Subgroups — #17948319

Order 4352.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#63216	C₂	1	1	yes	maximal, characteristic
2	C₈×#63216	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic
4	C₄×#63216	C₄	1	2	yes	—
5	C₄×#63216	C₄	1	2	yes	—
6	C₄×#63214	C₄	1	1	yes	characteristic
7	C₈×#3622	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
9	C₂×#63216	C₈	1	4	yes	—
10	C₂×#63216	C₈	1	4	yes	—
11	C₂×C₁₆×C₁₇	C₈	1	2	yes	—
12	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×#63214	C₈	1	2	yes	—
14	C₄×#3622	C₂×C₄	1	1	yes	characteristic
15	C₄×#3622	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₁₇	C₄²	1	1	yes	characteristic
19	C₂×#3622	C₂×C₈	1	2	yes	—
20	C₂×#3622	C₂×C₈	1	2	yes	—
21	#63216	—	2	4	no	—
22	#63216	C₁₆	1	8	yes	complemented
23	C₁₆×C₁₇	—	2	2	no	—
24	#63214	—	2	2	no	—
25	#7052	—	17	1	no	maximal, Sylow
26	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
29	C₈×C₁₇	M₃₂	1	2	yes	—
30	C₈×C₁₇	M₃₂	1	2	yes	—
31	#3622	C₂×C₁₆	1	4	yes	—
32	#3622	M₃₂	1	4	yes	—
33	C₈×C₁₆	—	17	1	no	—
34	C₈×C₁₆	—	17	1	no	—
35	C₈×C₁₆	—	17	1	no	—
36	C₂²×C₁₇	C₈²	1	1	yes	characteristic
37	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
38	C₄×C₁₇	#346	1	2	yes	—
39	C₄×C₁₇	#346	1	1	yes	characteristic
40	C₄×C₁₇	#346	1	1	yes	characteristic
41	C₄×C₁₆	—	17	2	no	—
42	C₄×C₁₆	—	17	2	no	—
43	C₄×C₁₆	—	17	1	no	—
44	C₄×C₁₆	#595	1	1	yes	characteristic
45	C₈²	—	17	1	no	—
46	C₂×C₁₇	C₈×C₁₆	1	1	yes	characteristic
47	C₂×C₁₇	#1311	1	1	yes	characteristic
48	C₂×C₁₇	#1311	1	1	yes	characteristic
49	C₂×C₁₆	—	17	4	no	—
50	C₂×C₁₆	—	17	4	no	—
51	C₂×C₁₆	—	17	2	no	—
52	C₂×C₁₆	#3624	1	2	yes	—
53	C₄×C₈	—	17	1	no	—
54	C₄×C₈	—	17	1	no	—
55	C₄×C₈	C₂×#595	1	1	yes	characteristic
56	C₁₇	#7052	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₈	—	17	2	no	—
58	C₂×C₈	—	17	2	no	—
59	C₂×C₈	#63242	1	1	yes	characteristic
60	C₂×C₈	C₂×#3624	1	1	yes	characteristic
61	C₄²	C₄×#595	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	17	8	no	—
64	C₁₆	—	34	4	no	—
65	C₁₆	—	34	2	no	—
66	C₂×C₄	C₈×#595	1	1	yes	characteristic
67	C₂×C₄	C₈×#595	1	1	yes	characteristic
68	C₂×C₄	C₄×#3624	1	1	yes	characteristic
69	C₈	—	17	4	no	—
70	C₈	—	17	4	no	—
71	C₈	#10587530	1	2	yes	—
72	C₈	#10587538	1	2	yes	—
73	C₂²	C₈×#3624	1	1	yes	characteristic
74	C₄	C₁₆×#595	1	2	yes	—
75	C₄	#11763696	1	2	yes	—
76	C₄	#11763697	1	1	yes	characteristic
77	C₄	#11763698	1	1	yes	characteristic
78	C₂	C₁₆×#3624	1	1	yes	characteristic
79	C₂	#14543695	1	1	yes	characteristic
80	C₂	#14543696	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#595
commutator	C₂×C₁₇	C₈×C₁₆
Fitting	C₄×C₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₈²
cosocle	C₈×#3622	C₂²
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#7052
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
22. #63216	C₁₆	semidirect
56. C₁₇	#7052	semidirect

Maximal subgroups

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

subgroup	index
C₈×#63216	2
C₁₆×#3622	2
#7052	17