₂

yes

maximal, complemented

#427×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

C₄

yes

complemented

C₂×Dic₅₀₈

C₂²

yes

—

#16861149

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

—

#107×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

—

Dic₅₀₈

—

Dic₅₀₈

C₂×C₄

yes

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

D₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

D₈

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#102

yes

—

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#102

yes

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#102

yes

characteristic

C₂×C₁₂₇

#510

yes

—

#2141

—

127

maximal, Sylow

C₁₂₇

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

C₂×M₃₂

—

127

—

#427

D₁₂₇

yes

characteristic

#358

—

127

—

C₂²×Q₈

—

254

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

M₃₂

—

254

—

C₂×C₁₆

—

127

—

#107

C₂×D₁₂₇

yes

—

#106

—

C₂×Q₈

—

508

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

508

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

Q₁₆

—

127

—

Q₁₆

—

254

—

Q₁₆

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

254

—

C₂³

#14523233

yes

characteristic

Q₈

—

508

—

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

100

C₂×C₄

#14523233

yes

characteristic

101

C₂×C₄

#14523235

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

103

C₈

(C₂×D₁₂₇)⋊C₄

yes

—

104

C₈

C₂×C₄×D₁₂₇

yes

characteristic

105

C₈

C₂×D₅₀₈

yes

characteristic

106

C₂²

—

107

C₂²

#16861129

yes

—

108

C₂²

C₂×#14523233

yes

characteristic

109

C₄

—

110

C₄

—

508

—

111

C₄

#16861129

yes

—

112

C₄

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

113

C₄

C₂×#14523235

yes

characteristic

114

C₂

—

115

C₂

C₂×#16861129

yes

characteristic

116

C₂

#22550316

yes

—

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#427×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂³×C₁₂₇	D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#427	D₁₂₇
hypocenter	C₁₂₇	#2141
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#14523233

commutator

C₈×C₁₂₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#427×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₈

prime radical

C₂³×C₁₂₇

D₈

cosocle

C₂×C₈×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #22549679	C₂	semidirect
8. C₂×Dic₅₀₈	C₄	semidirect
59. C₁₂₇	#2141	semidirect

H ≅ G/N

type

3. #22549679

C₂

semidirect

8. C₂×Dic₅₀₈

C₄

semidirect

59. C₁₂₇

#2141

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₅₀₈	2
C₂×#16861149	2
#22549679	2
#427×C₁₂₇	2
#2141	127

subgroup

index

C₂²×Dic₅₀₈

C₂×#16861149

#22549679

#427×C₁₂₇

#2141

127

Subgroups — #30640449

Conjugacy classes of subgroups

Complements

Maximal subgroups