Group #17949182 — subgroups

Subgroups — #17949182

Order 4352.

136 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 36 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	#14543730	C₂	1	2	yes	maximal
2	#14528921	C₂	1	1	yes	maximal, characteristic
3	#11763705	—	2	2	no	—
4	#11762833	C₂²	1	1	yes	characteristic
5	C₂×C₄×#3622	C₄	1	1	yes	characteristic
6	#11762573	—	2	2	no	—
7	C₁₇×#423	C₄	1	1	yes	characteristic
8	C₂×M₁₆×C₁₇	—	2	1	no	—
9	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂×#63223	D₄	1	2	yes	—
11	C₂²×#3622	—	2	1	no	—
12	#10587538	—	4	2	no	—
13	C₁₇×C₄⋊C₈	—	4	1	no	—
14	C₄⋊#3622	—	4	1	no	—
15	C₄×#3622	—	2	1	no	—
16	C₄×#3622	C₈	1	2	yes	—
17	C₂²×C₄×C₁₇	—	2	1	no	—
18	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
19	M₁₆×C₁₇	—	4	2	no	—
20	C₂×C₈×C₁₇	—	4	1	no	—
21	C₄²×C₁₇	—	2	1	no	—
22	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
23	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
24	#63223	—	4	2	no	—
25	C₂×#3622	—	2	2	no	—
26	C₂×#3622	—	2	2	no	—
27	C₂×#3622	—	4	1	no	—
28	C₂×#3622	—	4	1	no	—
29	#63216	—	8	2	no	—
30	#7368	—	17	1	no	maximal, Sylow
31	C₂³×C₁₇	#101	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	1	no	—
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	—	2	1	no	—
36	C₂×C₄×C₁₇	—	4	2	no	—
37	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
39	C₈×C₁₇	—	8	1	no	—
40	#3622	—	4	2	no	—
41	#3622	—	4	2	no	—
42	#1837	—	17	1	no	—
43	#1327	—	17	2	no	—
44	C₂²×C₁₇	—	2	1	no	—
45	C₂²×C₁₇	—	4	1	no	—
46	C₂²×C₁₇	#324	1	1	yes	characteristic
47	C₄×C₁₇	—	2	2	no	—
48	C₄×C₁₇	—	2	1	no	—
49	C₄×C₁₇	—	4	1	no	—
50	C₄×C₁₇	#354	1	1	yes	characteristic
51	C₄×C₁₇	#355	1	1	yes	characteristic
52	#423	—	17	1	no	—
53	#423	#595	1	1	yes	characteristic
54	C₂×C₄×C₈	—	17	1	no	—
55	#349	—	34	2	no	—
56	#341	—	34	2	no	—
57	C₂×C₁₇	—	2	1	no	—
58	C₂×C₁₇	—	4	1	no	—
59	C₂×C₁₇	#1324	1	1	yes	characteristic
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	17	2	no	—
62	C₂²×C₈	—	34	1	no	—
63	C₂×C₄²	C₂×#595	1	1	yes	characteristic
64	M₃₂	—	68	2	no	—
65	C₄⋊C₈	—	4	1	no	—
66	C₄⋊C₈	—	68	1	no	—
67	C₄×C₈	—	17	2	no	—
68	C₄×C₈	—	34	1	no	—
69	C₁₇	#7368	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
72	M₁₆	—	4	2	no	—
73	M₁₆	—	68	2	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	34	2	no	—
76	C₂×C₈	—	34	2	no	—
77	C₂×C₈	—	68	1	no	—
78	C₂×C₈	—	68	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	#63242	1	1	yes	characteristic
81	C₄²	#63243	1	1	yes	characteristic
82	C₁₆	—	136	2	no	—
83	C₂³	#10587545	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	#10587543	1	1	yes	characteristic
90	C₂×C₄	#10587551	1	1	yes	characteristic
91	C₈	—	8	1	no	—
92	C₈	—	68	2	no	—
93	C₈	—	68	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	#11763655	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	#11763715	1	1	yes	characteristic
101	C₄	#11763717	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	4	1	no	—
104	C₂	#14543725	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14543730

C₂

yes

maximal

#14528921

C₂

yes

maximal, characteristic

#11763705

—

#11762833

C₂²

yes

characteristic

C₂×C₄×#3622

C₄

yes

characteristic

#11762573

—

C₁₇×#423

C₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×#63223

D₄

yes

—

C₂²×#3622

—

#10587538

—

C₁₇×C₄⋊C₈

—

C₄⋊#3622

—

C₄×#3622

—

C₄×#3622

C₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

—

—

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

#99

yes

characteristic

C₈×C₁₇

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#324

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

yes

C₂×C₄×C₈

—

#349

—

#341

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1324

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7368

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²⋊#595

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

136

—

C₂³

#10587545

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#11763655

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#11763715

yes

characteristic

101

C₄

#11763717

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#14543725

yes

characteristic

	subgroup	G/H
center	C₄	#11763715
commutator	C₄²×C₁₇	C₂×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₁₇×#423	C₄
socle	C₂×C₁₇	#1324
prime radical	C₂³×C₁₇	#101
cosocle	#11762833	C₂²
hypercenter	#423	#595
hypocenter	C₁₇	#7368
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#11763715

commutator

C₄²×C₁₇

C₂×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₁₇×#423

C₄

C₂×C₁₇

C₂³×C₁₇

C₂²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
69. C₁₇	#7368	semidirect

H ≅ G/N

type

69. C₁₇

#7368

semidirect

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

subgroup	index
#14543730	2
#14528921	2
#7368	17

subgroup

index

#14543730

#14528921

#7368

Subgroups — #17949182

Conjugacy classes of subgroups

Complements

Maximal subgroups