Group #30594870 — subgroups

Subgroups — #30594870

Order 15616.

136 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 38 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	#21460765	C₂	1	2	yes	maximal
2	#21445957	C₂	1	1	yes	maximal, characteristic
3	#16840658	—	2	2	no	—
4	#16839786	C₄	1	1	yes	characteristic
5	#16839786	C₂²	1	1	yes	characteristic
6	#16839527	—	2	2	no	—
7	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
8	#14519851	—	4	2	no	—
9	C₂×#11758194	—	2	1	no	—
10	C₂×#11758194	D₄	1	2	yes	—
11	C₄⋊#92418	—	4	1	no	—
12	C₄⋊#92418	—	4	1	no	—
13	C₂²×C₈×C₆₁	—	2	1	no	—
14	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	C₈	1	2	yes	—
17	C₂²×C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₁	—	2	2	no	—
20	C₂×C₈×C₆₁	—	4	1	no	—
21	C₂×C₈×C₆₁	—	4	1	no	—
22	C₄²×C₆₁	—	2	1	no	—
23	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
24	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
25	#11758194	—	4	2	no	—
26	#11758194	—	4	2	no	—
27	C₂×#92418	—	2	2	no	—
28	C₂×#92418	—	4	1	no	—
29	#11758187	—	8	2	no	—
30	C₂³×C₆₁	#101	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	1	no	—
32	C₂×C₄×C₆₁	—	2	1	no	—
33	C₂×C₄×C₆₁	—	2	1	no	—
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	4	2	no	—
36	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
38	C₈×C₆₁	—	4	2	no	—
39	#92418	—	4	2	no	—
40	#92418	—	8	1	no	—
41	#7368	—	61	1	no	maximal, Sylow
42	C₂²×C₆₁	—	2	1	no	—
43	C₂²×C₆₁	—	4	1	no	—
44	C₂²×C₆₁	#324	1	1	yes	characteristic
45	C₄×C₆₁	—	2	2	no	—
46	C₄×C₆₁	—	2	1	no	—
47	C₄×C₆₁	—	4	1	no	—
48	C₄×C₆₁	#354	1	1	yes	characteristic
49	C₄×C₆₁	#355	1	1	yes	characteristic
50	#1837	—	61	1	no	—
51	#1327	—	61	2	no	—
52	C₂×C₆₁	—	2	1	no	—
53	C₂×C₆₁	—	4	1	no	—
54	C₂×C₆₁	#1324	1	1	yes	characteristic
55	#423	—	61	1	no	—
56	#423	—	61	1	no	—
57	C₂×C₄×C₈	#6923	1	1	yes	characteristic
58	#349	—	122	2	no	—
59	#341	—	122	2	no	—
60	C₆₁	#7368	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	61	2	no	—
62	C₂×M₁₆	—	122	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
65	M₃₂	—	244	2	no	—
66	C₄⋊C₈	—	244	1	no	—
67	C₄⋊C₈	—	244	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	#92420	1	2	yes	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
72	M₁₆	—	244	2	no	—
73	M₁₆	—	244	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	122	2	no	—
78	C₂×C₈	—	244	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	C₂×#92420	1	1	yes	characteristic
81	C₄²	#11758217	1	1	yes	characteristic
82	C₁₆	—	488	2	no	—
83	C₂³	#14519838	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
90	C₂×C₄	#14519836	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	488	1	no	—
93	C₈	—	244	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	#16840607	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	#16840669	1	1	yes	characteristic
101	C₄	#16840671	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	4	1	no	—
104	C₂	#21460759	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460765

C₂

yes

maximal

#21445957

C₂

yes

maximal, characteristic

#16840658

—

#16839786

C₄

yes

characteristic

#16839786

C₂²

yes

characteristic

#16839527

—

C₂×C₄×C₈×C₆₁

C₄

yes

—

—

D₄

yes

—

C₄⋊#92418

—

C₄⋊#92418

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₈

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

—

—

—

—

—

C₂³×C₆₁

#101

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

#99

yes

characteristic

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#324

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

C₂×C₄×C₈

#6923

yes

characteristic

#349

—

122

—

#341

—

122

—

C₆₁

#7368

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

244

—

C₄⋊C₈

—

244

—

C₄⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

#92420

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#6923

yes

characteristic

M₁₆

—

244

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

488

—

C₂³

#14519838

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

488

—

C₈

—

244

—

C₂²

—

C₂²

—

C₂²

#16840607

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#16840669

yes

characteristic

101

C₄

#16840671

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#21460759

yes

characteristic

	subgroup	G/H
center	C₄	#16840669
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle	C₂×C₆₁	#1324
prime radical	C₂³×C₆₁	#101
cosocle	#16839786	C₂²
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#7368
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄

#16840669

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×C₄×C₈×C₆₁

C₄

C₂×C₆₁

C₂³×C₆₁

C₂²

C₂×C₄×C₈

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
60. C₆₁	#7368	semidirect

H ≅ G/N

type

60. C₆₁

#7368

semidirect

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

subgroup	index
#21460765	2
#21445957	2
#7368	61

subgroup

index

#21460765

#21445957

#7368

Subgroups — #30594870

Conjugacy classes of subgroups

Complements

Maximal subgroups