Subgroups — #17960779

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545024	C₂	1	1	yes	maximal, characteristic
2	#14544992	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11763696	C₂	1	1	yes	maximal, characteristic, complemented
4	#14544558	C₂	1	1	yes	maximal, characteristic
5	#14543840	C₂	1	1	yes	maximal, characteristic, complemented
6	#14543827	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₇×#504	C₂	1	1	yes	maximal, characteristic
8	#11763865	C₂²	1	1	yes	characteristic
9	C₂×#10587529	C₂²	1	1	yes	characteristic
10	D₁₇⋊#104	C₄	1	1	yes	characteristic
11	#11763799	C₄	1	1	yes	characteristic
12	#11763772	C₂²	1	1	yes	characteristic
13	C₂×C₈×#595	C₂²	1	1	yes	characteristic
14	#11763708	C₄	1	2	yes	complemented
15	#11763707	C₄	1	2	yes	complemented
16	#11763696	—	2	1	no	—
17	#11763696	—	2	1	no	—
18	#11763696	—	2	1	no	—
19	C₁₆×#595	—	2	1	no	—
20	#11763684	C₄	1	1	yes	characteristic
21	#11763681	C₄	1	1	yes	characteristic
22	#131×D₁₇	C₂²	1	1	yes	characteristic
23	#11763227	—	2	1	no	—
24	#11763227	C₄	1	1	yes	characteristic, complemented
25	#11763226	C₄	1	1	yes	characteristic, complemented
26	#11763225	—	2	1	no	—
27	#11763225	C₄	1	1	yes	characteristic, complemented
28	M₃₂×D₁₇	—	2	1	no	—
29	M₃₂×D₁₇	C₂²	1	1	yes	characteristic
30	C₂×C₁₆×D₁₇	—	2	1	no	—
31	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
32	C₁₇×#504	C₄	1	1	yes	characteristic, complemented
33	#41×D₁₇	C₂×C₄	1	1	yes	characteristic
34	#10587677	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
36	D₁₇⋊M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#63242	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#131	C₂×C₄	1	1	yes	characteristic
39	#10587634	C₂×C₄	1	1	yes	characteristic
40	#10587593	—	2	1	no	—
41	#10587593	C₂×C₄	1	1	yes	characteristic
42	M₁₆×D₁₇	—	2	1	no	—
43	M₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
44	#10587587	—	2	1	no	—
45	#10587587	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₇	—	2	1	no	—
47	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
48	#10587550	C₈	1	2	yes	—
49	#10587549	C₈	1	2	yes	—
50	#10587548	C₈	1	2	yes	—
51	#10587547	C₈	1	2	yes	—
52	#10587538	C₂×C₄	1	2	yes	—
53	C₂×#63216	C₂×C₄	1	2	yes	—
54	#10587532	—	2	1	no	—
55	C₈×#595	—	2	1	no	—
56	C₈×#595	—	2	1	no	—
57	C₈×#595	—	2	1	no	—
58	#10587530	—	2	1	no	—
59	#10587529	—	2	1	no	—
60	#10587529	—	2	1	no	—
61	#10587529	—	2	1	no	—
62	C₁₇×M₃₂	—	2	1	no	—
63	C₁₇×M₃₂	C₂×C₄	1	1	yes	characteristic
64	C₂×C₁₆×C₁₇	—	2	1	no	—
65	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
66	#10587483	—	2	1	no	—
67	#10587483	C₂×C₄	1	1	yes	characteristic
68	C₂×#63214	—	2	1	no	—
69	C₂×#63214	C₂×C₄	1	1	yes	characteristic
70	#10587469	—	2	1	no	—
71	#10587469	—	2	1	no	—
72	#10587469	—	2	1	no	—
73	C₁₆×D₁₇	—	2	1	no	—
74	C₁₆×D₁₇	—	2	1	no	—
75	C₁₆×D₁₇	—	2	1	no	—
76	C₁₆×D₁₇	D₄	1	1	yes	characteristic
77	C₁₆×D₁₇	D₄	1	1	yes	characteristic
78	C₁₆×D₁₇	D₄	1	1	yes	characteristic
79	C₁₆×D₁₇	D₄	1	1	yes	characteristic
80	C₂²×#595	—	2	1	no	—
81	#41×C₁₇	C₄²	1	1	yes	characteristic
82	#63256	C₂×C₈	1	1	yes	characteristic
83	Q₈×D₁₇	C₂×C₈	1	1	yes	characteristic
84	#63254	—	2	1	no	—
85	#63254	C₂×C₈	1	1	yes	characteristic
86	D₄×D₁₇	—	2	1	no	—
87	D₄×D₁₇	C₂×C₈	1	1	yes	characteristic
88	#63252	—	2	1	no	—
89	#63252	C₄²	1	1	yes	characteristic
90	C₂×C₄×D₁₇	—	2	1	no	—
91	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²⋊#595	—	2	1	no	—
93	#63247	—	2	1	no	—
94	C₂×#3624	—	2	1	no	—
95	C₄⋊#595	C₂×C₈	1	2	yes	—
96	C₄×#595	—	2	1	no	—
97	C₄×#595	—	2	1	no	—
98	C₄×#595	C₂×C₈	1	2	yes	—
99	#63243	C₂×C₈	1	2	yes	—
100	#63242	—	2	1	no	—
101	#63242	—	2	1	no	—
102	#63242	C₂×C₈	1	2	yes	—
103	M₁₆×C₁₇	—	2	1	no	—
104	M₁₆×C₁₇	C₄²	1	1	yes	characteristic
105	C₂×C₈×C₁₇	—	2	1	no	—
106	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
107	#63223	—	2	1	no	—
108	#63223	C₄²	1	1	yes	characteristic
109	C₂×#3622	—	2	1	no	—
110	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
111	#63218	—	2	1	no	—
112	#63218	—	2	1	no	—
113	#63218	—	2	1	no	—
114	C₈×D₁₇	—	2	1	no	—
115	C₈×D₁₇	—	2	1	no	—
116	C₈×D₁₇	—	2	1	no	—
117	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
118	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
119	C₈×D₁₇	#41	1	1	yes	characteristic
120	C₈×D₁₇	#41	1	1	yes	characteristic
121	#63216	—	2	2	no	—
122	#63216	—	2	2	no	—
123	C₁₆×C₁₇	—	2	1	no	—
124	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
125	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
126	#63214	—	2	1	no	—
127	#63214	C₂²⋊C₄	1	1	yes	characteristic
128	#63214	C₂²⋊C₄	1	1	yes	characteristic
129	#12615	—	17	1	no	maximal, Sylow
130	C₂²×D₁₇	—	2	1	no	—
131	C₂²×D₁₇	#131	1	1	yes	characteristic
132	C₂×#595	—	2	2	no	—
133	C₂×#595	—	2	2	no	—
134	C₂×#595	—	4	2	no	—
135	Q₈×C₁₇	C₄×C₈	1	1	yes	characteristic
136	D₄×C₁₇	—	2	1	no	—
137	D₄×C₁₇	C₄×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₇	—	2	1	no	—
139	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
140	C₂²⋊D₁₇	—	2	1	no	—
141	C₂²⋊D₁₇	—	2	1	no	—
142	C₂²⋊D₁₇	—	2	1	no	—
143	C₂×Dic₁₇	—	2	1	no	—
144	C₂×Dic₁₇	#131	1	1	yes	characteristic
145	D₆₈	—	2	1	no	—
146	D₆₈	C₄×C₈	1	1	yes	characteristic
147	C₄×D₁₇	—	2	1	no	—
148	C₄×D₁₇	—	2	1	no	—
149	C₄×D₁₇	—	2	1	no	—
150	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
151	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
152	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
153	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
154	Dic₃₄	—	2	1	no	—
155	Dic₃₄	C₄×C₈	1	1	yes	characteristic
156	#3624	—	2	2	no	—
157	#3624	—	2	2	no	—
158	C₈×C₁₇	—	2	1	no	—
159	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₈×C₁₇	#117	1	1	yes	characteristic
161	#3622	—	2	1	no	—
162	#3622	C₄×D₄	1	1	yes	characteristic
163	#3622	C₄×D₄	1	1	yes	characteristic
164	C₂×#504	—	17	1	no	—
165	C₂×#350	—	17	1	no	—
166	#2108	—	17	1	no	—
167	C₂×#346	—	17	1	no	—
168	#1764	—	17	1	no	—
169	#1379	—	17	1	no	—
170	#1376	—	17	1	no	—
171	C₂²×C₁₇	—	2	1	no	—
172	C₂²×C₁₇	#405	1	1	yes	characteristic
173	C₂×D₁₇	—	2	2	no	—
174	C₂×D₁₇	—	2	1	no	—
175	C₂×D₁₇	—	4	2	no	—
176	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
177	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
178	#595	—	4	2	no	—
179	#595	—	4	2	no	—
180	C₄×C₁₇	—	2	1	no	—
181	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
182	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
183	Dic₁₇	—	2	1	no	—
184	Dic₁₇	C₈×D₄	1	1	yes	characteristic
185	Dic₁₇	C₈×D₄	1	1	yes	characteristic
186	C₂×#131	—	17	1	no	—
187	#504	—	17	1	no	—
188	#504	—	17	1	no	—
189	#504	—	17	1	no	—
190	#504	—	34	1	no	—
191	#504	—	34	1	no	—
192	#504	#595	1	1	yes	characteristic, complemented
193	C₂×M₃₂	—	17	1	no	—
194	C₂×M₃₂	—	17	1	no	—
195	C₂×M₃₂	—	34	1	no	—
196	C₂²×C₁₆	—	17	1	no	—
197	C₂²×C₁₆	—	17	1	no	—
198	C₂²×C₁₆	—	34	1	no	—
199	C₂×#104	—	17	1	no	—
200	#416	—	17	1	no	—
201	#405	—	17	1	no	—
202	C₂×C₄×C₈	—	17	1	no	—
203	#350	—	17	2	no	—
204	#350	—	17	2	no	—
205	#346	—	34	1	no	—
206	#346	—	34	1	no	—
207	#346	—	34	1	no	—
208	C₄×C₁₆	—	34	1	no	—
209	#341	—	17	1	no	—
210	#339	—	17	1	no	—
211	C₂×C₁₇	—	2	1	no	—
212	C₂×C₁₇	—	4	1	no	—
213	C₂×C₁₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
214	D₁₇	—	2	1	no	—
215	D₁₇	—	4	1	no	—
216	D₁₇	#2171	1	2	yes	—
217	C₂×#41	—	17	1	no	—
218	#131	—	17	1	no	—
219	#131	—	17	1	no	—
220	#131	—	17	1	no	—
221	#131	—	34	1	no	—
222	#131	—	34	1	no	—
223	#131	C₂×#595	1	1	yes	characteristic
224	C₂×M₁₆	—	17	1	no	—
225	C₂×M₁₆	—	17	1	no	—
226	C₂×M₁₆	—	34	1	no	—
227	C₂²×C₈	—	17	1	no	—
228	C₂²×C₈	—	17	1	no	—
229	C₂²×C₈	—	34	1	no	—
230	#117	—	17	1	no	—
231	C₂×C₄²	—	17	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	17	2	no	—
234	M₃₂	—	17	1	no	—
235	M₃₂	—	34	1	no	—
236	M₃₂	—	34	1	no	—
237	M₃₂	—	34	1	no	—
238	M₃₂	—	34	1	no	—
239	M₃₂	—	34	1	no	—
240	M₃₂	C₂×#595	1	1	yes	characteristic
241	C₂×C₁₆	—	2	1	no	—
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	—	34	1	no	—
251	C₂×C₁₆	—	34	1	no	—
252	C₂×C₁₆	—	34	1	no	—
253	C₂×C₁₆	—	34	1	no	—
254	C₂×C₁₆	—	34	1	no	—
255	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
256	#104	—	17	2	no	—
257	#104	—	17	2	no	—
258	#103	—	17	2	no	—
259	#102	—	17	2	no	—
260	#97	—	34	1	no	—
261	C₄×C₈	—	34	1	no	—
262	C₄×C₈	—	34	1	no	—
263	C₄×C₈	—	34	1	no	—
264	C₁₇	#12615	1	1	yes	characteristic, complemented, Sylow
265	#41	—	17	1	no	—
266	#41	—	17	1	no	—
267	#41	—	17	1	no	—
268	#41	—	34	1	no	—
269	#41	—	34	1	no	—
270	#41	C₄×#595	1	1	yes	characteristic
271	C₂×Q₈	—	17	1	no	—
272	C₂×D₄	—	17	1	no	—
273	C₂×D₄	—	34	1	no	—
274	C₂²×C₄	—	17	1	no	—
275	C₂²×C₄	—	34	1	no	—
276	C₂²×C₄	—	34	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	17	2	no	—
279	M₁₆	—	17	1	no	—
280	M₁₆	—	34	1	no	—
281	M₁₆	—	34	1	no	—
282	M₁₆	—	34	1	no	—
283	M₁₆	—	34	1	no	—
284	M₁₆	—	34	1	no	—
285	M₁₆	C₄×#595	1	1	yes	characteristic
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	17	1	no	—
288	C₂×C₈	—	17	1	no	—
289	C₂×C₈	—	17	1	no	—
290	C₂×C₈	—	17	1	no	—
291	C₂×C₈	—	17	1	no	—
292	C₂×C₈	—	17	2	no	—
293	C₂×C₈	—	34	1	no	—
294	C₂×C₈	—	34	1	no	—
295	C₂×C₈	—	34	1	no	—
296	C₂×C₈	—	34	1	no	—
297	C₂×C₈	—	34	1	no	—
298	C₂×C₈	—	34	1	no	—
299	C₂×C₈	—	34	1	no	—
300	C₂×C₈	C₂²×#595	1	1	yes	characteristic
301	C₄⋊C₄	—	17	2	no	—
302	C₂²⋊C₄	—	34	1	no	—
303	C₄²	—	17	2	no	—
304	C₄²	—	34	1	no	—
305	C₄²	—	34	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	17	1	no	—
308	C₁₆	—	17	1	no	—
309	C₁₆	—	34	2	no	—
310	C₁₆	—	34	2	no	—
311	C₁₆	—	34	1	no	—
312	C₁₆	C₂²⋊#595	1	1	yes	characteristic
313	C₁₆	C₂²⋊#595	1	1	yes	characteristic
314	C₂³	—	17	1	no	—
315	C₂³	—	34	1	no	—
316	Q₈	—	17	1	no	—
317	Q₈	—	34	1	no	—
318	Q₈	C₈×#595	1	1	yes	characteristic
319	D₄	—	2	1	no	—
320	D₄	—	17	1	no	—
321	D₄	—	34	1	no	—
322	D₄	—	34	1	no	—
323	D₄	—	34	1	no	—
324	D₄	—	34	1	no	—
325	D₄	C₈×#595	1	1	yes	characteristic
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	17	1	no	—
328	C₂×C₄	—	17	1	no	—
329	C₂×C₄	—	17	1	no	—
330	C₂×C₄	—	17	1	no	—
331	C₂×C₄	—	17	1	no	—
332	C₂×C₄	—	34	1	no	—
333	C₂×C₄	—	34	1	no	—
334	C₂×C₄	—	34	1	no	—
335	C₂×C₄	—	34	1	no	—
336	C₂×C₄	—	34	2	no	—
337	C₂×C₄	—	34	2	no	—
338	C₂×C₄	—	68	2	no	—
339	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	17	1	no	—
342	C₈	—	17	1	no	—
343	C₈	—	34	2	no	—
344	C₈	—	34	2	no	—
345	C₈	—	34	1	no	—
346	C₈	#10587677	1	1	yes	characteristic
347	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
348	C₂²	—	2	1	no	—
349	C₂²	—	17	1	no	—
350	C₂²	—	17	1	no	—
351	C₂²	—	34	2	no	—
352	C₂²	—	34	1	no	—
353	C₂²	—	68	2	no	—
354	C₂²	#11763772	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	17	1	no	—
357	C₄	—	17	1	no	—
358	C₄	—	34	1	no	—
359	C₄	—	68	2	no	—
360	C₄	—	68	2	no	—
361	C₄	C₄×C₂²⋊#595	1	1	yes	characteristic
362	C₄	C₂×C₈×#595	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	4	1	no	—
365	C₂	—	17	2	no	—
366	C₂	—	34	1	no	—
367	C₂	—	68	1	no	—
368	C₂	C₈×C₂²⋊#595	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#504	C₄
socle	C₂×C₁₇	C₈×C₂²⋊C₄
prime radical	#41×D₁₇	C₂×C₄
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#504	#595
hypocenter	C₁₇	#12615
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
2. #14544992	C₂	semidirect
3. C₂×#11763696	C₂	semidirect
5. #14543840	C₂	semidirect
6. #14543827	C₂	semidirect
14. #11763708	C₄	semidirect
15. #11763707	C₄	semidirect
24. #11763227	C₄	semidirect
25. #11763226	C₄	semidirect
27. #11763225	C₄	semidirect
32. C₁₇×#504	C₄	semidirect
192. #504	#595	semidirect
264. C₁₇	#12615	semidirect

Maximal subgroups

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

subgroup	index
#14545024	2
#14544992	2
C₂×#11763696	2
#14544558	2
#14543840	2
#14543827	2
D₁₇×#504	2
#12615	17