Subgroups — #14545751

Order 2176.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763867	C₂	1	8	yes	maximal, complemented
2	C₂×#10587538	C₂	1	6	yes	maximal, complemented
3	C₂×#10587634	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₁₇	C₄	1	1	yes	characteristic
5	#10587634	C₄	1	4	yes	—
6	#10587634	C₂²	1	4	yes	—
7	C₂×#63223	C₄	1	3	yes	—
8	C₂²×#3622	C₂²	1	3	yes	—
9	#10587538	C₂²	1	4	yes	—
10	#10587538	C₂²	1	12	yes	complemented
11	C₂×#63216	C₂²	1	12	yes	complemented
12	#41×C₁₇	C₈	1	4	yes	—
13	#41×C₁₇	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
15	C₂×D₄×C₁₇	C₈	1	3	yes	—
16	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
17	#63223	C₂×C₄	1	12	yes	—
18	C₂×#3622	C₂×C₄	1	3	yes	—
19	C₂×#3622	C₂×C₄	1	6	yes	—
20	C₂×#3622	C₂³	1	6	yes	—
21	C₂×#3622	C₂³	1	1	yes	characteristic
22	#63216	C₂³	1	8	yes	—
23	C₂³×C₁₇	C₂×C₈	1	3	yes	—
24	Q₈×C₁₇	C₂×C₈	1	4	yes	—
25	D₄×C₁₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₇	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₁₇	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂²×C₄	1	6	yes	—
30	#3622	C₂²×C₄	1	1	yes	characteristic
31	#3622	C₂²×C₄	1	6	yes	—
32	#3622	C₂⁴	1	1	yes	characteristic
33	#3407	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	—	4	3	no	—
35	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
36	C₂²×C₁₇	C₂²×C₈	1	6	yes	—
37	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₁₇	C₂²×C₈	1	6	yes	—
39	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×#131	—	17	1	no	—
41	#504	—	17	8	no	—
42	C₂×M₃₂	—	17	6	no	—
43	C₂×C₁₇	—	2	1	no	—
44	C₂×C₁₇	—	2	6	no	—
45	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
46	C₂×#41	#595	1	1	yes	characteristic
47	#131	—	17	4	no	—
48	#131	—	17	4	no	—
49	C₂×M₁₆	—	17	3	no	—
50	C₂²×C₈	—	17	3	no	—
51	M₃₂	—	17	4	no	—
52	M₃₂	—	17	12	no	—
53	C₂×C₁₆	—	17	12	no	—
54	C₁₇	#3407	1	1	yes	characteristic, complemented, Sylow
55	#41	#3624	1	4	yes	—
56	#41	C₂×#595	1	4	yes	—
57	C₂×Q₈	#3624	1	1	yes	characteristic
58	C₂×D₄	#3624	1	3	yes	—
59	C₂²×C₄	C₂×#595	1	3	yes	—
60	M₁₆	—	17	12	no	—
61	C₂×C₈	—	17	6	no	—
62	C₂×C₈	—	17	1	no	—
63	C₂×C₈	—	17	3	no	—
64	C₂×C₈	—	17	6	no	—
65	C₁₆	—	17	8	no	—
66	C₂³	C₂×#3624	1	3	yes	—
67	Q₈	C₂×#3624	1	4	yes	—
68	D₄	C₂×#3624	1	12	yes	—
69	C₂×C₄	C₂×#3624	1	3	yes	—
70	C₂×C₄	C₂×#3624	1	6	yes	—
71	C₂×C₄	C₂²×#595	1	1	yes	characteristic
72	C₂×C₄	C₂²×#595	1	6	yes	—
73	C₈	—	17	1	no	—
74	C₈	—	17	1	no	—
75	C₈	—	17	6	no	—
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×#3624	1	1	yes	characteristic
78	C₂²	C₂²×#3624	1	6	yes	—
79	C₄	C₂²×#3624	1	1	yes	characteristic
80	C₄	C₂²×#3624	1	6	yes	—
81	C₄	C₂³×#595	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×#3624	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#595
commutator, socle	C₂×C₁₇	C₂³×C₈
Fitting, prime radical	C₂×#41×C₁₇	C₄
cosocle	#3622	C₂⁴
hypercenter	C₂×#41	#595
hypocenter	C₁₇	#3407
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #11763867	C₂	semidirect
2. C₂×#10587538	C₂	semidirect
10. #10587538	C₂²	semidirect
11. C₂×#63216	C₂²	semidirect
54. C₁₇	#3407	semidirect

Maximal subgroups

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

subgroup	index
#11763867	2
C₂×#10587538	2
C₂×#10587634	2
#3407	17