Subgroups — #16950738

Order 4352.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63214	C₂	1	6	yes	maximal
2	C₁₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63214	C₂²	1	4	yes	—
4	C₄⋊#63214	—	2	12	no	—
5	C₂×C₁₇×#97	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
7	C₁₇×#376	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
11	C₁₇×#97	—	2	6	no	—
12	C₄×C₈×C₁₇	D₄	1	6	yes	—
13	C₄×C₈×C₁₇	Q₈	1	6	yes	—
14	C₁₇×#95	C₈	1	4	yes	—
15	C₂×#63214	—	2	12	no	—
16	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇	—	2	6	no	—
19	C₂×C₈×C₁₇	C₄⋊C₄	1	6	yes	—
20	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
22	C₄²×C₁₇	C₄⋊C₄	1	6	yes	—
23	C₄²×C₁₇	M₁₆	1	6	yes	—
24	#63214	—	4	4	no	—
25	#12899	—	17	1	no	maximal, Sylow
26	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	—	2	12	no	—
28	C₂×C₄×C₁₇	C₄⋊C₈	1	12	yes	—
29	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
30	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
31	C₈×C₁₇	—	2	6	no	—
32	C₈×C₁₇	#127	1	1	yes	characteristic
33	C₈×C₁₇	C₄⋊Q₈	1	3	yes	—
34	C₂×C₄⋊C₁₆	—	17	6	no	—
35	#1726	D₁₇	1	1	yes	characteristic
36	C₂²×C₁₇	C₂×C₄⋊C₈	1	3	yes	—
37	C₂²×C₁₇	#504	1	4	yes	—
38	C₄×C₁₇	—	2	12	no	—
39	C₄×C₁₇	#384	1	1	yes	characteristic
40	C₄×C₁₇	#423	1	3	yes	—
41	C₂²×C₁₆	—	17	4	no	—
42	C₂×#97	Dic₁₇	1	3	yes	—
43	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
44	#376	Dic₁₇	1	1	yes	characteristic
45	C₄⋊C₁₆	—	34	12	no	—
46	C₂×C₁₇	#1842	1	1	yes	characteristic
47	C₂×C₁₇	#2151	1	6	yes	—
48	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
49	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
50	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
51	C₂×C₁₆	—	34	12	no	—
52	#97	—	2	6	no	—
53	C₄×C₈	Dic₃₄	1	6	yes	—
54	C₄×C₈	D₆₈	1	6	yes	—
55	#95	#3622	1	4	yes	—
56	C₁₇	#12899	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₄	C₂×#3622	1	6	yes	—
58	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊Dic₁₇	1	6	yes	—
61	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
62	C₂×C₈	C₂×D₆₈	1	3	yes	—
63	C₄²	#63223	1	6	yes	—
64	C₄²	C₄⋊Dic₁₇	1	6	yes	—
65	C₁₆	—	68	4	no	—
66	C₂³	C₂²×#3622	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#3622	1	12	yes	—
69	C₂×C₄	C₂×#63223	1	3	yes	—
70	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#10587555	1	3	yes	—
73	C₈	#10587560	1	1	yes	characteristic
74	C₂²	C₂×C₄⋊#3622	1	3	yes	—
75	C₂²	#11763226	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#11762679	1	1	yes	characteristic
78	C₄	#11762833	1	3	yes	—
79	C₂	#14528936	1	1	yes	characteristic
80	C₂	#14530543	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₁₇×#1726	C₂
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1726	D₁₇
hypocenter	C₁₇	#12899
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
56. C₁₇	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63214	2
C₁₇×#1726	2
#12899	17