Subgroups — D₁₇⋊#2178 #17978611

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇⋊#443	C₂	1	2	yes	maximal, complemented
2	#14545185	C₂	1	2	yes	maximal, complemented
3	#14545184	C₂	1	4	yes	maximal, complemented
4	#14545183	C₂	1	2	yes	maximal, complemented
5	D₁₇⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₇⋊#359	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×#595	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₇×#508	C₂	1	1	yes	maximal, characteristic
9	D₁₇⋊#131	—	2	2	no	—
10	C₂×#10587529	C₂²	1	1	yes	characteristic
11	#11763853	C₂²	1	4	yes	complemented
12	#11763852	—	2	4	no	—
13	#11763851	C₂²	1	4	yes	complemented
14	D₁₇⋊#108	C₂²	1	2	yes	—
15	D₁₇⋊#104	—	2	2	no	—
16	C₂×C₈×#595	C₂²	1	1	yes	characteristic
17	#11763738	C₂²	1	2	yes	complemented
18	#11763737	C₂²	1	2	yes	complemented
19	#11763724	C₂²	1	4	yes	complemented
20	#11763723	C₂²	1	4	yes	complemented
21	C₁₆×#595	C₂²	1	2	yes	complemented
22	C₁₆×#595	C₂²	1	2	yes	complemented
23	#135×D₁₇	C₂²	1	2	yes	—
24	#11763242	C₂²	1	1	yes	characteristic
25	#11763241	C₄	1	1	yes	characteristic, complemented
26	#11763240	C₂²	1	2	yes	—
27	#11763239	C₄	1	2	yes	complemented
28	#11763238	C₂²	1	1	yes	characteristic
29	Q₃₂×D₁₇	C₄	1	1	yes	characteristic, complemented
30	SD₃₂×D₁₇	C₄	1	2	yes	complemented
31	D₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
32	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
33	C₁₇×#508	C₄	1	1	yes	characteristic, complemented
34	#41×D₁₇	—	2	2	no	—
35	#10587680	—	2	4	no	—
36	#10587678	—	2	4	no	—
37	#10587678	—	4	4	no	—
38	C₂×C₄×#595	D₄	1	1	yes	characteristic
39	D₁₇⋊M₁₆	—	2	2	no	—
40	D₁₇⋊M₁₆	—	4	2	no	—
41	C₂×#63242	—	4	2	no	—
42	C₂×#63242	D₄	1	1	yes	characteristic
43	C₁₇×#135	C₂×C₄	1	2	yes	—
44	#10587636	—	2	2	no	—
45	#10587605	C₂³	1	2	yes	—
46	Q₁₆×D₁₇	C₂×C₄	1	2	yes	—
47	#10587602	—	2	2	no	—
48	SD₁₆×D₁₇	—	2	2	no	—
49	#10587598	C₂³	1	2	yes	—
50	D₈×D₁₇	C₂×C₄	1	2	yes	—
51	#10587590	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
53	#10587550	—	2	4	no	—
54	#10587548	—	2	4	no	—
55	C₂×#63216	—	2	1	no	—
56	#10587536	C₂³	1	4	yes	—
57	#10587535	—	2	2	no	—
58	C₈×#595	D₄	1	2	yes	—
59	C₈×#595	C₂³	1	2	yes	—
60	#10587529	—	2	1	no	—
61	#10587529	C₂³	1	2	yes	—
62	C₁₇×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
63	C₁₇×SD₃₂	C₂×C₄	1	2	yes	complemented
64	C₁₇×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
66	#10587500	C₂×C₄	1	2	yes	complemented
67	#10587499	C₂×C₄	1	2	yes	complemented
68	#10587498	C₂×C₄	1	2	yes	complemented
69	#10587497	C₂×C₄	1	2	yes	complemented
70	C₂×#63214	C₂×C₄	1	1	yes	characteristic
71	Dic₁₃₆	C₂×C₄	1	1	yes	characteristic, complemented
72	#10587471	C₂×C₄	1	2	yes	complemented
73	D₂₇₂	C₂×C₄	1	1	yes	characteristic, complemented
74	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
75	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
76	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
77	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
78	C₂²×#595	—	2	1	no	—
79	#41×C₁₇	—	2	2	no	—
80	#63256	—	2	2	no	—
81	Q₈×D₁₇	—	2	2	no	—
82	#63254	—	2	2	no	—
83	#63254	—	4	2	no	—
84	D₄×D₁₇	—	2	2	no	—
85	D₄×D₁₇	—	4	2	no	—
86	#63252	—	2	2	no	—
87	#63252	—	4	2	no	—
88	C₂×C₄×D₁₇	—	4	2	no	—
89	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
90	#63247	—	4	2	no	—
91	#63247	—	4	4	no	—
92	C₂×#3624	—	4	4	no	—
93	C₂×#3624	D₈	1	2	yes	—
94	C₄×#595	—	2	1	no	—
95	C₄×#595	C₂×D₄	1	2	yes	—
96	#63243	—	2	4	no	—
97	#63243	—	4	4	no	—
98	#63242	—	4	2	no	—
99	#63242	—	4	4	no	—
100	#63242	D₈	1	1	yes	characteristic
101	#63242	D₈	1	1	yes	characteristic
102	#63242	C₂×D₄	1	2	yes	—
103	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₁₇	—	2	2	no	—
105	D₈×C₁₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
107	#63231	—	2	2	no	—
108	#63230	—	2	2	no	—
109	#63229	—	2	2	no	—
110	#63228	—	2	2	no	—
111	C₂×#3622	#41	1	1	yes	characteristic
112	Dic₆₈	C₂²×C₄	1	2	yes	—
113	D₁₃₆	C₂²×C₄	1	2	yes	—
114	#63219	—	2	2	no	—
115	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
117	C₈×D₁₇	#41	1	1	yes	characteristic
118	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
119	#63216	—	2	2	no	—
120	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
122	#63214	C₂²×C₄	1	1	yes	characteristic
123	#63214	C₂²×C₄	1	1	yes	characteristic
124	C₂×#2178	—	17	1	no	maximal, Sylow
125	C₂²×D₁₇	—	4	2	no	—
126	C₂²×D₁₇	#135	1	1	yes	characteristic
127	C₂×#595	—	2	4	no	—
128	C₂×#595	—	2	2	no	—
129	Q₈×C₁₇	—	2	2	no	—
130	D₄×C₁₇	—	2	2	no	—
131	D₄×C₁₇	—	4	2	no	—
132	C₂×C₄×C₁₇	—	4	2	no	—
133	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
134	C₂²⋊D₁₇	—	4	2	no	—
135	C₂²⋊D₁₇	—	4	2	no	—
136	C₂²⋊D₁₇	—	4	2	no	—
137	C₂×Dic₁₇	—	4	2	no	—
138	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
139	D₆₈	—	2	2	no	—
140	D₆₈	—	4	2	no	—
141	C₄×D₁₇	—	4	2	no	—
142	C₄×D₁₇	—	4	2	no	—
143	C₄×D₁₇	—	4	2	no	—
144	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
145	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
146	C₄×D₁₇	#135	1	1	yes	characteristic
147	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
148	Dic₃₄	—	2	2	no	—
149	Dic₃₄	—	4	2	no	—
150	#3624	—	4	4	no	—
151	#3624	C₂×D₈	1	2	yes	—
152	#3624	C₂×D₈	1	2	yes	—
153	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
155	#3622	C₄×D₄	1	1	yes	characteristic
156	#3622	C₂×#41	1	1	yes	characteristic
157	C₂×#508	—	17	1	no	—
158	C₂×#443	—	17	2	no	—
159	#2178	—	17	2	no	—
160	#2178	—	17	4	no	—
161	#2178	—	17	2	no	—
162	C₂×#368	—	17	1	no	—
163	C₂×#359	—	17	2	no	—
164	C₂×C₄×C₁₆	—	17	1	no	—
165	C₂²×C₁₇	—	4	2	no	—
166	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
167	C₂×D₁₇	—	2	2	no	—
168	C₂×D₁₇	—	4	2	no	—
169	C₂×D₁₇	—	8	4	no	—
170	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
171	C₂×D₁₇	C₂×#135	1	1	yes	characteristic
172	#595	—	2	4	no	—
173	C₄×C₁₇	—	4	2	no	—
174	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
175	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
176	Dic₁₇	—	4	2	no	—
177	Dic₁₇	C₄×D₈	1	1	yes	characteristic
178	Dic₁₇	C₂²×D₈	1	1	yes	characteristic
179	C₂×#135	—	17	2	no	—
180	C₂×#131	—	34	2	no	—
181	#508	—	17	2	no	—
182	#508	—	17	1	no	—
183	#508	—	17	2	no	—
184	#508	—	17	1	no	—
185	#508	—	17	1	no	—
186	#508	#595	1	1	yes	characteristic, complemented
187	C₂×Q₃₂	—	17	1	no	—
188	C₂×SD₃₂	—	17	2	no	—
189	C₂×D₁₆	—	17	1	no	—
190	C₂²×C₁₆	—	17	1	no	—
191	C₂²×C₁₆	—	17	1	no	—
192	#443	—	17	4	no	—
193	#443	—	17	4	no	—
194	#443	—	34	4	no	—
195	C₂×#108	—	17	2	no	—
196	C₂×#104	—	34	2	no	—
197	C₂×C₄×C₈	—	17	1	no	—
198	#368	—	17	2	no	—
199	#368	—	17	2	no	—
200	#359	—	17	4	no	—
201	#359	—	17	4	no	—
202	C₄×C₁₆	—	17	2	no	—
203	C₄×C₁₆	—	17	2	no	—
204	C₂×C₁₇	—	2	1	no	—
205	C₂×C₁₇	—	8	2	no	—
206	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
207	D₁₇	—	2	1	no	—
208	D₁₇	—	8	2	no	—
209	D₁₇	#2178	1	2	yes	complemented
210	C₂×#41	—	34	2	no	—
211	#135	—	17	2	no	—
212	#135	—	17	2	no	—
213	#135	—	17	2	no	—
214	#135	—	34	2	no	—
215	#135	—	34	2	no	—
216	#135	C₂×#595	1	2	yes	—
217	C₂×Q₁₆	—	17	2	no	—
218	C₂×SD₁₆	—	34	2	no	—
219	C₂×D₈	—	17	2	no	—
220	#131	—	34	4	no	—
221	#131	—	34	4	no	—
222	#131	—	68	4	no	—
223	C₂×M₁₆	—	34	2	no	—
224	C₂×M₁₆	—	68	2	no	—
225	C₂²×C₈	—	17	1	no	—
226	C₂²×C₈	—	17	1	no	—
227	C₂²×C₈	—	68	2	no	—
228	C₂×C₄²	—	17	1	no	—
229	Q₃₂	—	17	2	no	—
230	Q₃₂	—	17	1	no	—
231	Q₃₂	C₂×#595	1	1	yes	characteristic, complemented
232	SD₃₂	—	17	2	no	—
233	SD₃₂	—	17	2	no	—
234	SD₃₂	—	17	2	no	—
235	SD₃₂	C₂×#595	1	2	yes	complemented
236	D₁₆	—	17	2	no	—
237	D₁₆	—	17	1	no	—
238	D₁₆	C₂×#595	1	1	yes	characteristic, complemented
239	C₂×C₁₆	—	17	1	no	—
240	C₂×C₁₆	—	17	1	no	—
241	C₂×C₁₆	—	17	1	no	—
242	C₂×C₁₆	—	17	1	no	—
243	C₂×C₁₆	—	17	1	no	—
244	C₂×C₁₆	—	17	2	no	—
245	C₂×C₁₆	—	34	1	no	—
246	C₂×C₁₆	—	34	1	no	—
247	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
248	#108	—	17	4	no	—
249	#108	—	34	2	no	—
250	#104	—	34	4	no	—
251	#104	—	34	4	no	—
252	C₄×C₈	—	17	2	no	—
253	C₄×C₈	—	17	2	no	—
254	C₁₇	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
255	#41	—	2	2	no	—
256	#41	—	34	2	no	—
257	#41	—	34	2	no	—
258	#41	—	34	2	no	—
259	#41	—	68	2	no	—
260	#41	—	68	2	no	—
261	C₂×Q₈	—	34	2	no	—
262	C₂×D₄	—	34	2	no	—
263	C₂×D₄	—	68	2	no	—
264	C₂²×C₄	—	17	1	no	—
265	C₂²×C₄	—	34	1	no	—
266	C₂²×C₄	—	68	2	no	—
267	Q₁₆	—	17	2	no	—
268	Q₁₆	—	34	2	no	—
269	Q₁₆	C₂²×#595	1	2	yes	—
270	SD₁₆	—	2	2	no	—
271	SD₁₆	—	34	2	no	—
272	SD₁₆	—	34	2	no	—
273	SD₁₆	—	34	2	no	—
274	D₈	—	17	2	no	—
275	D₈	—	34	2	no	—
276	D₈	C₂²×#595	1	2	yes	—
277	M₁₆	—	34	4	no	—
278	M₁₆	—	68	4	no	—
279	M₁₆	—	68	4	no	—
280	M₁₆	—	68	2	no	—
281	C₂×C₈	—	17	1	no	—
282	C₂×C₈	—	17	1	no	—
283	C₂×C₈	—	17	2	no	—
284	C₂×C₈	—	17	2	no	—
285	C₂×C₈	—	17	1	no	—
286	C₂×C₈	—	17	1	no	—
287	C₂×C₈	—	17	1	no	—
288	C₂×C₈	—	17	1	no	—
289	C₂×C₈	—	17	1	no	—
290	C₂×C₈	—	68	4	no	—
291	C₂×C₈	—	68	4	no	—
292	C₂×C₈	—	68	2	no	—
293	C₂×C₈	C₂²×#595	1	1	yes	characteristic
294	C₄²	—	17	2	no	—
295	C₄²	—	34	1	no	—
296	C₁₆	—	17	1	no	—
297	C₁₆	—	17	1	no	—
298	C₁₆	—	34	2	no	—
299	C₁₆	C₂²×#595	1	1	yes	characteristic
300	C₁₆	C₂²×#595	1	1	yes	characteristic
301	C₂³	—	17	1	no	—
302	C₂³	—	68	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	34	2	no	—
305	Q₈	—	68	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	34	2	no	—
309	D₄	—	68	2	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	68	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	17	1	no	—
317	C₂×C₄	—	17	1	no	—
318	C₂×C₄	—	17	1	no	—
319	C₂×C₄	—	34	2	no	—
320	C₂×C₄	—	34	4	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	—	68	2	no	—
324	C₂×C₄	—	68	2	no	—
325	C₂×C₄	D₄×#595	1	1	yes	characteristic
326	C₈	—	17	2	no	—
327	C₈	—	17	2	no	—
328	C₈	—	17	1	no	—
329	C₈	—	17	1	no	—
330	C₈	—	68	4	no	—
331	C₈	D₄×#595	1	1	yes	characteristic
332	C₈	C₂³×#595	1	1	yes	characteristic
333	C₂²	—	4	2	no	—
334	C₂²	—	17	1	no	—
335	C₂²	—	17	1	no	—
336	C₂²	—	34	2	no	—
337	C₂²	—	68	2	no	—
338	C₂²	—	136	4	no	—
339	C₂²	D₈×#595	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	17	1	no	—
342	C₄	—	17	1	no	—
343	C₄	—	34	4	no	—
344	C₄	—	68	2	no	—
345	C₄	D₈×#595	1	1	yes	characteristic
346	C₄	C₂×D₄×#595	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	8	2	no	—
349	C₂	—	17	2	no	—
350	C₂	—	34	1	no	—
351	C₂	—	136	2	no	—
352	C₂	C₂×D₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×#595
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₈	C₂³×#595
Fitting	C₁₇×#508	C₄
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	D₁₇×#508	C₂
cosocle	C₈×D₁₇	C₂⁴
hypercenter	#508	#595
hypocenter	C₁₇	C₂×#2178
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. D₁₇⋊#443	C₂	semidirect
2. #14545185	C₂	semidirect
3. #14545184	C₂	semidirect
4. #14545183	C₂	semidirect
5. D₁₇⋊#368	C₂	semidirect
6. D₁₇⋊#359	C₂	semidirect
7. C₂×C₁₆×#595	C₂	semidirect
11. #11763853	C₂²	semidirect
13. #11763851	C₂²	semidirect
17. #11763738	C₂²	semidirect
18. #11763737	C₂²	semidirect
19. #11763724	C₂²	semidirect
20. #11763723	C₂²	semidirect
21. C₁₆×#595	C₂²	semidirect
22. C₁₆×#595	C₂²	semidirect
25. #11763241	C₄	semidirect
27. #11763239	C₄	semidirect
29. Q₃₂×D₁₇	C₄	semidirect
30. SD₃₂×D₁₇	C₄	semidirect
31. D₁₆×D₁₇	C₄	semidirect
33. C₁₇×#508	C₄	semidirect
62. C₁₇×Q₃₂	C₂×C₄	semidirect
63. C₁₇×SD₃₂	C₂×C₄	semidirect
64. C₁₇×D₁₆	C₂×C₄	semidirect
66. #10587500	C₂×C₄	semidirect
67. #10587499	C₂×C₄	semidirect
68. #10587498	C₂×C₄	semidirect
69. #10587497	C₂×C₄	semidirect
71. Dic₁₃₆	C₂×C₄	semidirect
72. #10587471	C₂×C₄	semidirect
73. D₂₇₂	C₂×C₄	semidirect
186. #508	#595	semidirect
209. D₁₇	#2178	semidirect
231. Q₃₂	C₂×#595	semidirect
235. SD₃₂	C₂×#595	semidirect
238. D₁₆	C₂×#595	semidirect
254. C₁₇	C₂×#2178	semidirect

Maximal subgroups

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

subgroup	index
D₁₇⋊#443	2
#14545185	2
#14545184	2
#14545183	2
D₁₇⋊#368	2
D₁₇⋊#359	2
C₂×C₁₆×#595	2
D₁₇×#508	2
C₂×#2178	17

Subgroups — D17⋊#2178 #17978611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇⋊#2178 #17978611