Subgroups — #14530365

Order 2176.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762858	C₂	1	1	yes	maximal, characteristic
2	#11762837	C₂	1	1	yes	maximal, characteristic
3	#11762581	C₂	1	2	yes	maximal
4	#11762573	C₂	1	2	yes	maximal
5	C₁₇×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	#10587612	C₂²	1	1	yes	characteristic
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	#10587490	—	2	2	no	—
12	#10587485	—	2	2	no	—
13	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
14	M₁₆×C₁₇	—	2	2	no	—
15	M₁₆×C₁₇	—	2	2	no	—
16	M₁₆×C₁₇	—	2	2	no	—
17	C₂×C₈×C₁₇	—	2	2	no	—
18	C₂×C₈×C₁₇	D₄	1	2	yes	—
19	C₂²⋊Dic₁₇	Q₈	1	2	yes	—
20	C₄⋊Dic₁₇	—	2	1	no	—
21	C₄×Dic₁₇	—	2	1	no	—
22	#63223	—	2	2	no	—
23	#63223	—	2	2	no	—
24	C₂×#3622	—	2	1	no	—
25	C₂×#3622	—	2	2	no	—
26	C₂×#3622	—	2	2	no	—
27	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	#41	1	2	yes	—
30	C₂×C₄×C₁₇	#41	1	2	yes	—
31	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
32	C₂×Dic₁₇	—	2	2	no	—
33	C₈×C₁₇	—	2	2	no	—
34	C₈×C₁₇	—	4	2	no	—
35	#3622	—	4	1	no	—
36	#3622	—	4	2	no	—
37	#2103	—	17	1	no	maximal, Sylow
38	C₂²×C₁₇	—	4	1	no	—
39	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂²×C₁₇	#125	1	2	yes	—
41	C₄×C₁₇	#123	1	1	yes	characteristic
42	C₄×C₁₇	#124	1	1	yes	characteristic
43	C₄×C₁₇	#126	1	2	yes	—
44	Dic₁₇	—	4	2	no	—
45	#430	—	17	1	no	—
46	#424	—	17	1	no	—
47	#344	—	17	2	no	—
48	#341	—	17	2	no	—
49	#338	D₁₇	1	1	yes	characteristic
50	C₂×C₁₇	—	2	1	no	—
51	C₂×C₁₇	—	2	2	no	—
52	C₂×C₁₇	#400	1	1	yes	characteristic
53	C₂×M₁₆	—	17	2	no	—
54	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
55	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
56	C₂²×C₈	—	17	1	no	—
57	#117	—	17	1	no	—
58	#108	—	34	2	no	—
59	C₄⋊C₈	—	34	2	no	—
60	C₁₇	#2103	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	2	2	no	—
64	M₁₆	—	2	2	no	—
65	M₁₆	—	34	2	no	—
66	M₁₆	—	34	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	34	1	no	—
69	C₂×C₈	—	34	2	no	—
70	C₂×C₈	—	34	2	no	—
71	C₂×C₈	D₆₈	1	2	yes	—
72	C₄⋊C₄	—	34	1	no	—
73	C₂²⋊C₄	—	17	2	no	—
74	C₄²	—	34	1	no	—
75	C₂³	Q₈×D₁₇	1	1	yes	characteristic
76	C₂×C₄	—	34	2	no	—
77	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
78	C₂×C₄	#63252	1	2	yes	—
79	C₂×C₄	#63254	1	1	yes	characteristic
80	C₂×C₄	#63254	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	68	2	no	—
84	C₈	—	68	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#10587575	1	2	yes	—
87	C₂²	#10587583	1	1	yes	characteristic
88	C₄	—	68	2	no	—
89	C₄	#10587561	1	1	yes	characteristic
90	C₄	#10587565	1	2	yes	—
91	C₄	#10587572	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#11762749	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10587561
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#338	C₂
socle	C₂×C₁₇	#400
prime radical	C₂³×C₁₇	C₂×Q₈
hypercenter	#338	D₁₇
hypocenter	C₁₇	#2103
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
60. C₁₇	#2103	semidirect

Maximal subgroups

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

subgroup	index
#11762858	2
#11762837	2
#11762581	2
#11762573	2
C₁₇×#338	2
#2103	17