Subgroups — #14530544

Order 2176.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63214	C₂	1	1	yes	maximal, characteristic
2	#11762764	C₂	1	1	yes	maximal, characteristic
3	#11762627	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
6	#10587612	C₄	1	1	yes	characteristic
7	C₂×#63223	C₄	1	1	yes	characteristic
8	C₁₇×M₃₂	—	2	2	no	—
9	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
10	#10587486	C₄	1	2	yes	—
11	C₈×Dic₁₇	C₂²	1	2	yes	—
12	C₂×#63214	—	2	2	no	—
13	C₂×#63214	C₂²	1	2	yes	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	D₄	1	2	yes	—
17	C₂×C₈×C₁₇	Q₈	1	2	yes	—
18	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²⋊Dic₁₇	C₈	1	2	yes	—
20	C₄⋊Dic₁₇	C₈	1	2	yes	—
21	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
22	#63223	—	2	2	no	—
23	C₂×#3622	C₂×C₄	1	2	yes	—
24	C₁₆×C₁₇	—	2	2	no	—
25	#63214	—	2	2	no	—
26	C₂³×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₇	C₂×C₈	1	4	yes	—
32	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
33	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	#3622	—	2	2	no	—
36	#2151	—	17	1	no	maximal, Sylow
37	C₂²×C₁₇	—	2	2	no	—
38	C₂²×C₁₇	C₄⋊C₈	1	2	yes	—
39	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₇	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
43	Dic₁₇	—	2	4	no	—
44	C₂×M₃₂	D₁₇	1	1	yes	characteristic
45	C₂²×C₁₆	—	17	1	no	—
46	#405	—	17	1	no	—
47	C₄⋊C₁₆	—	17	4	no	—
48	C₂×C₁₇	—	2	2	no	—
49	C₂×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₁₇	#504	1	2	yes	—
51	C₂×M₁₆	—	17	1	no	—
52	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
53	#117	—	17	1	no	—
54	M₃₂	—	2	2	no	—
55	C₂×C₁₆	—	17	2	no	—
56	C₂×C₁₆	—	34	2	no	—
57	C₂×C₁₆	C₂×D₁₇	1	2	yes	—
58	#97	—	17	2	no	—
59	C₄×C₈	—	17	2	no	—
60	C₁₇	#2151	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
62	M₁₆	—	34	2	no	—
63	C₂×C₈	—	17	2	no	—
64	C₂×C₈	Dic₃₄	1	2	yes	—
65	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
67	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
68	C₄⋊C₄	—	17	2	no	—
69	C₂²⋊C₄	—	17	2	no	—
70	C₄²	—	17	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	34	2	no	—
73	C₂³	C₈×D₁₇	1	1	yes	characteristic
74	C₂×C₄	—	17	4	no	—
75	C₂×C₄	C₈×D₁₇	1	1	yes	characteristic
76	C₂×C₄	#63218	1	2	yes	—
77	C₂×C₄	#63225	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
79	C₈	—	34	2	no	—
80	C₈	#63225	1	2	yes	—
81	C₈	C₂×Dic₃₄	1	1	yes	characteristic
82	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#10587485	1	2	yes	—
85	C₂²	C₂×C₈×D₁₇	1	1	yes	characteristic
86	C₄	—	34	4	no	—
87	C₄	#10587485	1	2	yes	—
88	C₄	C₂×#63218	1	1	yes	characteristic
89	C₄	C₂×#63225	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#10587485	1	1	yes	characteristic
92	C₂	#11763227	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₈
Fitting	C₂×C₁₇×M₃₂	C₂
prime radical	C₂³×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	C₂×M₃₂	D₁₇
hypocenter	C₁₇	#2151
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #11762627	C₂	semidirect
60. C₁₇	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#63214	2
#11762764	2
#11762627	2
C₂×C₁₇×M₃₂	2
#2151	17