Subgroups — #16950724

Order 4352.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14526862	C₂	1	6	yes	maximal
2	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63214	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₁₇	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
6	C₄³×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
8	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
9	C₂×C₄²×C₁₇	C₈	1	3	yes	—
10	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
12	C₄×C₈×C₁₇	—	2	6	no	—
13	C₄×C₈×C₁₇	—	2	6	no	—
14	C₂×#63214	—	2	12	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	C₂×C₈×C₁₇	—	2	3	no	—
19	C₂×C₈×C₁₇	—	2	3	no	—
20	C₂×C₈×C₁₇	M₁₆	1	6	yes	—
21	C₂×C₈×C₁₇	#41	1	6	yes	—
22	C₄²×C₁₇	—	2	6	no	—
23	C₄²×C₁₇	—	2	3	no	—
24	C₄²×C₁₇	—	2	3	no	—
25	C₄²×C₁₇	C₁₆	1	4	yes	complemented
26	#63214	—	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	C₈×C₁₇	—	2	6	no	—
38	C₈×C₁₇	#126	1	4	yes	—
39	C₄²×C₈	D₁₇	1	1	yes	characteristic
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	Dic₁₇	1	3	yes	—
50	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
51	C₄³	Dic₁₇	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	C₂×Dic₁₇	1	3	yes	—
56	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
57	C₂×C₄²	#3622	1	3	yes	—
58	C₂×C₄²	#3622	1	1	yes	characteristic
59	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
60	C₂×C₁₆	—	34	12	no	—
61	C₄×C₈	—	2	6	no	—
62	C₄×C₈	—	2	6	no	—
63	C₁₇	#12894	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₄	C₂×#3622	1	3	yes	—
65	C₂²×C₄	C₂×#3622	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#63223	1	6	yes	—
70	C₂×C₈	#63252	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#63214	1	4	yes	complemented
75	C₁₆	—	68	4	no	—
76	C₂³	C₂²×#3622	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×#63214	1	6	yes	—
82	C₂×C₄	#10587483	1	6	yes	—
83	C₂×C₄	C₂×#63223	1	3	yes	—
84	C₂×C₄	#10587612	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#10587562	1	4	yes	—
87	C₂²	#11762862	1	3	yes	—
88	C₂²	C₂²×#63214	1	1	yes	characteristic
89	C₂²	C₂×#10587483	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#11762677	1	1	yes	characteristic
93	C₄	#11762866	1	3	yes	—
94	C₂	#14528927	1	1	yes	characteristic
95	C₂	#14530583	1	3	yes	—
96	C₂	#14530587	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₄²×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#12894
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄²×C₁₇	C₁₆	semidirect
63. C₁₇	#12894	semidirect
74. C₄²	#63214	semidirect

Maximal subgroups

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

subgroup	index
#14526862	2
C₄²×C₈×C₁₇	2
#12894	17