Subgroups — #508×#595 #17978595

Order 4352.

690 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×#595	C₂	1	2	yes	maximal, complemented
2	Q₃₂×#595	C₂	1	2	yes	maximal, complemented
3	SD₃₂×#595	C₂	1	4	yes	maximal, complemented
4	D₁₆×#595	C₂	1	2	yes	maximal, complemented
5	#14545135	C₂	1	1	yes	maximal, characteristic, complemented
6	#14545091	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	#41×#595	—	2	2	no	—
10	C₂×#10587529	C₂²	1	1	yes	characteristic
11	Q₁₆×#595	C₂²	1	4	yes	—
12	SD₁₆×#595	—	2	4	no	—
13	D₈×#595	C₂²	1	4	yes	—
14	#11763819	C₂²	1	2	yes	—
15	#11763799	—	2	2	no	—
16	C₂×C₈×#595	C₂²	1	1	yes	characteristic
17	#11763736	C₂²	1	2	yes	—
18	#11763735	C₂²	1	2	yes	—
19	#11763722	C₂²	1	4	yes	—
20	#11763721	C₂²	1	4	yes	—
21	C₁₆×#595	C₂²	1	2	yes	—
22	C₁₆×#595	C₂²	1	2	yes	—
23	#135×D₁₇	C₂²	1	2	yes	—
24	#11763242	C₄	1	1	yes	characteristic, complemented
25	#11763241	C₄	1	1	yes	characteristic, complemented
26	#11763240	C₄	1	2	yes	complemented
27	#11763239	C₄	1	2	yes	complemented
28	#11763238	C₄	1	1	yes	characteristic, complemented
29	Q₃₂×D₁₇	C₂²	1	1	yes	characteristic
30	SD₃₂×D₁₇	C₂²	1	2	yes	—
31	D₁₆×D₁₇	C₂²	1	1	yes	characteristic
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	Q₈×#595	—	2	4	no	—
36	D₄×#595	—	2	4	no	—
37	D₄×#595	—	4	4	no	—
38	#10587677	—	2	2	no	—
39	#10587677	—	4	2	no	—
40	C₂×C₄×#595	—	4	2	no	—
41	C₂×C₄×#595	D₄	1	1	yes	characteristic
42	C₂×#63242	D₄	1	1	yes	characteristic
43	C₁₇×#135	C₂×C₄	1	2	yes	complemented
44	#10587636	—	2	2	no	—
45	#10587605	C₂×C₄	1	2	yes	complemented
46	Q₁₆×D₁₇	C₂³	1	2	yes	—
47	#10587602	—	2	2	no	—
48	SD₁₆×D₁₇	—	2	2	no	—
49	#10587598	C₂×C₄	1	2	yes	complemented
50	D₈×D₁₇	C₂³	1	2	yes	—
51	#10587590	C₂×C₄	1	2	yes	complemented
52	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
53	#10587549	—	2	4	no	—
54	#10587547	—	2	4	no	—
55	C₂×#63216	—	2	1	no	—
56	#10587534	C₂³	1	4	yes	—
57	#10587533	—	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, complemented
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, complemented
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, complemented
75	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic, complemented
76	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
77	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
78	C₂²×#595	—	4	4	no	—
79	C₂²×#595	D₈	1	2	yes	—
80	#41×C₁₇	—	2	2	no	—
81	#63256	—	2	2	no	—
82	Q₈×D₁₇	—	2	2	no	—
83	#63254	—	2	2	no	—
84	#63254	—	4	2	no	—
85	D₄×D₁₇	—	2	2	no	—
86	D₄×D₁₇	—	4	2	no	—
87	#63252	—	2	2	no	—
88	#63252	—	4	2	no	—
89	C₂×C₄×D₁₇	—	4	2	no	—
90	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊#595	—	4	2	no	—
92	C₂²⋊#595	—	4	4	no	—
93	C₂×#3624	—	2	1	no	—
94	C₄⋊#595	—	2	4	no	—
95	C₄⋊#595	—	4	4	no	—
96	C₄×#595	—	4	4	no	—
97	C₄×#595	—	4	2	no	—
98	C₄×#595	D₈	1	1	yes	characteristic
99	C₄×#595	D₈	1	1	yes	characteristic
100	C₄×#595	C₂×D₄	1	2	yes	—
101	#63242	—	2	1	no	—
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	C₄×#508	—	17	1	no	maximal, Sylow
125	C₂²×D₁₇	—	4	2	no	—
126	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
127	C₂×#595	—	2	4	no	—
128	C₂×#595	—	4	4	no	—
129	C₂×#595	—	8	8	no	—
130	C₂×#595	C₂×D₈	1	2	yes	—
131	C₂×#595	C₂×D₈	1	2	yes	—
132	Q₈×C₁₇	—	2	2	no	—
133	D₄×C₁₇	—	2	2	no	—
134	D₄×C₁₇	—	4	2	no	—
135	C₂×C₄×C₁₇	—	4	2	no	—
136	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
137	C₂²⋊D₁₇	—	4	2	no	—
138	C₂²⋊D₁₇	—	4	2	no	—
139	C₂²⋊D₁₇	—	4	2	no	—
140	C₂×Dic₁₇	—	4	2	no	—
141	C₂×Dic₁₇	#135	1	1	yes	characteristic
142	D₆₈	—	2	2	no	—
143	D₆₈	—	4	2	no	—
144	C₄×D₁₇	—	4	2	no	—
145	C₄×D₁₇	—	4	2	no	—
146	C₄×D₁₇	—	4	2	no	—
147	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
148	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
149	C₄×D₁₇	#135	1	1	yes	characteristic
150	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
151	Dic₃₄	—	2	2	no	—
152	Dic₃₄	—	4	2	no	—
153	#3624	—	2	2	no	—
154	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
156	#3622	C₄×D₄	1	1	yes	characteristic
157	#3622	C₂×#41	1	1	yes	characteristic
158	C₂×#508	—	17	1	no	—
159	C₄×#135	—	17	2	no	—
160	C₄×Q₃₂	—	17	2	no	—
161	C₄×SD₃₂	—	17	4	no	—
162	C₄×D₁₆	—	17	2	no	—
163	#2158	—	17	1	no	—
164	#2138	—	17	2	no	—
165	C₂×C₄×C₁₆	—	17	1	no	—
166	C₂²×C₁₇	—	4	2	no	—
167	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
168	C₂×D₁₇	—	2	2	no	—
169	C₂×D₁₇	—	4	2	no	—
170	C₂×D₁₇	—	8	4	no	—
171	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
172	C₂×D₁₇	C₂²×D₈	1	1	yes	characteristic
173	#595	—	2	2	no	—
174	#595	—	8	4	no	—
175	#595	#508	1	4	yes	direct product factor
176	C₄×C₁₇	—	4	2	no	—
177	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
178	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
179	Dic₁₇	—	4	2	no	—
180	Dic₁₇	C₄×D₈	1	1	yes	characteristic
181	Dic₁₇	C₂×#135	1	1	yes	characteristic
182	C₂×#135	—	17	2	no	—
183	C₄×#41	—	34	2	no	—
184	#508	—	17	2	no	—
185	#508	—	17	1	no	—
186	#508	—	17	2	no	—
187	#508	—	17	1	no	—
188	#508	—	17	1	no	—
189	#508	#595	1	1	yes	characteristic, direct product factor
190	C₂×Q₃₂	—	17	1	no	—
191	C₂×SD₃₂	—	17	2	no	—
192	C₂×D₁₆	—	17	1	no	—
193	C₂²×C₁₆	—	17	1	no	—
194	C₂²×C₁₆	—	17	1	no	—
195	C₄×Q₁₆	—	17	4	no	—
196	C₄×SD₁₆	—	34	4	no	—
197	C₄×D₈	—	17	4	no	—
198	#427	—	17	2	no	—
199	#416	—	34	2	no	—
200	C₂×C₄×C₈	—	17	1	no	—
201	#367	—	17	2	no	—
202	#366	—	17	2	no	—
203	#358	—	17	4	no	—
204	#357	—	17	4	no	—
205	C₄×C₁₆	—	17	2	no	—
206	C₄×C₁₆	—	17	2	no	—
207	C₂×C₁₇	—	2	1	no	—
208	C₂×C₁₇	—	8	2	no	—
209	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
210	D₁₇	—	2	1	no	—
211	D₁₇	—	8	2	no	—
212	D₁₇	C₂×#508	1	2	yes	—
213	C₂×#41	—	34	2	no	—
214	#135	—	17	2	no	—
215	#135	—	17	2	no	—
216	#135	—	17	2	no	—
217	#135	—	34	2	no	—
218	#135	—	34	2	no	—
219	#135	C₂×#595	1	2	yes	complemented
220	C₂×Q₁₆	—	17	2	no	—
221	C₂×SD₁₆	—	34	2	no	—
222	C₂×D₈	—	17	2	no	—
223	C₂²×C₈	—	17	1	no	—
224	C₂²×C₈	—	17	1	no	—
225	C₄×Q₈	—	34	4	no	—
226	C₄×D₄	—	34	4	no	—
227	C₄×D₄	—	68	4	no	—
228	#117	—	34	2	no	—
229	#117	—	68	2	no	—
230	C₂×C₄²	—	17	1	no	—
231	C₂×C₄²	—	68	2	no	—
232	Q₃₂	—	17	2	no	—
233	Q₃₂	—	17	1	no	—
234	Q₃₂	C₂×#595	1	1	yes	characteristic, complemented
235	SD₃₂	—	17	2	no	—
236	SD₃₂	—	17	2	no	—
237	SD₃₂	—	17	2	no	—
238	SD₃₂	C₂×#595	1	2	yes	complemented
239	D₁₆	—	17	2	no	—
240	D₁₆	—	17	1	no	—
241	D₁₆	C₂×#595	1	1	yes	characteristic, complemented
242	C₂×C₁₆	—	17	1	no	—
243	C₂×C₁₆	—	17	1	no	—
244	C₂×C₁₆	—	17	1	no	—
245	C₂×C₁₆	—	17	1	no	—
246	C₂×C₁₆	—	17	1	no	—
247	C₂×C₁₆	—	17	2	no	—
248	C₂×C₁₆	—	34	1	no	—
249	C₂×C₁₆	—	34	1	no	—
250	C₂×C₁₆	C₂×#595	1	1	yes	characteristic, complemented
251	#107	—	17	4	no	—
252	#106	—	34	2	no	—
253	#103	—	34	4	no	—
254	#102	—	34	4	no	—
255	C₄×C₈	—	17	2	no	—
256	C₄×C₈	—	17	2	no	—
257	C₁₇	C₄×#508	1	1	yes	characteristic, complemented, Sylow
258	#41	—	2	2	no	—
259	#41	—	34	2	no	—
260	#41	—	34	2	no	—
261	#41	—	34	2	no	—
262	#41	—	68	2	no	—
263	#41	—	68	2	no	—
264	C₂×Q₈	—	34	2	no	—
265	C₂×D₄	—	34	2	no	—
266	C₂×D₄	—	68	2	no	—
267	C₂²×C₄	—	17	1	no	—
268	C₂²×C₄	—	17	2	no	—
269	C₂²×C₄	—	68	2	no	—
270	C₂²×C₄	—	68	4	no	—
271	Q₁₆	—	17	2	no	—
272	Q₁₆	—	34	2	no	—
273	Q₁₆	C₂²×#595	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	34	2	no	—
276	SD₁₆	—	34	2	no	—
277	SD₁₆	—	34	2	no	—
278	D₈	—	17	2	no	—
279	D₈	—	34	2	no	—
280	D₈	C₂²×#595	1	2	yes	—
281	C₂×C₈	—	17	1	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	2	no	—
287	C₂×C₈	—	34	1	no	—
288	C₂×C₈	—	34	1	no	—
289	C₂×C₈	C₂²×#595	1	1	yes	characteristic
290	C₄⋊C₄	—	34	4	no	—
291	C₄⋊C₄	—	68	4	no	—
292	C₂²⋊C₄	—	68	4	no	—
293	C₂²⋊C₄	—	68	2	no	—
294	C₄²	—	17	1	no	—
295	C₄²	—	17	1	no	—
296	C₄²	—	17	2	no	—
297	C₄²	—	68	2	no	—
298	C₄²	—	68	4	no	—
299	C₁₆	—	17	1	no	—
300	C₁₆	—	17	1	no	—
301	C₁₆	—	34	2	no	—
302	C₁₆	C₂²×#595	1	1	yes	characteristic
303	C₁₆	C₂²×#595	1	1	yes	characteristic
304	C₂³	—	17	1	no	—
305	C₂³	—	68	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	34	2	no	—
308	Q₈	—	68	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	34	2	no	—
312	D₄	—	68	2	no	—
313	D₄	—	68	2	no	—
314	D₄	—	68	2	no	—
315	D₄	—	68	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	17	2	no	—
318	C₂×C₄	—	17	1	no	—
319	C₂×C₄	—	17	1	no	—
320	C₂×C₄	—	17	1	no	—
321	C₂×C₄	—	17	1	no	—
322	C₂×C₄	—	17	1	no	—
323	C₂×C₄	—	17	2	no	—
324	C₂×C₄	—	34	4	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	—	68	4	no	—
330	C₂×C₄	—	136	8	no	—
331	C₂×C₄	D₄×#595	1	1	yes	characteristic
332	C₈	—	17	1	no	—
333	C₈	—	17	1	no	—
334	C₈	—	34	2	no	—
335	C₈	D₄×#595	1	1	yes	characteristic
336	C₈	C₂³×#595	1	1	yes	characteristic
337	C₂²	—	4	2	no	—
338	C₂²	—	17	1	no	—
339	C₂²	—	17	1	no	—
340	C₂²	—	34	2	no	—
341	C₂²	—	68	2	no	—
342	C₂²	—	136	4	no	—
343	C₂²	D₈×#595	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	17	4	no	—
346	C₄	—	17	1	no	—
347	C₄	—	17	1	no	—
348	C₄	—	34	2	no	—
349	C₄	—	68	2	no	—
350	C₄	—	136	4	no	—
351	C₄	D₈×#595	1	1	yes	characteristic
352	C₄	C₂×D₄×#595	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	8	2	no	—
355	C₂	—	17	2	no	—
356	C₂	—	34	1	no	—
357	C₂	—	136	2	no	—
358	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	C₄×#508
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. #135×#595	C₂	semidirect
2. Q₃₂×#595	C₂	semidirect
3. SD₃₂×#595	C₂	semidirect
4. D₁₆×#595	C₂	semidirect
5. #14545135	C₂	semidirect
6. #14545091	C₂	semidirect
7. C₂×C₁₆×#595	C₂	semidirect
24. #11763242	C₄	semidirect
25. #11763241	C₄	semidirect
26. #11763240	C₄	semidirect
27. #11763239	C₄	semidirect
28. #11763238	C₄	semidirect
33. C₁₇×#508	C₄	semidirect
43. C₁₇×#135	C₂×C₄	semidirect
45. #10587605	C₂×C₄	semidirect
49. #10587598	C₂×C₄	semidirect
51. #10587590	C₂×C₄	semidirect
62. C₁₇×Q₃₂	C₂×C₄	semidirect
63. C₁₇×SD₃₂	C₂×C₄	semidirect
64. C₁₇×D₁₆	C₂×C₄	semidirect
65. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
66. #10587500	C₂×C₄	semidirect
67. #10587499	C₂×C₄	semidirect
68. #10587498	C₂×C₄	semidirect
69. #10587497	C₂×C₄	semidirect
70. C₂×#63214	C₂×C₄	semidirect
71. Dic₁₃₆	C₂×C₄	semidirect
72. #10587471	C₂×C₄	semidirect
73. D₂₇₂	C₂×C₄	semidirect
74. C₁₆×D₁₇	C₂×C₄	semidirect
75. C₁₆×D₁₇	C₂×C₄	semidirect
175. #595	#508	direct (N × H)
189. #508	#595	direct (N × H)
219. #135	C₂×#595	semidirect
234. Q₃₂	C₂×#595	semidirect
238. SD₃₂	C₂×#595	semidirect
241. D₁₆	C₂×#595	semidirect
250. C₂×C₁₆	C₂×#595	semidirect
257. C₁₇	C₄×#508	semidirect

Maximal subgroups

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

subgroup	index
#135×#595	2
Q₃₂×#595	2
SD₃₂×#595	2
D₁₆×#595	2
#14545135	2
#14545091	2
C₂×C₁₆×#595	2
D₁₇×#508	2
C₄×#508	17