Subgroups — #16950704

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528944	C₂	1	2	yes	maximal
2	#14526876	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587489	—	2	2	no	—
5	C₂×#10587489	C₂²	1	4	yes	—
6	#11762679	—	2	2	no	—
7	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₁₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₇	—	2	4	no	—
14	C₂×C₄⋊Dic₁₇	—	4	2	no	—
15	C₂×C₄⋊Dic₁₇	—	4	2	no	—
16	C₂×C₁₆×C₁₇	—	2	4	no	—
17	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₇	Q₈	1	2	yes	—
20	#10587489	—	2	8	no	—
21	#10587489	—	4	4	no	—
22	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₁₇	—	4	4	no	—
26	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
28	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₇	#41	1	4	yes	—
31	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₇	Q₁₆	1	2	yes	—
33	C₄⋊Dic₁₇	—	4	4	no	—
34	C₄⋊Dic₁₇	—	4	4	no	—
35	C₄⋊Dic₁₇	—	8	4	no	—
36	C₄⋊Dic₁₇	—	8	4	no	—
37	C₁₆×C₁₇	—	2	4	no	—
38	#12888	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	#107	1	2	yes	—
41	C₂×C₄×C₁₇	#107	1	2	yes	—
42	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
43	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
44	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#124	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×Dic₁₇	—	8	4	no	—
50	C₂×Dic₁₇	—	8	4	no	—
51	C₂×Dic₁₇	—	8	4	no	—
52	C₈×C₁₇	#106	1	4	yes	—
53	C₈×C₁₇	#117	1	2	yes	—
54	C₈×C₁₇	#125	1	2	yes	—
55	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
56	#1845	—	17	2	no	—
57	#1380	—	17	4	no	—
58	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
59	C₂²×C₁₇	#486	1	2	yes	—
60	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
61	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
62	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
63	C₄×C₁₇	#366	1	4	yes	—
64	C₄×C₁₇	#367	1	4	yes	—
65	C₄×C₁₇	#382	1	1	yes	characteristic
66	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
67	C₄×C₁₇	#487	1	2	yes	—
68	Dic₁₇	—	16	4	no	—
69	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
70	C₂×#107	—	17	4	no	—
71	C₂×#107	—	34	2	no	—
72	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
73	#384	—	34	2	no	—
74	C₄×C₁₆	Dic₁₇	1	4	yes	complemented
75	C₂×C₁₇	#1834	1	1	yes	characteristic
76	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
77	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
78	C₂×C₁₇	#2240	1	2	yes	—
79	C₂×C₁₇	#2241	1	2	yes	—
80	C₂²×C₈	D₆₈	1	1	yes	characteristic
81	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	34	4	no	—
83	C₂×C₄⋊C₄	—	68	2	no	—
84	C₂×C₄⋊C₄	—	68	2	no	—
85	C₂×C₄²	D₆₈	1	1	yes	characteristic
86	C₂×C₁₆	—	2	4	no	—
87	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
88	#107	—	34	8	no	—
89	#107	—	68	4	no	—
90	C₄×C₈	Dic₃₄	1	2	yes	—
91	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
92	C₁₇	#12888	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	68	4	no	—
94	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
95	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
97	C₂×C₈	#63219	1	4	yes	—
98	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
99	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
100	C₂×C₈	#63252	1	4	yes	—
101	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₄⋊C₄	—	68	4	no	—
103	C₄⋊C₄	—	68	4	no	—
104	C₄⋊C₄	—	136	4	no	—
105	C₄⋊C₄	—	136	4	no	—
106	C₄²	Dic₆₈	1	2	yes	—
107	C₄²	C₄⋊Dic₁₇	1	2	yes	—
108	C₁₆	—	2	4	no	—
109	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
110	C₂×C₄	—	136	4	no	—
111	C₂×C₄	—	136	4	no	—
112	C₂×C₄	—	136	4	no	—
113	C₂×C₄	D₂₇₂	1	2	yes	—
114	C₂×C₄	#10587471	1	4	yes	—
115	C₂×C₄	Dic₁₃₆	1	2	yes	—
116	C₂×C₄	#10587489	1	2	yes	—
117	C₂×C₄	#10587489	1	2	yes	—
118	C₂×C₄	#10587561	1	2	yes	—
119	C₂×C₄	C₂×#63219	1	2	yes	—
120	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
122	C₈	#10587488	1	4	yes	—
123	C₈	#10587556	1	2	yes	—
124	C₈	#10587612	1	2	yes	—
125	C₂²	C₂×#10587489	1	1	yes	characteristic
126	C₂²	#11763156	1	2	yes	—
127	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
128	C₂²	C₂×#10587471	1	2	yes	—
129	C₂²	C₂×Dic₁₃₆	1	1	yes	characteristic
130	C₄	—	272	4	no	—
131	C₄	#11762635	1	4	yes	—
132	C₄	#11762637	1	4	yes	—
133	C₄	#11762675	1	1	yes	characteristic
134	C₄	C₂×#10587488	1	1	yes	characteristic
135	C₄	#11763160	1	2	yes	—
136	C₂	#14528912	1	1	yes	characteristic
137	C₂	C₂×#11762635	1	1	yes	characteristic
138	C₂	C₂×#11762637	1	1	yes	characteristic
139	C₂	#14531008	1	2	yes	—
140	C₂	#14531012	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#12888
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₆×C₁₇	C₄	semidirect
74. C₄×C₁₆	Dic₁₇	semidirect
92. C₁₇	#12888	semidirect

Maximal subgroups

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

subgroup	index
#14528944	2
#14526876	2
C₂×C₄×C₁₆×C₁₇	2
#12888	17