Subgroups — D₁₇×#2262 #17192448

Order 4352.

470 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇×#508	C₂	1	2	yes	maximal, complemented
2	#14531094	C₂	1	1	yes	maximal, characteristic, complemented
3	#14531093	C₂	1	1	yes	maximal, characteristic, complemented
4	#14531092	C₂	1	2	yes	maximal, complemented
5	#14531091	C₂	1	2	yes	maximal, complemented
6	#14531090	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₇×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₇×SD₆₄	C₂	1	2	yes	maximal, complemented
9	D₁₇×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₂×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#2262	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₁₇	—	2	2	no	—
13	#11763242	C₂²	1	2	yes	complemented
14	#11763241	C₂²	1	2	yes	complemented
15	#11763240	—	2	2	no	—
16	#11763239	—	2	2	no	—
17	#11763238	C₂²	1	2	yes	complemented
18	Q₃₂×D₁₇	C₂²	1	2	yes	—
19	SD₃₂×D₁₇	—	2	2	no	—
20	D₁₆×D₁₇	C₂²	1	2	yes	—
21	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
22	Dic₂₇₂	C₂²	1	1	yes	characteristic, complemented
23	#11762649	C₂²	1	2	yes	complemented
24	#11762648	C₂²	1	2	yes	complemented
25	#11762647	C₂²	1	2	yes	complemented
26	#11762646	C₂²	1	2	yes	complemented
27	D₅₄₄	C₂²	1	1	yes	characteristic, complemented
28	#11762644	C₂²	1	2	yes	complemented
29	C₂×#10587465	C₂²	1	1	yes	characteristic, complemented
30	C₃₂×D₁₇	C₂²	1	1	yes	characteristic
31	C₃₂×D₁₇	C₂²	1	1	yes	characteristic
32	C₃₂×D₁₇	C₂²	1	1	yes	characteristic, complemented
33	C₃₂×D₁₇	C₂²	1	1	yes	characteristic, complemented
34	C₁₇×#508	C₂²	1	2	yes	complemented
35	C₁₇×Q₆₄	C₂²	1	1	yes	characteristic, complemented
36	C₁₇×SD₆₄	C₂²	1	2	yes	complemented
37	C₁₇×D₃₂	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic, complemented
39	#41×D₁₇	—	4	2	no	—
40	C₁₇×#135	—	2	2	no	—
41	#10587636	—	4	2	no	—
42	#10587605	—	2	2	no	—
43	Q₁₆×D₁₇	—	2	2	no	—
44	#10587602	—	4	2	no	—
45	SD₁₆×D₁₇	—	4	2	no	—
46	#10587598	—	2	2	no	—
47	D₈×D₁₇	—	2	2	no	—
48	#10587590	—	2	2	no	—
49	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
50	C₁₇×Q₃₂	C₂³	1	2	yes	—
51	C₁₇×SD₃₂	—	2	2	no	—
52	C₁₇×D₁₆	C₂³	1	2	yes	—
53	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
54	#10587500	—	2	2	no	—
55	#10587499	—	2	2	no	—
56	#10587498	—	2	2	no	—
57	#10587497	—	2	2	no	—
58	C₂×#63214	D₄	1	1	yes	characteristic
59	Dic₁₃₆	C₂³	1	2	yes	—
60	#10587471	—	2	2	no	—
61	D₂₇₂	C₂³	1	2	yes	—
62	C₁₆×D₁₇	D₄	1	1	yes	characteristic
63	C₁₆×D₁₇	D₄	1	1	yes	characteristic
64	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
65	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
66	C₁₇×C₃₂	C₂³	1	1	yes	characteristic
67	C₁₇×C₃₂	C₂³	1	1	yes	characteristic
68	#10587465	C₂³	1	1	yes	characteristic
69	#10587465	C₂³	1	1	yes	characteristic
70	#41×C₁₇	—	4	2	no	—
71	#63256	—	4	2	no	—
72	Q₈×D₁₇	—	4	2	no	—
73	#63254	—	4	2	no	—
74	#63254	—	8	2	no	—
75	D₄×D₁₇	—	4	2	no	—
76	D₄×D₁₇	—	8	2	no	—
77	#63252	—	4	2	no	—
78	#63252	—	8	2	no	—
79	C₂×C₄×D₁₇	—	8	2	no	—
80	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
81	Q₁₆×C₁₇	—	2	2	no	—
82	SD₁₆×C₁₇	—	4	2	no	—
83	D₈×C₁₇	—	2	2	no	—
84	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
85	#63231	—	4	2	no	—
86	#63230	—	4	2	no	—
87	#63229	—	4	2	no	—
88	#63228	—	4	2	no	—
89	C₂×#3622	D₈	1	1	yes	characteristic
90	Dic₆₈	—	2	2	no	—
91	D₁₃₆	—	2	2	no	—
92	#63219	—	4	2	no	—
93	C₈×D₁₇	D₈	1	1	yes	characteristic
94	C₈×D₁₇	D₈	1	1	yes	characteristic
95	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
96	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
97	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
99	#63214	C₂×D₄	1	1	yes	characteristic
100	#63214	C₂×D₄	1	1	yes	characteristic
101	C₂×#2262	—	17	1	no	maximal, Sylow
102	C₂²×D₁₇	—	8	2	no	—
103	C₂²×D₁₇	D₁₆	1	1	yes	characteristic
104	Q₈×C₁₇	—	4	2	no	—
105	D₄×C₁₇	—	4	2	no	—
106	D₄×C₁₇	—	8	2	no	—
107	C₂×C₄×C₁₇	—	8	2	no	—
108	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
109	C₂²⋊D₁₇	—	8	2	no	—
110	C₂²⋊D₁₇	—	8	2	no	—
111	C₂²⋊D₁₇	—	8	2	no	—
112	C₂×Dic₁₇	—	8	2	no	—
113	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
114	D₆₈	—	4	2	no	—
115	D₆₈	—	8	2	no	—
116	C₄×D₁₇	—	8	2	no	—
117	C₄×D₁₇	—	8	2	no	—
118	C₄×D₁₇	—	8	2	no	—
119	C₄×D₁₇	D₁₆	1	1	yes	characteristic
120	C₄×D₁₇	D₁₆	1	1	yes	characteristic
121	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
122	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
123	Dic₃₄	—	4	2	no	—
124	Dic₃₄	—	8	2	no	—
125	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
126	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
127	#3622	C₂×D₈	1	1	yes	characteristic
128	#3622	C₂×D₈	1	1	yes	characteristic
129	C₂×#508	—	17	2	no	—
130	#2262	—	17	2	no	—
131	#2262	—	17	1	no	—
132	#2262	—	17	2	no	—
133	#2262	—	17	1	no	—
134	#2262	—	17	1	no	—
135	#2262	D₁₇	1	1	yes	characteristic, direct product factor
136	C₂×Q₆₄	—	17	1	no	—
137	C₂×SD₆₄	—	17	2	no	—
138	C₂×D₃₂	—	17	1	no	—
139	C₂²×C₃₂	—	17	1	no	—
140	C₂²×C₁₇	—	8	2	no	—
141	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
142	C₂×D₁₇	—	2	2	no	—
143	C₂×D₁₇	—	8	2	no	—
144	C₂×D₁₇	—	16	4	no	—
145	C₂×D₁₇	C₂×D₁₆	1	1	yes	characteristic
146	C₂×D₁₇	C₂×D₁₆	1	1	yes	characteristic
147	C₄×C₁₇	—	8	2	no	—
148	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
149	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
150	Dic₁₇	—	8	2	no	—
151	Dic₁₇	C₂×D₁₆	1	1	yes	characteristic
152	Dic₁₇	C₂×D₁₆	1	1	yes	characteristic
153	C₂×#135	—	34	2	no	—
154	#508	—	17	2	no	—
155	#508	—	17	2	no	—
156	#508	—	17	2	no	—
157	#508	—	34	2	no	—
158	#508	—	34	2	no	—
159	#508	C₂×D₁₇	1	2	yes	complemented
160	C₂×Q₃₂	—	17	2	no	—
161	C₂×SD₃₂	—	34	2	no	—
162	C₂×D₁₆	—	17	2	no	—
163	C₂²×C₁₆	—	17	1	no	—
164	Q₆₄	—	17	2	no	—
165	Q₆₄	—	17	1	no	—
166	Q₆₄	C₂×D₁₇	1	1	yes	characteristic, complemented
167	SD₆₄	—	17	2	no	—
168	SD₆₄	—	17	2	no	—
169	SD₆₄	—	17	2	no	—
170	SD₆₄	C₂×D₁₇	1	2	yes	complemented
171	D₃₂	—	17	2	no	—
172	D₃₂	—	17	1	no	—
173	D₃₂	C₂×D₁₇	1	1	yes	characteristic, complemented
174	C₂×C₃₂	—	17	1	no	—
175	C₂×C₃₂	—	17	1	no	—
176	C₂×C₃₂	—	17	1	no	—
177	C₂×C₃₂	—	17	1	no	—
178	C₂×C₃₂	—	17	1	no	—
179	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic, complemented
180	C₂×C₁₇	—	2	1	no	—
181	C₂×C₁₇	—	16	2	no	—
182	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
183	D₁₇	—	2	1	no	—
184	D₁₇	—	16	2	no	—
185	D₁₇	#2262	1	2	yes	direct product factor
186	C₂×#41	—	68	2	no	—
187	#135	—	2	2	no	—
188	#135	—	34	2	no	—
189	#135	—	34	2	no	—
190	#135	—	34	2	no	—
191	#135	—	68	2	no	—
192	#135	—	68	2	no	—
193	C₂×Q₁₆	—	34	2	no	—
194	C₂×SD₁₆	—	68	2	no	—
195	C₂×D₈	—	34	2	no	—
196	C₂²×C₈	—	17	1	no	—
197	Q₃₂	—	17	2	no	—
198	Q₃₂	—	34	2	no	—
199	Q₃₂	C₂²×D₁₇	1	2	yes	—
200	SD₃₂	—	2	2	no	—
201	SD₃₂	—	34	2	no	—
202	SD₃₂	—	34	2	no	—
203	SD₃₂	—	34	2	no	—
204	D₁₆	—	17	2	no	—
205	D₁₆	—	34	2	no	—
206	D₁₆	C₂²×D₁₇	1	2	yes	—
207	C₂×C₁₆	—	17	1	no	—
208	C₂×C₁₆	—	17	1	no	—
209	C₂×C₁₆	—	17	1	no	—
210	C₂×C₁₆	—	17	1	no	—
211	C₂×C₁₆	—	17	1	no	—
212	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
213	C₃₂	—	17	1	no	—
214	C₃₂	—	17	1	no	—
215	C₃₂	C₂²×D₁₇	1	1	yes	characteristic
216	C₃₂	C₂²×D₁₇	1	1	yes	characteristic
217	C₁₇	C₂×#2262	1	1	yes	characteristic, complemented, Sylow
218	#41	—	4	2	no	—
219	#41	—	68	2	no	—
220	#41	—	68	2	no	—
221	#41	—	68	2	no	—
222	#41	—	136	2	no	—
223	#41	—	136	2	no	—
224	C₂×Q₈	—	68	2	no	—
225	C₂×D₄	—	68	2	no	—
226	C₂×D₄	—	136	2	no	—
227	C₂²×C₄	—	17	1	no	—
228	C₂²×C₄	—	136	2	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	34	2	no	—
231	Q₁₆	—	68	2	no	—
232	SD₁₆	—	4	2	no	—
233	SD₁₆	—	68	2	no	—
234	SD₁₆	—	68	2	no	—
235	SD₁₆	—	68	2	no	—
236	D₈	—	2	2	no	—
237	D₈	—	34	2	no	—
238	D₈	—	68	2	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	17	1	no	—
242	C₂×C₈	—	17	1	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
245	C₁₆	—	17	1	no	—
246	C₁₆	—	17	1	no	—
247	C₁₆	D₄×D₁₇	1	1	yes	characteristic
248	C₁₆	C₂³×D₁₇	1	1	yes	characteristic
249	C₂³	—	17	1	no	—
250	C₂³	—	136	2	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	68	2	no	—
253	Q₈	—	136	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	68	2	no	—
257	D₄	—	136	2	no	—
258	D₄	—	136	2	no	—
259	D₄	—	136	2	no	—
260	D₄	—	136	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	17	1	no	—
263	C₂×C₄	—	17	1	no	—
264	C₂×C₄	—	17	1	no	—
265	C₂×C₄	—	17	1	no	—
266	C₂×C₄	—	17	1	no	—
267	C₂×C₄	—	136	2	no	—
268	C₂×C₄	—	136	2	no	—
269	C₂×C₄	—	136	2	no	—
270	C₂×C₄	—	136	2	no	—
271	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
272	C₈	—	17	1	no	—
273	C₈	—	17	1	no	—
274	C₈	D₈×D₁₇	1	1	yes	characteristic
275	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
276	C₂²	—	8	2	no	—
277	C₂²	—	272	4	no	—
278	C₂²	—	17	1	no	—
279	C₂²	—	17	1	no	—
280	C₂²	—	34	2	no	—
281	C₂²	—	136	2	no	—
282	C₂²	D₁₆×D₁₇	1	1	yes	characteristic
283	C₄	—	8	2	no	—
284	C₄	—	17	1	no	—
285	C₄	—	17	1	no	—
286	C₄	—	136	2	no	—
287	C₄	D₁₆×D₁₇	1	1	yes	characteristic
288	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	16	2	no	—
291	C₂	—	272	2	no	—
292	C₂	—	17	2	no	—
293	C₂	—	34	1	no	—
294	C₂	C₂×D₁₆×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₁₇
commutator, cosocle	C₁₆×C₁₇	C₂⁴
Frattini	C₁₆	C₂³×D₁₇
Fitting	C₁₇×#2262	C₂
socle	C₂×C₁₇	C₂²×D₁₆
hypercenter	#2262	D₁₇
hypocenter	C₁₇	C₂×#2262
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. D₁₇×#508	C₂	semidirect
2. #14531094	C₂	semidirect
3. #14531093	C₂	semidirect
4. #14531092	C₂	semidirect
5. #14531091	C₂	semidirect
6. #14531090	C₂	semidirect
7. D₁₇×Q₆₄	C₂	semidirect
8. D₁₇×SD₆₄	C₂	semidirect
9. D₁₇×D₃₂	C₂	semidirect
10. C₂×C₃₂×D₁₇	C₂	semidirect
11. C₁₇×#2262	C₂	semidirect
13. #11763242	C₂²	semidirect
14. #11763241	C₂²	semidirect
17. #11763238	C₂²	semidirect
22. Dic₂₇₂	C₂²	semidirect
23. #11762649	C₂²	semidirect
24. #11762648	C₂²	semidirect
25. #11762647	C₂²	semidirect
26. #11762646	C₂²	semidirect
27. D₅₄₄	C₂²	semidirect
28. #11762644	C₂²	semidirect
29. C₂×#10587465	C₂²	semidirect
32. C₃₂×D₁₇	C₂²	semidirect
33. C₃₂×D₁₇	C₂²	semidirect
34. C₁₇×#508	C₂²	semidirect
35. C₁₇×Q₆₄	C₂²	semidirect
36. C₁₇×SD₆₄	C₂²	semidirect
37. C₁₇×D₃₂	C₂²	semidirect
38. C₂×C₁₇×C₃₂	C₂²	semidirect
135. #2262	D₁₇	direct (N × H)
159. #508	C₂×D₁₇	semidirect
166. Q₆₄	C₂×D₁₇	semidirect
170. SD₆₄	C₂×D₁₇	semidirect
173. D₃₂	C₂×D₁₇	semidirect
179. C₂×C₃₂	C₂×D₁₇	semidirect
185. D₁₇	#2262	direct (N × H)
217. C₁₇	C₂×#2262	semidirect

Maximal subgroups

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

subgroup	index
D₁₇×#508	2
#14531094	2
#14531093	2
#14531092	2
#14531091	2
#14531090	2
D₁₇×Q₆₄	2
D₁₇×SD₆₄	2
D₁₇×D₃₂	2
C₂×C₃₂×D₁₇	2
C₁₇×#2262	2
C₂×#2262	17

Subgroups — D17×#2262 #17192448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇×#2262 #17192448