Subgroups — #17961656

Order 4352.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63216	C₂	1	6	yes	maximal
2	#14528471	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₂²	1	4	yes	—
4	C₄⋊#63216	—	2	12	no	—
5	C₂×#10587474	C₄	1	3	yes	—
6	C₂×C₄×#3622	C₂²	1	3	yes	—
7	C₁₇×#376	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
9	C₂²×#3622	C₂×C₄	1	3	yes	—
10	C₂²×#3622	C₂³	1	1	yes	characteristic
11	C₂×#63216	—	2	12	no	—
12	C₁₇×#95	C₈	1	4	yes	—
13	#10587474	—	2	6	no	—
14	C₄×#3622	D₄	1	6	yes	—
15	C₄×#3622	Q₈	1	6	yes	—
16	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₁₇	C₄⋊C₄	1	6	yes	—
19	C₄²×C₁₇	M₁₆	1	6	yes	—
20	C₂×#3622	—	2	6	no	—
21	C₂×#3622	C₄⋊C₄	1	6	yes	—
22	C₂×#3622	C₂×D₄	1	3	yes	—
23	C₂×#3622	C₂×Q₈	1	3	yes	—
24	#63216	—	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	#3622	—	2	6	no	—
32	#3622	#127	1	1	yes	characteristic
33	#3622	C₄⋊Q₈	1	3	yes	—
34	C₂×C₄⋊C₁₆	—	17	6	no	—
35	#1726	—	17	1	no	—
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	—	17	3	no	—
43	C₂×C₄×C₈	—	17	3	no	—
44	#376	#595	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	—	17	1	no	—
49	C₂²×C₈	—	17	3	no	—
50	C₂×C₄²	C₂×#595	1	3	yes	—
51	C₂×C₁₆	—	34	12	no	—
52	#97	—	34	6	no	—
53	C₄×C₈	—	17	6	no	—
54	C₄×C₈	—	17	6	no	—
55	#95	#3624	1	4	yes	—
56	C₁₇	#12899	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₄	C₂×#3624	1	6	yes	—
58	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
59	C₂×C₈	—	17	3	no	—
60	C₂×C₈	—	17	3	no	—
61	C₂×C₈	—	17	6	no	—
62	C₂×C₈	—	34	6	no	—
63	C₄²	#63243	1	6	yes	—
64	C₄²	C₄⋊#595	1	6	yes	—
65	C₁₆	—	68	4	no	—
66	C₂³	C₂²×#3624	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#3624	1	12	yes	—
69	C₂×C₄	C₂×#63243	1	3	yes	—
70	C₂×C₄	C₂×C₄⋊#595	1	3	yes	—
71	C₈	—	17	1	no	—
72	C₈	—	17	3	no	—
73	C₈	—	34	6	no	—
74	C₂²	C₂×C₄⋊#3624	1	3	yes	—
75	C₂²	#11763867	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#11763755	1	1	yes	characteristic
78	C₄	#11763813	1	3	yes	—
79	C₂	#14544716	1	1	yes	characteristic
80	C₂	#14545119	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₁₇×#376	C₄
cosocle	C₂²×#3622	C₂³
hypercenter	#376	#595
hypocenter	C₁₇	#12899
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

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⋊#63216	2
#14528471	2
#12899	17