Group #30636846 — subgroups

Subgroups — #30636846

Order 16256.

128 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 69 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×#16861149	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861145	C₂	1	1	yes	maximal, characteristic
3	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	—	2	1	no	—
5	#16861149	—	2	1	no	—
6	#16861149	C₄	1	2	yes	complemented
7	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
8	#16861145	—	2	1	no	—
9	#16861145	C₄	1	2	yes	complemented
10	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#117×C₁₂₇	—	2	1	no	—
12	#107×C₁₂₇	—	2	1	no	—
13	#106×C₁₂₇	C₄	1	2	yes	—
14	#14523232	—	2	2	no	—
15	#14523232	—	4	1	no	—
16	C₂×C₁₂₇⋊C₈	—	4	1	no	—
17	C₁₂₇⋊C₁₆	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₄⋊C₄×C₁₂₇	—	2	2	no	—
26	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
27	C₄²×C₁₂₇	—	4	1	no	—
28	C₁₂₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
37	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	#102	1	1	yes	characteristic
43	C₂²×C₁₂₇	#103	1	1	yes	characteristic
44	C₂²×C₁₂₇	#106	1	1	yes	characteristic
45	C₄×C₁₂₇	—	8	2	no	—
46	C₄×C₁₂₇	#95	1	1	yes	characteristic
47	C₄×C₁₂₇	#102	1	1	yes	characteristic
48	C₄×C₁₂₇	#103	1	1	yes	characteristic
49	C₄×C₁₂₇	#107	1	1	yes	characteristic
50	C₂×C₁₂₇	—	2	1	no	—
51	C₂×C₁₂₇	—	2	1	no	—
52	C₂×C₁₂₇	—	2	1	no	—
53	C₂×C₁₂₇	#340	1	1	yes	characteristic
54	#1389	—	127	1	no	maximal, Sylow
55	C₁₂₇	#1389	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	127	1	no	—
57	C₂×#108	—	127	1	no	—
58	#427	D₁₂₇	1	1	yes	characteristic
59	C₂×M₁₆	—	254	1	no	—
60	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	M₃₂	—	127	2	no	—
63	M₃₂	—	254	1	no	—
64	C₂×C₁₆	—	254	1	no	—
65	#108	—	127	2	no	—
66	#108	—	254	1	no	—
67	#107	—	2	1	no	—
68	#106	Dic₁₂₇	1	2	yes	—
69	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
70	M₁₆	—	508	1	no	—
71	M₁₆	—	254	2	no	—
72	C₂×C₈	—	508	1	no	—
73	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
74	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
75	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
77	C₂×C₈	D₅₀₈	1	1	yes	characteristic
78	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	4	1	no	—
82	C₁₆	—	254	2	no	—
83	C₂³	#14523236	1	1	yes	characteristic
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
87	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
88	C₂×C₄	#14523233	1	1	yes	characteristic
89	C₂×C₄	#14523235	1	1	yes	characteristic
90	C₂×C₄	#14523238	1	1	yes	characteristic
91	C₈	—	508	2	no	—
92	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
93	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
94	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
95	C₈	C₄⋊Dic₁₂₇	1	1	yes	characteristic
96	C₂²	—	4	1	no	—
97	C₂²	#16861130	1	1	yes	characteristic
98	C₂²	#16861132	1	1	yes	characteristic
99	C₂²	#16861140	1	1	yes	characteristic
100	C₄	—	8	2	no	—
101	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
102	C₄	#16861129	1	1	yes	characteristic
103	C₄	#16861133	1	1	yes	characteristic
104	C₄	#16861142	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	#22549638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861149

C₂

yes

maximal, characteristic

C₂×#16861145

C₂

yes

maximal, characteristic

#427×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

—

#16861149

—

#16861149

C₄

yes

complemented

C₂×C₁₂₇⋊C₁₆

—

#16861145

—

#16861145

C₄

yes

complemented

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

—

#107×C₁₂₇

—

#106×C₁₂₇

C₄

yes

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

D₈

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

Q₁₆

yes

characteristic

C₈×C₁₂₇

C₄²

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#340

yes

characteristic

#1389

—

127

maximal, Sylow

C₁₂₇

#1389

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

C₂×#108

—

127

—

#427

D₁₂₇

yes

characteristic

C₂×M₁₆

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

M₃₂

—

127

—

M₃₂

—

254

—

C₂×C₁₆

—

254

—

#108

—

127

—

#108

—

254

—

#107

—

#106

Dic₁₂₇

yes

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

M₁₆

—

508

—

M₁₆

—

254

—

C₂×C₈

—

508

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

254

—

C₂³

#14523236

yes

characteristic

C₂×C₄

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

C₂×C₄

Dic₁₂₇⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

508

—

C₈

C₄×Dic₁₂₇

yes

characteristic

C₈

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₈

Dic₁₂₇⋊C₄

yes

characteristic

C₈

C₄⋊Dic₁₂₇

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

C₁₂₇⋊#95

yes

characteristic

102

C₄

#16861129

yes

characteristic

103

C₄

#16861133

yes

characteristic

104

C₄

#16861142

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

#22549638

yes

characteristic

	subgroup	G/H
center	C₄	#16861129
commutator	C₈×C₁₂₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	#427×C₁₂₇	C₂
socle	C₂×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	SD₁₆
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	#427	D₁₂₇
hypocenter	C₁₂₇	#1389
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#16861129

commutator

C₈×C₁₂₇

C₄²

Frattini

C₂²×C₈

C₂×D₁₂₇

Fitting

#427×C₁₂₇

C₂

socle

C₂×C₁₂₇

#340

prime radical

C₂³×C₁₂₇

SD₁₆

cosocle

C₂²×C₈×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
6. #16861149	C₄	semidirect
9. #16861145	C₄	semidirect
55. C₁₂₇	#1389	semidirect

H ≅ G/N

type

6. #16861149

C₄

semidirect

9. #16861145

C₄

semidirect

55. C₁₂₇

#1389

semidirect

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

subgroup	index
C₂×#16861149	2
C₂×#16861145	2
#427×C₁₂₇	2
#1389	127

subgroup

index

C₂×#16861149

C₂×#16861145

#427×C₁₂₇

#1389

127

Subgroups — #30636846

Conjugacy classes of subgroups

Complements

Maximal subgroups