Subgroups — #17981027

Order 4352.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545751	C₂	1	15	yes	maximal, complemented
2	C₂×#11763867	C₂	1	15	yes	maximal, complemented
3	#14543496	C₂	1	1	yes	maximal, characteristic
4	#11763867	C₂²	1	20	yes	complemented
5	#11763867	C₂²	1	60	yes	complemented
6	C₂×#10587538	C₂²	1	45	yes	complemented
7	C₂²×#63216	C₂²	1	15	yes	complemented
8	#11763576	C₄	1	15	yes	—
9	C₂×#10587634	C₂²	1	15	yes	—
10	C₁₇×#585	C₄	1	1	yes	characteristic
11	C₁₇×2₋¹⁺⁴	C₈	1	6	yes	—
12	C₁₇×H(5,𝔽₂)	C₈	1	10	yes	—
13	C₂×#41×C₁₇	C₂×C₄	1	15	yes	—
14	#10587634	C₂×C₄	1	60	yes	—
15	#10587634	C₂³	1	20	yes	—
16	C₂×#63223	C₂×C₄	1	45	yes	—
17	C₂²×#3622	C₂³	1	15	yes	—
18	#10587538	C₂³	1	60	yes	—
19	C₂×#63216	C₂³	1	15	yes	—
20	C₂×#63216	C₂³	1	45	yes	—
21	#41×C₁₇	C₂×C₈	1	60	yes	—
22	#41×C₁₇	C₂²×C₄	1	20	yes	—
23	C₂×Q₈×C₁₇	C₂×C₈	1	15	yes	—
24	C₂×D₄×C₁₇	C₂×C₈	1	45	yes	—
25	C₂²×C₄×C₁₇	C₂²×C₄	1	15	yes	—
26	#63223	C₂²×C₄	1	60	yes	—
27	C₂×#3622	C₂²×C₄	1	45	yes	—
28	C₂×#3622	C₂⁴	1	15	yes	—
29	#63216	C₂⁴	1	1	yes	characteristic
30	#63216	C₂⁴	1	15	yes	—
31	#62624	—	17	1	no	maximal, Sylow
32	C₂³×C₁₇	C₂²×C₈	1	15	yes	—
33	Q₈×C₁₇	C₂²×C₈	1	20	yes	—
34	D₄×C₁₇	C₂²×C₈	1	60	yes	—
35	C₂×C₄×C₁₇	C₂²×C₈	1	45	yes	—
36	C₂×C₄×C₁₇	C₂³×C₄	1	15	yes	—
37	#3622	C₂³×C₄	1	15	yes	—
38	#3622	C₂⁵	1	1	yes	characteristic
39	#3573	—	17	1	no	—
40	#3407	—	17	15	no	—
41	C₂×#504	—	17	15	no	—
42	C₂²×C₁₇	—	4	15	no	—
43	C₂²×C₁₇	C₂³×C₈	1	15	yes	—
44	C₄×C₁₇	C₂³×C₈	1	15	yes	—
45	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
46	#585	#595	1	1	yes	characteristic
47	#568	—	17	15	no	—
48	C₂×#131	—	17	15	no	—
49	#504	—	17	20	no	—
50	#504	—	17	60	no	—
51	C₂×M₃₂	—	17	45	no	—
52	C₂²×C₁₆	—	17	15	no	—
53	C₂×C₁₇	—	2	15	no	—
54	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
55	2₋¹⁺⁴	#3624	1	6	yes	—
56	H(5,𝔽₂)	#3624	1	10	yes	—
57	C₂×#41	C₂×#595	1	15	yes	—
58	#131	—	17	20	no	—
59	#131	—	17	60	no	—
60	C₂×M₁₆	—	17	45	no	—
61	C₂²×C₈	—	17	15	no	—
62	M₃₂	—	17	60	no	—
63	C₂×C₁₆	—	17	15	no	—
64	C₂×C₁₆	—	17	45	no	—
65	C₁₇	#62624	1	1	yes	characteristic, complemented, Sylow
66	#41	C₂×#3624	1	60	yes	—
67	#41	C₂²×#595	1	20	yes	—
68	C₂×Q₈	C₂×#3624	1	15	yes	—
69	C₂×D₄	C₂×#3624	1	45	yes	—
70	C₂²×C₄	C₂²×#595	1	15	yes	—
71	M₁₆	—	17	60	no	—
72	C₂×C₈	—	17	15	no	—
73	C₂×C₈	—	17	45	no	—
74	C₁₆	—	17	1	no	—
75	C₁₆	—	17	15	no	—
76	C₂³	C₂²×#3624	1	15	yes	—
77	Q₈	C₂²×#3624	1	20	yes	—
78	D₄	C₂²×#3624	1	60	yes	—
79	C₂×C₄	C₂²×#3624	1	45	yes	—
80	C₂×C₄	C₂³×#595	1	15	yes	—
81	C₈	—	17	1	no	—
82	C₈	—	17	15	no	—
83	C₂²	—	4	15	no	—
84	C₂²	C₂³×#3624	1	15	yes	—
85	C₄	C₂³×#3624	1	15	yes	—
86	C₄	C₂⁴×#595	1	1	yes	characteristic
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×#3624	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×#595
commutator, socle	C₂×C₁₇	C₂⁴×C₈
Fitting, prime radical	C₁₇×#585	C₄
cosocle	#3622	C₂⁵
hypercenter	#585	#595
hypocenter	C₁₇	#62624
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #14545751	C₂	semidirect
2. C₂×#11763867	C₂	semidirect
4. #11763867	C₂²	semidirect
5. #11763867	C₂²	semidirect
6. C₂×#10587538	C₂²	semidirect
7. C₂²×#63216	C₂²	semidirect
65. C₁₇	#62624	semidirect

Maximal subgroups

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

subgroup	index
#14545751	2
C₂×#11763867	2
#14543496	2
#62624	17