Subgroups — #14531011

Order 2176.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763214	C₂	1	1	yes	maximal, characteristic
2	#11762613	C₂	1	4	yes	maximal
3	C₄×#63214	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₁₇	—	2	1	no	—
6	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
7	C₁₇×C₄⋊Q₈	—	2	1	no	—
8	#10587555	—	2	1	no	—
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587489	—	2	1	no	—
11	#10587489	C₂²	1	2	yes	—
12	C₂×#63214	C₂²	1	2	yes	—
13	C₂×Q₈×C₁₇	—	2	2	no	—
14	C₂×Dic₃₄	—	4	1	no	—
15	Q₁₆×C₁₇	—	2	4	no	—
16	Q₁₆×C₁₇	—	4	2	no	—
17	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₇	—	4	1	no	—
20	C₄⋊C₄×C₁₇	—	4	1	no	—
21	C₄²×C₁₇	D₄	1	1	yes	characteristic
22	C₄⋊Dic₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	—	4	1	no	—
24	#63214	—	2	2	no	—
25	Q₈×C₁₇	—	4	2	no	—
26	Q₈×C₁₇	—	4	2	no	—
27	C₂×C₄×C₁₇	—	4	2	no	—
28	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₇	—	4	2	no	—
32	Dic₃₄	—	8	2	no	—
33	C₈×C₁₇	SD₁₆	1	2	yes	—
34	C₈×C₁₇	#41	1	2	yes	—
35	#2241	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
37	C₄×C₁₇	—	8	2	no	—
38	C₄×C₁₇	SD₃₂	1	2	yes	—
39	C₄×C₁₇	Q₃₂	1	2	yes	—
40	C₄×C₁₇	#124	1	1	yes	characteristic
41	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
42	Dic₁₇	—	8	2	no	—
43	#500	—	17	1	no	—
44	#494	D₁₇	1	1	yes	characteristic
45	#358	—	17	4	no	—
46	C₄×C₁₆	—	17	1	no	—
47	C₂×C₁₇	#486	1	1	yes	characteristic
48	C₂×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
49	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×Q₁₆	C₂×D₁₇	1	2	yes	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	34	1	no	—
54	C₂×C₁₆	—	17	2	no	—
55	#107	—	17	2	no	—
56	#107	—	34	1	no	—
57	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
58	C₁₇	#2241	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	68	1	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	34	2	no	—
68	C₄⋊C₄	—	68	1	no	—
69	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
70	C₁₆	—	34	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	136	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	68	2	no	—
76	C₂×C₄	#63228	1	1	yes	characteristic
77	C₂×C₄	#63228	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
79	C₈	#63229	1	2	yes	—
80	C₈	#63254	1	2	yes	—
81	C₂²	C₂×#63228	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	136	2	no	—
84	C₄	#10587499	1	2	yes	—
85	C₄	#10587500	1	2	yes	—
86	C₄	C₂×#63229	1	1	yes	characteristic
87	C₄	#10587622	1	1	yes	characteristic
88	C₂	#11763155	1	1	yes	characteristic
89	C₂	C₂×#10587499	1	1	yes	characteristic
90	C₂	C₂×#10587500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#494	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#494	D₁₇
hypocenter	C₁₇	#2241
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
58. C₁₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
#11763214	2
#11762613	2
C₄×#63214	2
C₁₇×#494	2
#2241	17