Subgroups — #14530656

Order 2176.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₆₈	C₂	1	2	yes	maximal
3	#11762637	C₂	1	1	yes	maximal, characteristic
4	#11762614	C₂	1	2	yes	maximal
5	C₁₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₈	C₂²	1	2	yes	—
7	C₄×Dic₃₄	—	2	2	no	—
8	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	#10587489	C₂²	1	2	yes	—
12	#10587487	—	2	2	no	—
13	Dic₁₃₆	C₄	1	4	yes	complemented
14	C₂×Dic₃₄	—	2	2	no	—
15	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₄⋊Dic₁₇	—	2	2	no	—
19	#63225	—	4	2	no	—
20	C₄×Dic₁₇	—	4	2	no	—
21	Dic₆₈	—	2	2	no	—
22	Dic₆₈	C₂×C₄	1	4	yes	—
23	C₁₆×C₁₇	—	2	1	no	—
24	C₁₆×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₇	D₈	1	2	yes	—
26	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₁₇	—	4	2	no	—
28	C₂×Dic₁₇	—	4	2	no	—
29	Dic₃₄	—	2	4	no	—
30	Dic₃₄	—	4	2	no	—
31	C₈×C₁₇	—	2	1	no	—
32	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇	#41	1	1	yes	characteristic
34	#2176	—	17	1	no	maximal, Sylow
35	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₇	—	2	2	no	—
37	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₇	#135	1	1	yes	characteristic
39	Dic₁₇	—	4	4	no	—
40	Dic₁₇	—	8	2	no	—
41	C₂×Q₃₂	—	17	1	no	—
42	C₄×Q₁₆	—	17	2	no	—
43	#367	—	17	1	no	—
44	#358	—	17	2	no	—
45	#346	D₁₇	1	1	yes	characteristic
46	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
47	C₂×C₁₇	#510	1	2	yes	—
48	C₂×Q₁₆	—	17	2	no	—
49	C₄×Q₈	—	34	2	no	—
50	Q₃₂	—	17	4	no	—
51	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
52	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
53	#107	—	17	2	no	—
54	#103	—	34	2	no	—
55	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
56	C₁₇	#2176	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₈	—	34	2	no	—
58	Q₁₆	—	17	4	no	—
59	Q₁₆	—	34	2	no	—
60	C₂×C₈	D₆₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
62	C₄⋊C₄	—	34	2	no	—
63	C₄⋊C₄	—	68	2	no	—
64	C₄²	—	68	2	no	—
65	C₄²	D₆₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₇	1	2	yes	—
68	Q₈	—	34	4	no	—
69	Q₈	—	68	2	no	—
70	C₂×C₄	—	68	2	no	—
71	C₂×C₄	—	68	2	no	—
72	C₂×C₄	D₁₃₆	1	2	yes	—
73	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
76	C₈	#63252	1	1	yes	characteristic
77	C₂²	C₂×D₁₃₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	68	4	no	—
80	C₄	—	136	2	no	—
81	C₄	C₄×D₆₈	1	1	yes	characteristic
82	C₄	#10587590	1	1	yes	characteristic
83	C₂	C₄×D₁₃₆	1	1	yes	characteristic
84	C₂	#11763252	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₆
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#346	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	#346	D₁₇
hypocenter	C₁₇	#2176
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₁₃₆	C₄	semidirect
56. C₁₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₆	2
C₄×Dic₆₈	2
#11762637	2
#11762614	2
C₁₇×#346	2
#2176	17