Subgroups — D₁₇⋊#2179 #17978616

Order 4352.

654 conjugacy classes of subgroups.

The subgroup lattice has 350 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	#14545188	C₂	1	2	yes	maximal, complemented
3	#14545187	C₂	1	2	yes	maximal, complemented
4	#14545186	C₂	1	4	yes	maximal, complemented
5	D₁₇⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₇⋊#359	C₂	1	2	yes	maximal, complemented
7	C₂×#11763696	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	#11763696	C₂²	1	2	yes	complemented
22	#11763696	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	2	yes	—
101	#63242	C₂×D₄	1	2	yes	—
102	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
103	SD₁₆×C₁₇	—	2	2	no	—
104	D₈×C₁₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	#63231	—	2	2	no	—
107	#63230	—	2	2	no	—
108	#63229	—	2	2	no	—
109	#63228	—	2	2	no	—
110	C₂×#3622	#41	1	1	yes	characteristic
111	Dic₆₈	C₂²×C₄	1	2	yes	—
112	D₁₃₆	C₂²×C₄	1	2	yes	—
113	#63219	—	2	2	no	—
114	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
116	C₈×D₁₇	#41	1	1	yes	characteristic
117	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
118	#63216	—	2	2	no	—
119	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
121	#63214	C₂²×C₄	1	1	yes	characteristic
122	#63214	C₂²×C₄	1	1	yes	characteristic
123	C₂×#2179	—	17	1	no	maximal, Sylow
124	C₂²×D₁₇	—	4	2	no	—
125	C₂²×D₁₇	#135	1	1	yes	characteristic
126	C₂×#595	—	2	2	no	—
127	C₂×#595	—	4	2	no	—
128	Q₈×C₁₇	—	2	2	no	—
129	D₄×C₁₇	—	2	2	no	—
130	D₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	—	4	2	no	—
132	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
133	C₂²⋊D₁₇	—	4	2	no	—
134	C₂²⋊D₁₇	—	4	2	no	—
135	C₂²⋊D₁₇	—	4	2	no	—
136	C₂×Dic₁₇	—	4	2	no	—
137	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
138	D₆₈	—	2	2	no	—
139	D₆₈	—	4	2	no	—
140	C₄×D₁₇	—	4	2	no	—
141	C₄×D₁₇	—	4	2	no	—
142	C₄×D₁₇	—	4	2	no	—
143	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
144	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
145	C₄×D₁₇	#135	1	1	yes	characteristic
146	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
147	Dic₃₄	—	2	2	no	—
148	Dic₃₄	—	4	2	no	—
149	#3624	—	4	4	no	—
150	#3624	C₂×D₈	1	4	yes	—
151	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
152	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
153	#3622	C₄×D₄	1	1	yes	characteristic
154	#3622	C₂×#41	1	1	yes	characteristic
155	C₂×#508	—	17	1	no	—
156	C₂×#443	—	17	2	no	—
157	#2179	—	17	4	no	—
158	#2179	—	17	2	no	—
159	#2179	—	17	2	no	—
160	C₂×#368	—	17	1	no	—
161	C₂×#359	—	17	2	no	—
162	C₂×#346	—	17	1	no	—
163	C₂²×C₁₇	—	4	2	no	—
164	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
165	C₂×D₁₇	—	2	2	no	—
166	C₂×D₁₇	—	4	2	no	—
167	C₂×D₁₇	—	8	4	no	—
168	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
169	C₂×D₁₇	C₂×#135	1	1	yes	characteristic
170	#595	—	4	2	no	—
171	C₄×C₁₇	—	4	2	no	—
172	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
173	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
174	Dic₁₇	—	4	2	no	—
175	Dic₁₇	C₄×D₈	1	1	yes	characteristic
176	Dic₁₇	C₂²×D₈	1	1	yes	characteristic
177	C₂×#135	—	17	2	no	—
178	C₂×#131	—	34	2	no	—
179	#508	—	17	2	no	—
180	#508	—	17	1	no	—
181	#508	—	17	2	no	—
182	#508	—	17	1	no	—
183	#508	—	17	1	no	—
184	#508	#595	1	1	yes	characteristic, complemented
185	C₂×Q₃₂	—	17	1	no	—
186	C₂×SD₃₂	—	17	2	no	—
187	C₂×D₁₆	—	17	1	no	—
188	C₂²×C₁₆	—	17	1	no	—
189	C₂²×C₁₆	—	17	1	no	—
190	#443	—	17	4	no	—
191	#443	—	17	4	no	—
192	#443	—	34	4	no	—
193	C₂×#108	—	17	2	no	—
194	C₂×#104	—	34	2	no	—
195	C₂×C₄×C₈	—	17	1	no	—
196	#368	—	17	2	no	—
197	#368	—	17	2	no	—
198	#359	—	17	4	no	—
199	#359	—	17	4	no	—
200	#346	—	17	2	no	—
201	#346	—	17	2	no	—
202	C₂×C₁₇	—	2	1	no	—
203	C₂×C₁₇	—	8	2	no	—
204	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
205	D₁₇	—	2	1	no	—
206	D₁₇	—	8	2	no	—
207	D₁₇	#2179	1	2	yes	complemented
208	C₂×#41	—	34	2	no	—
209	#135	—	17	2	no	—
210	#135	—	17	2	no	—
211	#135	—	17	2	no	—
212	#135	—	34	2	no	—
213	#135	—	34	2	no	—
214	#135	C₂×#595	1	2	yes	—
215	C₂×Q₁₆	—	17	2	no	—
216	C₂×SD₁₆	—	34	2	no	—
217	C₂×D₈	—	17	2	no	—
218	#131	—	34	4	no	—
219	#131	—	34	4	no	—
220	#131	—	68	4	no	—
221	C₂×M₁₆	—	34	2	no	—
222	C₂×M₁₆	—	68	2	no	—
223	C₂²×C₈	—	17	1	no	—
224	C₂²×C₈	—	17	1	no	—
225	C₂²×C₈	—	68	2	no	—
226	C₂×C₄²	—	17	1	no	—
227	Q₃₂	—	17	2	no	—
228	Q₃₂	—	17	1	no	—
229	Q₃₂	C₂×#595	1	1	yes	characteristic, complemented
230	SD₃₂	—	17	2	no	—
231	SD₃₂	—	17	2	no	—
232	SD₃₂	—	17	2	no	—
233	SD₃₂	C₂×#595	1	2	yes	complemented
234	D₁₆	—	17	2	no	—
235	D₁₆	—	17	1	no	—
236	D₁₆	C₂×#595	1	1	yes	characteristic, complemented
237	C₂×C₁₆	—	17	1	no	—
238	C₂×C₁₆	—	17	1	no	—
239	C₂×C₁₆	—	17	1	no	—
240	C₂×C₁₆	—	17	1	no	—
241	C₂×C₁₆	—	17	1	no	—
242	C₂×C₁₆	—	17	2	no	—
243	C₂×C₁₆	—	34	1	no	—
244	C₂×C₁₆	—	34	1	no	—
245	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
246	#108	—	17	4	no	—
247	#108	—	34	2	no	—
248	#104	—	34	4	no	—
249	#104	—	34	4	no	—
250	C₄×C₈	—	17	2	no	—
251	C₄×C₈	—	17	2	no	—
252	C₁₇	C₂×#2179	1	1	yes	characteristic, complemented, Sylow
253	#41	—	2	2	no	—
254	#41	—	34	2	no	—
255	#41	—	34	2	no	—
256	#41	—	34	2	no	—
257	#41	—	68	2	no	—
258	#41	—	68	2	no	—
259	C₂×Q₈	—	34	2	no	—
260	C₂×D₄	—	34	2	no	—
261	C₂×D₄	—	68	2	no	—
262	C₂²×C₄	—	17	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	C₂²×C₄	—	68	2	no	—
265	Q₁₆	—	17	2	no	—
266	Q₁₆	—	34	2	no	—
267	Q₁₆	C₂²×#595	1	2	yes	—
268	SD₁₆	—	2	2	no	—
269	SD₁₆	—	34	2	no	—
270	SD₁₆	—	34	2	no	—
271	SD₁₆	—	34	2	no	—
272	D₈	—	17	2	no	—
273	D₈	—	34	2	no	—
274	D₈	C₂²×#595	1	2	yes	—
275	M₁₆	—	34	4	no	—
276	M₁₆	—	68	4	no	—
277	M₁₆	—	68	4	no	—
278	M₁₆	—	68	2	no	—
279	C₂×C₈	—	17	2	no	—
280	C₂×C₈	—	17	2	no	—
281	C₂×C₈	—	17	2	no	—
282	C₂×C₈	—	17	1	no	—
283	C₂×C₈	—	17	1	no	—
284	C₂×C₈	—	17	1	no	—
285	C₂×C₈	—	17	1	no	—
286	C₂×C₈	—	17	1	no	—
287	C₂×C₈	—	68	4	no	—
288	C₂×C₈	—	68	4	no	—
289	C₂×C₈	—	68	2	no	—
290	C₂×C₈	C₂²×#595	1	1	yes	characteristic
291	C₄²	—	17	2	no	—
292	C₄²	—	34	1	no	—
293	C₁₆	—	17	1	no	—
294	C₁₆	—	17	1	no	—
295	C₁₆	—	34	2	no	—
296	C₁₆	C₂²×#595	1	1	yes	characteristic
297	C₁₆	C₂²×#595	1	1	yes	characteristic
298	C₂³	—	17	1	no	—
299	C₂³	—	68	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	34	2	no	—
302	Q₈	—	68	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	34	2	no	—
306	D₄	—	68	2	no	—
307	D₄	—	68	2	no	—
308	D₄	—	68	2	no	—
309	D₄	—	68	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	34	2	no	—
317	C₂×C₄	—	68	2	no	—
318	C₂×C₄	—	68	2	no	—
319	C₂×C₄	—	68	2	no	—
320	C₂×C₄	—	68	2	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	D₄×#595	1	1	yes	characteristic
323	C₈	—	17	4	no	—
324	C₈	—	17	1	no	—
325	C₈	—	17	1	no	—
326	C₈	—	68	4	no	—
327	C₈	D₄×#595	1	1	yes	characteristic
328	C₈	C₂³×#595	1	1	yes	characteristic
329	C₂²	—	4	2	no	—
330	C₂²	—	17	1	no	—
331	C₂²	—	17	1	no	—
332	C₂²	—	34	2	no	—
333	C₂²	—	68	2	no	—
334	C₂²	—	136	4	no	—
335	C₂²	D₈×#595	1	1	yes	characteristic
336	C₄	—	4	2	no	—
337	C₄	—	17	1	no	—
338	C₄	—	17	1	no	—
339	C₄	—	68	2	no	—
340	C₄	—	68	2	no	—
341	C₄	D₈×#595	1	1	yes	characteristic
342	C₄	C₂×D₄×#595	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	8	2	no	—
345	C₂	—	17	2	no	—
346	C₂	—	34	1	no	—
347	C₂	—	136	2	no	—
348	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×#2179
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. #14545188	C₂	semidirect
3. #14545187	C₂	semidirect
4. #14545186	C₂	semidirect
5. D₁₇⋊#368	C₂	semidirect
6. D₁₇⋊#359	C₂	semidirect
7. C₂×#11763696	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. #11763696	C₂²	semidirect
22. #11763696	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
184. #508	#595	semidirect
207. D₁₇	#2179	semidirect
229. Q₃₂	C₂×#595	semidirect
233. SD₃₂	C₂×#595	semidirect
236. D₁₆	C₂×#595	semidirect
252. C₁₇	C₂×#2179	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
#14545188	2
#14545187	2
#14545186	2
D₁₇⋊#368	2
D₁₇⋊#359	2
C₂×#11763696	2
D₁₇×#508	2
C₂×#2179	17

Subgroups — D17⋊#2179 #17978616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇⋊#2179 #17978616