Subgroups — #350×#595 #17960718

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×#595	C₂	1	1	yes	maximal, characteristic
2	#14545024	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#595	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×#595	C₂	1	1	yes	maximal, characteristic, complemented
5	#14543840	C₂	1	1	yes	maximal, characteristic, complemented
6	#14543837	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₇×#350	C₂	1	1	yes	maximal, characteristic
8	D₁₇⋊#131	C₄	1	1	yes	characteristic
9	#41×#595	C₄	1	1	yes	characteristic
10	#11763865	C₂²	1	1	yes	characteristic
11	C₂×#10587529	C₂²	1	1	yes	characteristic
12	#11763775	—	2	1	no	—
13	#11763775	C₄	1	1	yes	characteristic
14	#11763772	—	2	1	no	—
15	#11763772	C₂²	1	1	yes	characteristic
16	M₁₆×#595	—	2	1	no	—
17	M₁₆×#595	C₄	1	1	yes	characteristic
18	C₂×C₈×#595	—	2	1	no	—
19	C₂×C₈×#595	C₂²	1	1	yes	characteristic
20	#11763708	C₄	1	2	yes	complemented
21	#11763707	C₄	1	2	yes	complemented
22	#11763696	—	2	1	no	—
23	C₁₆×#595	—	2	1	no	—
24	C₁₆×#595	—	2	1	no	—
25	C₁₆×#595	—	2	1	no	—
26	#131×D₁₇	C₂²	1	1	yes	characteristic
27	M₃₂×D₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
29	#11762594	C₄	1	1	yes	characteristic, complemented
30	#11762593	C₄	1	1	yes	characteristic, complemented
31	#11762592	C₄	1	1	yes	characteristic, complemented
32	C₁₇×#350	C₄	1	1	yes	characteristic, complemented
33	#41×D₁₇	C₂×C₄	1	1	yes	characteristic
34	Q₈×#595	C₈	1	2	yes	—
35	#10587680	C₈	1	2	yes	—
36	D₄×#595	—	2	2	no	—
37	D₄×#595	C₈	1	2	yes	—
38	#10587678	—	2	2	no	—
39	#10587678	C₈	1	2	yes	—
40	#10587677	—	2	1	no	—
41	#10587677	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×#595	—	2	1	no	—
43	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
44	D₁₇⋊M₁₆	—	2	1	no	—
45	D₁₇⋊M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#63242	—	2	1	no	—
47	C₂×#63242	C₂×C₄	1	1	yes	characteristic
48	C₁₇×#131	C₂×C₄	1	1	yes	characteristic
49	#10587634	C₂×C₄	1	1	yes	characteristic
50	#10587593	—	2	1	no	—
51	#10587593	C₂×C₄	1	1	yes	characteristic
52	M₁₆×D₁₇	—	2	1	no	—
53	M₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
54	#10587587	—	2	1	no	—
55	#10587587	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₁₇	—	2	1	no	—
57	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
58	#10587538	C₂×C₄	1	2	yes	complemented
59	C₂×#63216	C₂×C₄	1	2	yes	complemented
60	#10587532	—	2	1	no	—
61	#10587532	—	2	1	no	—
62	#10587532	—	2	1	no	—
63	C₈×#595	—	2	1	no	—
64	C₈×#595	—	2	1	no	—
65	C₈×#595	—	2	1	no	—
66	C₈×#595	D₄	1	1	yes	characteristic
67	C₈×#595	D₄	1	1	yes	characteristic
68	C₈×#595	D₄	1	1	yes	characteristic
69	C₈×#595	D₄	1	1	yes	characteristic
70	#10587530	—	2	1	no	—
71	#10587529	—	2	1	no	—
72	#10587529	—	2	1	no	—
73	#10587529	—	2	1	no	—
74	C₁₇×M₃₂	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
76	#10587483	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×#63214	C₂×C₄	1	1	yes	characteristic, complemented
78	#10587469	—	2	1	no	—
79	C₁₆×D₁₇	—	2	1	no	—
80	C₁₆×D₁₇	—	2	1	no	—
81	C₁₆×D₁₇	—	2	1	no	—
82	C₂²×#595	—	2	2	no	—
83	C₂²×#595	M₁₆	1	2	yes	—
84	#41×C₁₇	C₄²	1	1	yes	characteristic
85	#63256	C₂×C₈	1	1	yes	characteristic
86	Q₈×D₁₇	C₂×C₈	1	1	yes	characteristic
87	#63254	—	2	1	no	—
88	#63254	C₂×C₈	1	1	yes	characteristic
89	D₄×D₁₇	—	2	1	no	—
90	D₄×D₁₇	C₂×C₈	1	1	yes	characteristic
91	#63252	—	2	1	no	—
92	#63252	C₄²	1	1	yes	characteristic
93	C₂×C₄×D₁₇	—	2	1	no	—
94	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²⋊#595	—	2	1	no	—
96	C₂²⋊#595	—	2	2	no	—
97	#63247	—	2	1	no	—
98	#63247	—	2	2	no	—
99	C₂×#3624	—	2	2	no	—
100	C₂×#3624	M₁₆	1	2	yes	—
101	C₄⋊#595	—	2	2	no	—
102	C₄⋊#595	C₂×C₈	1	2	yes	—
103	C₄×#595	—	2	2	no	—
104	C₄×#595	—	2	1	no	—
105	C₄×#595	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×#595	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×#595	C₂×C₈	1	2	yes	—
108	#63243	—	2	2	no	—
109	#63243	C₂×C₈	1	2	yes	—
110	#63242	—	2	1	no	—
111	#63242	—	2	2	no	—
112	#63242	C₂²⋊C₄	1	1	yes	characteristic
113	#63242	C₂²⋊C₄	1	1	yes	characteristic
114	#63242	C₂×C₈	1	2	yes	—
115	M₁₆×C₁₇	—	2	1	no	—
116	M₁₆×C₁₇	C₄²	1	1	yes	characteristic
117	C₂×C₈×C₁₇	—	2	1	no	—
118	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
119	#63223	—	2	1	no	—
120	#63223	C₄²	1	1	yes	characteristic
121	C₂×#3622	—	2	1	no	—
122	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
123	#63218	—	2	1	no	—
124	#63218	—	2	1	no	—
125	#63218	—	2	1	no	—
126	C₈×D₁₇	—	2	1	no	—
127	C₈×D₁₇	—	2	1	no	—
128	C₈×D₁₇	—	2	1	no	—
129	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
130	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
131	C₈×D₁₇	#41	1	1	yes	characteristic
132	C₈×D₁₇	#41	1	1	yes	characteristic
133	#63216	—	2	2	no	—
134	#63216	—	2	2	no	—
135	C₁₆×C₁₇	—	2	1	no	—
136	C₁₆×C₁₇	—	2	1	no	—
137	#63214	—	2	1	no	—
138	#63214	—	2	1	no	—
139	C₄×#350	—	17	1	no	maximal, Sylow
140	C₂²×D₁₇	—	2	1	no	—
141	C₂²×D₁₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂×#595	—	2	4	no	—
143	C₂×#595	—	2	2	no	—
144	C₂×#595	—	4	4	no	—
145	C₂×#595	C₂²⋊C₈	1	2	yes	—
146	C₂×#595	C₂²⋊C₈	1	2	yes	—
147	Q₈×C₁₇	C₄×C₈	1	1	yes	characteristic
148	D₄×C₁₇	—	2	1	no	—
149	D₄×C₁₇	C₄×C₈	1	1	yes	characteristic
150	C₂×C₄×C₁₇	—	2	1	no	—
151	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
152	C₂²⋊D₁₇	—	2	1	no	—
153	C₂²⋊D₁₇	—	2	1	no	—
154	C₂²⋊D₁₇	—	2	1	no	—
155	C₂×Dic₁₇	—	2	1	no	—
156	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
157	D₆₈	—	2	1	no	—
158	D₆₈	C₄×C₈	1	1	yes	characteristic
159	C₄×D₁₇	—	2	1	no	—
160	C₄×D₁₇	—	2	1	no	—
161	C₄×D₁₇	—	2	1	no	—
162	C₄×D₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×D₁₇	#117	1	1	yes	characteristic
164	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
165	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
166	Dic₃₄	—	2	1	no	—
167	Dic₃₄	C₄×C₈	1	1	yes	characteristic
168	#3624	—	2	2	no	—
169	#3624	C₂²⋊C₈	1	2	yes	—
170	#3624	C₂²⋊C₈	1	2	yes	—
171	C₈×C₁₇	—	2	1	no	—
172	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
173	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
174	#3622	—	2	1	no	—
175	#3622	C₄×D₄	1	1	yes	characteristic
176	#3622	C₄×D₄	1	1	yes	characteristic
177	C₄×#131	—	17	1	no	—
178	C₂×#350	—	17	1	no	—
179	C₂×#350	—	17	1	no	—
180	C₄×M₃₂	—	17	1	no	—
181	C₂×C₄×C₁₆	—	17	1	no	—
182	#1379	—	17	1	no	—
183	#1379	—	17	1	no	—
184	C₂²×C₁₇	—	2	1	no	—
185	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
186	C₂×D₁₇	—	2	2	no	—
187	C₂×D₁₇	—	2	1	no	—
188	C₂×D₁₇	—	4	2	no	—
189	C₂×D₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
190	C₂×D₁₇	#431	1	1	yes	characteristic
191	#595	—	2	2	no	—
192	#595	—	4	2	no	—
193	#595	#350	1	4	yes	direct product factor
194	C₄×C₁₇	—	2	1	no	—
195	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
197	Dic₁₇	—	2	1	no	—
198	Dic₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
199	Dic₁₇	#431	1	1	yes	characteristic
200	C₂×#131	—	17	1	no	—
201	C₂×#131	—	17	1	no	—
202	C₄×#41	—	17	1	no	—
203	C₂×M₃₂	—	17	1	no	—
204	C₂×M₃₂	—	17	1	no	—
205	C₂²×C₁₆	—	17	1	no	—
206	C₂²×C₁₆	—	17	1	no	—
207	#405	—	17	1	no	—
208	#405	—	17	1	no	—
209	#405	—	34	1	no	—
210	#405	—	34	1	no	—
211	C₄×M₁₆	—	17	1	no	—
212	C₄×M₁₆	—	34	1	no	—
213	C₂×C₄×C₈	—	17	1	no	—
214	C₂×C₄×C₈	—	34	1	no	—
215	#350	—	17	2	no	—
216	#350	—	17	2	no	—
217	#350	—	17	1	no	—
218	#350	—	17	1	no	—
219	#350	—	17	1	no	—
220	#350	#595	1	1	yes	characteristic, direct product factor
221	#346	—	34	1	no	—
222	C₄×C₁₆	—	34	1	no	—
223	C₄×C₁₆	—	34	1	no	—
224	C₄×C₁₆	—	34	1	no	—
225	C₂×C₁₇	—	2	1	no	—
226	C₂×C₁₇	—	4	1	no	—
227	C₂×C₁₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
228	D₁₇	—	2	1	no	—
229	D₁₇	—	4	1	no	—
230	D₁₇	C₂×#350	1	2	yes	—
231	C₂×#41	—	17	1	no	—
232	#131	—	17	2	no	—
233	#131	—	17	2	no	—
234	#131	—	17	1	no	—
235	#131	—	17	1	no	—
236	#131	—	17	1	no	—
237	#131	—	34	1	no	—
238	#131	—	34	1	no	—
239	#131	—	34	2	no	—
240	#131	C₂×#595	1	1	yes	characteristic
241	C₂×M₁₆	—	17	1	no	—
242	C₂×M₁₆	—	17	1	no	—
243	C₂×M₁₆	—	34	1	no	—
244	C₂×M₁₆	—	34	1	no	—
245	C₂²×C₈	—	17	1	no	—
246	C₂²×C₈	—	17	1	no	—
247	C₂²×C₈	—	34	1	no	—
248	C₂²×C₈	—	34	1	no	—
249	C₄×Q₈	—	17	2	no	—
250	C₄×D₄	—	17	2	no	—
251	C₄×D₄	—	34	2	no	—
252	#117	—	17	1	no	—
253	#117	—	34	1	no	—
254	C₂×C₄²	—	17	1	no	—
255	C₂×C₄²	—	34	1	no	—
256	M₃₂	—	17	2	no	—
257	M₃₂	—	17	1	no	—
258	M₃₂	—	34	1	no	—
259	M₃₂	—	34	1	no	—
260	M₃₂	C₂×#595	1	1	yes	characteristic, complemented
261	C₂×C₁₆	—	17	2	no	—
262	C₂×C₁₆	—	17	1	no	—
263	C₂×C₁₆	—	34	1	no	—
264	C₂×C₁₆	—	34	1	no	—
265	C₂×C₁₆	—	34	1	no	—
266	C₂×C₁₆	—	34	1	no	—
267	C₂×C₁₆	—	34	1	no	—
268	C₂×C₁₆	—	34	1	no	—
269	C₂×C₁₆	C₂×#595	1	1	yes	characteristic, complemented
270	#97	—	34	1	no	—
271	#97	—	34	1	no	—
272	#97	—	34	1	no	—
273	C₄×C₈	—	17	1	no	—
274	C₄×C₈	—	17	1	no	—
275	C₄×C₈	—	17	1	no	—
276	C₄×C₈	—	17	1	no	—
277	C₄×C₈	—	34	1	no	—
278	C₄×C₈	—	34	1	no	—
279	C₄×C₈	—	34	1	no	—
280	C₁₇	C₄×#350	1	1	yes	characteristic, complemented, Sylow
281	#41	—	17	1	no	—
282	#41	—	17	1	no	—
283	#41	—	17	1	no	—
284	#41	—	34	1	no	—
285	#41	—	34	1	no	—
286	#41	C₄×#595	1	1	yes	characteristic
287	C₂×Q₈	—	17	1	no	—
288	C₂×D₄	—	17	1	no	—
289	C₂×D₄	—	34	1	no	—
290	C₂²×C₄	—	17	1	no	—
291	C₂²×C₄	—	17	2	no	—
292	C₂²×C₄	—	34	1	no	—
293	C₂²×C₄	—	34	2	no	—
294	M₁₆	—	2	1	no	—
295	M₁₆	—	17	2	no	—
296	M₁₆	—	17	1	no	—
297	M₁₆	—	34	1	no	—
298	M₁₆	—	34	1	no	—
299	M₁₆	—	34	1	no	—
300	M₁₆	—	34	2	no	—
301	M₁₆	—	34	2	no	—
302	M₁₆	—	34	1	no	—
303	M₁₆	—	34	1	no	—
304	M₁₆	C₄×#595	1	1	yes	characteristic
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	17	1	no	—
307	C₂×C₈	—	17	1	no	—
308	C₂×C₈	—	17	2	no	—
309	C₂×C₈	—	17	2	no	—
310	C₂×C₈	—	17	1	no	—
311	C₂×C₈	—	17	1	no	—
312	C₂×C₈	—	17	1	no	—
313	C₂×C₈	—	17	1	no	—
314	C₂×C₈	—	17	1	no	—
315	C₂×C₈	—	34	1	no	—
316	C₂×C₈	—	34	1	no	—
317	C₂×C₈	—	34	1	no	—
318	C₂×C₈	—	34	2	no	—
319	C₂×C₈	—	34	2	no	—
320	C₂×C₈	—	34	1	no	—
321	C₂×C₈	—	34	1	no	—
322	C₂×C₈	C₂²×#595	1	1	yes	characteristic
323	C₄⋊C₄	—	17	2	no	—
324	C₄⋊C₄	—	34	2	no	—
325	C₂²⋊C₄	—	34	2	no	—
326	C₂²⋊C₄	—	34	1	no	—
327	C₄²	—	17	1	no	—
328	C₄²	—	17	1	no	—
329	C₄²	—	17	2	no	—
330	C₄²	—	34	1	no	—
331	C₄²	—	34	2	no	—
332	C₁₆	—	2	1	no	—
333	C₁₆	—	2	1	no	—
334	C₁₆	—	34	2	no	—
335	C₁₆	—	34	2	no	—
336	C₁₆	—	34	1	no	—
337	C₁₆	—	34	1	no	—
338	C₂³	—	17	1	no	—
339	C₂³	—	34	1	no	—
340	Q₈	—	17	1	no	—
341	Q₈	—	34	1	no	—
342	Q₈	C₈×#595	1	1	yes	characteristic
343	D₄	—	2	1	no	—
344	D₄	—	17	1	no	—
345	D₄	—	34	1	no	—
346	D₄	—	34	1	no	—
347	D₄	—	34	1	no	—
348	D₄	—	34	1	no	—
349	D₄	C₈×#595	1	1	yes	characteristic
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	17	2	no	—
352	C₂×C₄	—	17	1	no	—
353	C₂×C₄	—	17	1	no	—
354	C₂×C₄	—	17	1	no	—
355	C₂×C₄	—	17	1	no	—
356	C₂×C₄	—	17	1	no	—
357	C₂×C₄	—	17	2	no	—
358	C₂×C₄	—	34	1	no	—
359	C₂×C₄	—	34	1	no	—
360	C₂×C₄	—	34	1	no	—
361	C₂×C₄	—	34	4	no	—
362	C₂×C₄	—	34	1	no	—
363	C₂×C₄	—	34	2	no	—
364	C₂×C₄	—	68	4	no	—
365	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	17	2	no	—
368	C₈	—	17	2	no	—
369	C₈	—	17	1	no	—
370	C₈	—	17	1	no	—
371	C₈	—	34	2	no	—
372	C₈	—	34	1	no	—
373	C₈	D₄×#595	1	1	yes	characteristic
374	C₈	D₄×#595	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	17	1	no	—
377	C₂²	—	17	1	no	—
378	C₂²	—	34	2	no	—
379	C₂²	—	34	1	no	—
380	C₂²	—	68	2	no	—
381	C₂²	M₁₆×#595	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	17	4	no	—
384	C₄	—	17	1	no	—
385	C₄	—	17	1	no	—
386	C₄	—	34	2	no	—
387	C₄	—	34	1	no	—
388	C₄	—	68	2	no	—
389	C₄	C₂²⋊C₄×#595	1	1	yes	characteristic
390	C₄	C₂×C₈×#595	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	4	1	no	—
393	C₂	—	17	2	no	—
394	C₂	—	34	1	no	—
395	C₂	—	68	1	no	—
396	C₂	C₂²⋊C₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#350	C₄
socle	C₂×C₁₇	C₄×C₂²⋊C₈
prime radical	#41×D₁₇	C₂×C₄
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#350	#595
hypocenter	C₁₇	C₄×#350
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

Complements

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

N	H ≅ G/N	type
3. M₃₂×#595	C₂	semidirect
4. C₂×C₁₆×#595	C₂	semidirect
5. #14543840	C₂	semidirect
6. #14543837	C₂	semidirect
20. #11763708	C₄	semidirect
21. #11763707	C₄	semidirect
29. #11762594	C₄	semidirect
30. #11762593	C₄	semidirect
31. #11762592	C₄	semidirect
32. C₁₇×#350	C₄	semidirect
58. #10587538	C₂×C₄	semidirect
59. C₂×#63216	C₂×C₄	semidirect
74. C₁₇×M₃₂	C₂×C₄	semidirect
75. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
76. #10587483	C₂×C₄	semidirect
77. C₂×#63214	C₂×C₄	semidirect
193. #595	#350	direct (N × H)
220. #350	#595	direct (N × H)
260. M₃₂	C₂×#595	semidirect
269. C₂×C₁₆	C₂×#595	semidirect
280. C₁₇	C₄×#350	semidirect

Maximal subgroups

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

subgroup	index
#131×#595	2
#14545024	2
M₃₂×#595	2
C₂×C₁₆×#595	2
#14543840	2
#14543837	2
D₁₇×#350	2
C₄×#350	17