Subgroups — #17961642

Order 4352.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543822	C₂	1	6	yes	maximal
2	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₂²	1	4	yes	—
4	C₂×C₄×#3622	C₄	1	3	yes	—
5	C₂×C₄×#3622	C₂²	1	3	yes	—
6	C₄³×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₁₇	C₈	1	3	yes	—
8	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
9	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
10	C₂²×#3622	C₂×C₄	1	3	yes	—
11	C₂²×#3622	C₂³	1	1	yes	characteristic
12	C₂×#63216	—	2	12	no	—
13	C₄×#3622	—	2	6	no	—
14	C₄×#3622	—	2	6	no	—
15	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₁₇	—	2	6	no	—
19	C₄²×C₁₇	—	2	3	no	—
20	C₄²×C₁₇	—	2	3	no	—
21	C₄²×C₁₇	C₁₆	1	4	yes	complemented
22	C₂×#3622	—	2	3	no	—
23	C₂×#3622	—	2	3	no	—
24	C₂×#3622	M₁₆	1	6	yes	—
25	C₂×#3622	#41	1	6	yes	—
26	#63216	—	4	4	no	—
27	#12894	—	17	1	no	maximal, Sylow
28	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇	—	2	3	no	—
30	C₂×C₄×C₁₇	—	2	3	no	—
31	C₂×C₄×C₁₇	—	2	3	no	—
32	C₂×C₄×C₁₇	—	2	3	no	—
33	C₂×C₄×C₁₇	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₁₇	M₃₂	1	6	yes	—
35	C₂×C₄×C₁₇	#117	1	3	yes	—
36	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
37	#3622	—	2	6	no	—
38	#3622	#126	1	4	yes	—
39	C₄²×C₈	—	17	1	no	—
40	#1374	—	17	6	no	—
41	C₂²×C₁₇	#431	1	3	yes	—
42	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
43	C₂²×C₁₇	C₂×M₃₂	1	3	yes	—
44	C₄×C₁₇	—	2	6	no	—
45	C₄×C₁₇	—	2	6	no	—
46	C₄×C₁₇	#383	1	1	yes	characteristic
47	C₄×C₁₇	#432	1	3	yes	—
48	C₂²×C₁₆	—	17	4	no	—
49	C₂×C₄×C₈	—	17	3	no	—
50	C₂×C₄×C₈	—	17	3	no	—
51	C₄³	#595	1	1	yes	characteristic
52	C₂×C₁₇	#1839	1	1	yes	characteristic
53	C₂×C₁₇	#2162	1	3	yes	—
54	C₂×C₁₇	#2163	1	3	yes	—
55	C₂²×C₈	—	17	1	no	—
56	C₂²×C₈	—	17	3	no	—
57	C₂×C₄²	#3624	1	3	yes	—
58	C₂×C₄²	#3624	1	1	yes	characteristic
59	C₂×C₄²	C₂×#595	1	3	yes	—
60	C₂×C₁₆	—	34	12	no	—
61	C₄×C₈	—	34	6	no	—
62	C₄×C₈	—	34	6	no	—
63	C₁₇	#12894	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₄	C₂×#3624	1	3	yes	—
65	C₂²×C₄	C₂×#3624	1	3	yes	—
66	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
67	C₂×C₈	—	17	6	no	—
68	C₂×C₈	—	17	6	no	—
69	C₂×C₈	—	34	3	no	—
70	C₂×C₈	—	34	3	no	—
71	C₄²	—	2	3	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	6	no	—
74	C₄²	#63216	1	4	yes	complemented
75	C₁₆	—	68	4	no	—
76	C₂³	C₂²×#3624	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#63216	1	6	yes	—
82	C₂×C₄	#10587538	1	6	yes	—
83	C₂×C₄	D₁₇⋊M₁₆	1	3	yes	—
84	C₂×C₄	#10587677	1	3	yes	—
85	C₈	—	17	4	no	—
86	C₈	—	34	6	no	—
87	C₂²	#11763825	1	3	yes	—
88	C₂²	C₂²×#63216	1	1	yes	characteristic
89	C₂²	C₂×#10587538	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#11763754	1	1	yes	characteristic
93	C₄	#11763832	1	3	yes	—
94	C₂	#14544711	1	1	yes	characteristic
95	C₂	#14545141	1	3	yes	—
96	C₂	#14545145	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₄³×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₄³	#595
hypocenter	C₁₇	#12894
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
21. C₄²×C₁₇	C₁₆	semidirect
63. C₁₇	#12894	semidirect
74. C₄²	#63216	semidirect

Maximal subgroups

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

subgroup	index
#14543822	2
C₄²×#3622	2
#12894	17