Group #30639722 — subgroups

Subgroups — #30639722

Order 16256.

228 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#16861201	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
3	#22549644	C₂	1	4	yes	maximal
4	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	2	yes	—
6	#16861201	—	2	2	no	—
7	#16861201	—	2	1	no	—
8	#16861201	—	2	1	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×Dic₁₂₇⋊C₄	C₄	1	4	yes	—
11	#16861120	—	2	1	no	—
12	C₄×C₁₂₇⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₂₇	—	2	2	no	—
16	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
17	#14523232	—	2	2	no	—
18	#14523232	Q₈	1	4	yes	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	Dic₁₂₇⋊C₄	—	2	4	no	—
23	Dic₁₂₇⋊C₄	—	2	4	no	—
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₂₇	—	2	4	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₄²×C₁₂₇	D₄	1	2	yes	—
29	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×Dic₁₂₇	—	2	4	no	—
32	C₂×Dic₁₂₇	—	4	4	no	—
33	C₁₂₇⋊C₈	—	2	4	no	—
34	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	—	4	2	no	—
36	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₂₇	#41	1	2	yes	—
42	C₈×C₁₂₇	—	4	2	no	—
43	C₂²×C₁₂₇	—	2	2	no	—
44	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₂₇	—	2	2	no	—
47	C₄×C₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₂₇	#124	1	1	yes	characteristic
50	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₂₇	—	4	4	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	#391	1	1	yes	characteristic
54	C₂×C₁₂₇	#413	1	2	yes	—
55	#1998	—	127	1	no	maximal, Sylow
56	C₁₂₇	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	127	1	no	—
58	#423	D₁₂₇	1	1	yes	characteristic
59	C₄×M₁₆	—	127	1	no	—
60	#343	—	127	4	no	—
61	C₂×M₁₆	—	127	2	no	—
62	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
63	#125	—	254	2	no	—
64	#125	—	254	1	no	—
65	#125	—	254	1	no	—
66	C₂×C₄⋊C₄	—	127	4	no	—
67	C₂×C₄⋊C₄	—	127	2	no	—
68	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	254	1	no	—
71	C₄×C₈	—	254	1	no	—
72	C₂²×C₄	—	127	4	no	—
73	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	127	4	no	—
77	M₁₆	—	254	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	254	2	no	—
80	C₄⋊C₄	—	254	4	no	—
81	C₄⋊C₄	—	254	4	no	—
82	C₄⋊C₄	—	254	2	no	—
83	C₄⋊C₄	—	254	2	no	—
84	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
85	C₄²	D₅₀₈	1	1	yes	characteristic
86	C₄²	D₅₀₈	1	1	yes	characteristic
87	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	508	4	no	—
90	C₂×C₄	—	254	4	no	—
91	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
94	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
96	C₂×C₄	#14523247	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	254	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
101	C₂²	#16861174	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	508	4	no	—
105	C₄	#16861189	1	2	yes	—
106	C₄	#16861196	1	1	yes	characteristic
107	C₄	#16861207	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#22549781	1	1	yes	characteristic
110	C₂	#22549858	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861201

C₂

yes

maximal, characteristic

C₄×#14523232

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal

#423×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

#16861201

—

#16861201

—

#16861201

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₄

yes

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

#14523232

—

#14523232

Q₈

yes

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#391

yes

characteristic

C₂×C₁₂₇

#413

yes

—

#1998

—

127

maximal, Sylow

C₁₂₇

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

127

—

#423

D₁₂₇

yes

characteristic

C₄×M₁₆

—

127

—

#343

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

#125

—

254

—

#125

—

254

—

#125

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

#97

—

254

—

C₄×C₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

127

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₄²

D₅₀₈

yes

characteristic

C₄²

D₅₀₈

yes

characteristic

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

Q₈×D₁₂₇

yes

—

C₂×C₄

#14523247

yes

—

C₈

—

C₈

—

254

—

C₂²

—

100

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

101

C₂²

#16861174

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

508

—

105

C₄

#16861189

yes

—

106

C₄

#16861196

yes

characteristic

107

C₄

#16861207

yes

characteristic

108

C₂

—

109

C₂

#22549781

yes

characteristic

110

C₂

#22549858

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₂₇)⋊C₄
commutator, prime radical	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#1998
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

commutator, prime radical

C₂³×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#423×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
56. C₁₂₇	#1998	semidirect

H ≅ G/N

type

56. C₁₂₇

#1998

semidirect

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

subgroup	index
C₂×#16861201	2
C₄×#14523232	2
#22549644	2
#423×C₁₂₇	2
#1998	127

subgroup

index

#423×C₁₂₇

127

Subgroups — #30639722

Conjugacy classes of subgroups

Complements

Maximal subgroups