Subgroups — #17187739

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541917	C₂	1	1	yes	maximal, characteristic
2	#14537696	C₂	1	2	yes	maximal, complemented
3	#14530670	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530669	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530668	C₂	1	2	yes	maximal, complemented
6	#14530666	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530662	C₂	1	2	yes	maximal, complemented
8	#14530577	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₇×#359	C₂	1	2	yes	maximal, complemented
10	#14530379	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₁₇	C₂²	1	2	yes	—
13	#11763574	—	2	2	no	—
14	#11763251	C₄	1	2	yes	complemented
15	#11763248	C₄	1	2	yes	complemented
16	#11763246	C₄	1	2	yes	complemented
17	#11763243	C₄	1	2	yes	complemented
18	#11763241	C₂²	1	1	yes	characteristic
19	#11763239	C₂²	1	2	yes	—
20	C₂×#10587469	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
22	#11762929	—	2	2	no	—
23	#11762928	—	2	2	no	—
24	#11762927	C₂²	1	2	yes	complemented
25	#11762926	C₂²	1	2	yes	complemented
26	#11762925	C₂²	1	2	yes	complemented
27	#108×D₁₇	C₂²	1	2	yes	—
28	#11762824	—	2	2	no	—
29	#11762763	C₂²	1	1	yes	characteristic
30	#11762639	C₂²	1	1	yes	characteristic, complemented
31	#11762638	C₂²	1	2	yes	complemented
32	#11762617	C₂²	1	2	yes	complemented
33	#11762616	C₂²	1	2	yes	complemented
34	#11762615	C₂²	1	2	yes	complemented
35	#11762585	C₂²	1	1	yes	characteristic, complemented
36	#11762584	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
38	C₁₇×#508	C₂²	1	1	yes	characteristic
39	C₁₇×#443	C₂²	1	2	yes	complemented
40	C₁₇×#368	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×#359	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic, complemented
43	#41×D₁₇	—	2	2	no	—
44	C₁₇×#135	C₂³	1	2	yes	—
45	C₁₇×#131	—	2	2	no	—
46	#10587636	—	2	2	no	—
47	#10587634	—	2	2	no	—
48	#10587605	C₂×C₄	1	2	yes	—
49	Q₁₆×D₁₇	C₂×C₄	1	2	yes	—
50	#10587602	—	2	2	no	—
51	SD₁₆×D₁₇	—	2	2	no	—
52	#10587598	C₂×C₄	1	2	yes	—
53	D₈×D₁₇	C₂×C₄	1	2	yes	—
54	#10587593	—	2	2	no	—
55	#10587593	—	4	2	no	—
56	M₁₆×D₁₇	—	2	2	no	—
57	M₁₆×D₁₇	—	4	2	no	—
58	#10587590	C₂³	1	2	yes	—
59	#10587587	—	2	2	no	—
60	#10587587	—	4	2	no	—
61	C₂×#63218	—	4	2	no	—
62	C₂×#63218	D₄	1	1	yes	characteristic
63	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
64	#10587558	D₄	1	1	yes	characteristic
65	C₁₇×Q₃₂	C₂×C₄	1	1	yes	characteristic
66	C₁₇×SD₃₂	C₂×C₄	1	2	yes	—
67	C₁₇×D₁₆	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
70	C₁₇×#108	C₂³	1	2	yes	—
71	C₁₇×#104	—	2	2	no	—
72	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	#10587508	—	2	2	no	—
74	#10587500	C₂×C₄	1	2	yes	—
75	#10587499	C₂×C₄	1	2	yes	—
76	#10587498	C₂×C₄	1	2	yes	—
77	#10587497	C₂×C₄	1	2	yes	—
78	#10587496	—	2	2	no	—
79	#10587493	—	2	2	no	—
80	#10587490	C₂³	1	2	yes	—
81	#10587486	D₄	1	1	yes	characteristic
82	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
83	C₂×#63214	C₂³	1	1	yes	characteristic
84	C₂×#63214	C₂³	1	1	yes	characteristic
85	#10587476	—	2	2	no	—
86	#10587474	D₄	1	1	yes	characteristic
87	Dic₁₃₆	C₂×C₄	1	1	yes	characteristic
88	#10587471	C₂×C₄	1	2	yes	—
89	D₂₇₂	C₂×C₄	1	1	yes	characteristic
90	#10587469	C₂×C₄	1	2	yes	—
91	#10587469	C₂×C₄	1	2	yes	—
92	C₁₆×D₁₇	—	2	2	no	—
93	#41×C₁₇	—	2	2	no	—
94	#63256	—	2	2	no	—
95	Q₈×D₁₇	—	2	2	no	—
96	#63254	—	2	2	no	—
97	#63254	—	4	2	no	—
98	D₄×D₁₇	—	2	2	no	—
99	D₄×D₁₇	—	4	2	no	—
100	#63252	—	2	2	no	—
101	#63252	—	4	2	no	—
102	C₂×C₄×D₁₇	—	4	2	no	—
103	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
105	SD₁₆×C₁₇	—	2	2	no	—
106	D₈×C₁₇	C₂²×C₄	1	2	yes	—
107	M₁₆×C₁₇	—	2	2	no	—
108	M₁₆×C₁₇	—	4	2	no	—
109	C₂×C₈×C₁₇	—	4	2	no	—
110	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
112	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
113	#63231	—	2	2	no	—
114	#63230	—	2	2	no	—
115	#63229	—	2	2	no	—
116	#63228	—	2	2	no	—
117	#63227	—	2	1	no	—
118	#63225	—	2	1	no	—
119	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	#63223	—	2	2	no	—
121	#63223	—	4	2	no	—
122	C₂×#3622	—	4	2	no	—
123	C₂×#3622	C₂×D₄	1	1	yes	characteristic
124	C₂×#3622	C₂×D₄	1	1	yes	characteristic
125	Dic₆₈	C₂²×C₄	1	2	yes	—
126	D₁₃₆	C₂²×C₄	1	2	yes	—
127	#63219	—	2	2	no	—
128	#63218	—	4	2	no	—
129	#63218	—	4	2	no	—
130	#63218	—	4	2	no	—
131	#63218	D₈	1	2	yes	—
132	#63218	D₈	1	2	yes	—
133	C₈×D₁₇	—	4	2	no	—
134	C₈×D₁₇	—	4	2	no	—
135	C₈×D₁₇	—	4	2	no	—
136	C₈×D₁₇	C₂²×C₄	1	2	yes	—
137	C₈×D₁₇	#41	1	2	yes	—
138	C₁₆×C₁₇	—	2	1	no	—
139	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
140	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
141	#63214	—	2	1	no	—
142	#63214	C₂²×C₄	1	2	yes	—
143	#33464	—	17	1	no	maximal, Sylow
144	C₂²×D₁₇	—	4	2	no	—
145	C₂²×D₁₇	#135	1	1	yes	characteristic
146	Q₈×C₁₇	—	2	2	no	—
147	D₄×C₁₇	—	2	2	no	—
148	D₄×C₁₇	—	4	2	no	—
149	C₂×C₄×C₁₇	—	2	1	no	—
150	C₂×C₄×C₁₇	—	4	2	no	—
151	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₁₇	—	4	2	no	—
153	C₂²⋊D₁₇	—	4	2	no	—
154	C₂²⋊D₁₇	—	4	2	no	—
155	C₂×Dic₁₇	—	2	1	no	—
156	C₂×Dic₁₇	—	4	2	no	—
157	C₂×Dic₁₇	#135	1	1	yes	characteristic
158	D₆₈	—	2	2	no	—
159	D₆₈	—	4	2	no	—
160	C₄×D₁₇	—	4	2	no	—
161	C₄×D₁₇	—	4	2	no	—
162	C₄×D₁₇	—	4	2	no	—
163	C₄×D₁₇	C₄×D₄	1	2	yes	—
164	C₄×D₁₇	C₂×D₈	1	2	yes	—
165	Dic₃₄	—	2	2	no	—
166	Dic₃₄	—	4	2	no	—
167	C₈×C₁₇	—	4	2	no	—
168	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
169	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
170	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
171	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
172	#3622	—	4	2	no	—
173	#3622	C₄×D₄	1	2	yes	—
174	#3622	C₂×D₈	1	2	yes	—
175	#3414	—	17	1	no	—
176	#2957	—	17	2	no	—
177	#2179	—	17	1	no	—
178	#2179	—	17	2	no	—
179	#2179	—	17	1	no	—
180	#2178	—	17	2	no	—
181	#2178	—	17	1	no	—
182	#2178	D₁₇	1	1	yes	characteristic, complemented
183	#2161	—	17	1	no	—
184	C₂×#359	—	17	2	no	—
185	#2108	—	17	1	no	—
186	C₂²×C₁₇	—	4	2	no	—
187	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
188	C₂×D₁₇	—	4	1	no	—
189	C₂×D₁₇	—	4	2	no	—
190	C₂×D₁₇	—	16	2	no	—
191	C₂×D₁₇	C₄×D₈	1	2	yes	—
192	C₄×C₁₇	—	4	1	no	—
193	C₄×C₁₇	—	4	2	no	—
194	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
196	Dic₁₇	—	4	1	no	—
197	Dic₁₇	—	4	2	no	—
198	Dic₁₇	C₄×D₈	1	2	yes	—
199	C₂×#135	—	17	2	no	—
200	#568	—	34	2	no	—
201	#510	—	17	2	no	—
202	#510	—	17	2	no	—
203	#509	—	17	2	no	—
204	#509	—	17	2	no	—
205	#508	—	17	2	no	—
206	#508	—	17	1	no	—
207	#508	C₂×D₁₇	1	1	yes	characteristic
208	C₂×M₃₂	—	17	1	no	—
209	C₂²×C₁₆	—	17	1	no	—
210	#444	—	34	2	no	—
211	#444	—	34	2	no	—
212	#443	—	17	2	no	—
213	#443	—	17	2	no	—
214	#443	—	17	2	no	—
215	#443	C₂×D₁₇	1	2	yes	complemented
216	C₂×#108	—	17	2	no	—
217	#421	—	34	2	no	—
218	#405	—	17	1	no	—
219	#368	—	17	1	no	—
220	#368	—	17	2	no	—
221	#368	C₂×D₁₇	1	1	yes	characteristic, complemented
222	#359	—	17	2	no	—
223	#359	—	17	2	no	—
224	#359	—	17	2	no	—
225	#359	C₂×D₁₇	1	2	yes	complemented
226	#346	—	17	1	no	—
227	#346	—	17	1	no	—
228	C₄×C₁₆	—	17	1	no	—
229	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic, complemented
230	C₂×C₁₇	—	2	1	no	—
231	C₂×C₁₇	—	8	2	no	—
232	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
233	D₁₇	—	2	2	no	—
234	D₁₇	—	8	2	no	—
235	C₂×#41	—	34	2	no	—
236	#135	—	17	2	no	—
237	#135	—	17	2	no	—
238	#135	—	17	2	no	—
239	#135	—	34	2	no	—
240	#135	—	34	2	no	—
241	#135	C₂²×D₁₇	1	2	yes	—
242	C₂×Q₁₆	—	17	2	no	—
243	C₂×SD₁₆	—	34	2	no	—
244	C₂×D₈	—	17	2	no	—
245	#131	—	2	2	no	—
246	#131	—	34	2	no	—
247	#131	—	34	2	no	—
248	#131	—	34	2	no	—
249	#131	—	68	2	no	—
250	#131	—	68	2	no	—
251	C₂×M₁₆	—	17	1	no	—
252	C₂×M₁₆	—	34	2	no	—
253	C₂×M₁₆	—	68	2	no	—
254	C₂×M₁₆	—	68	2	no	—
255	C₂²×C₈	—	17	1	no	—
256	#117	—	17	1	no	—
257	Q₃₂	—	17	2	no	—
258	Q₃₂	—	17	1	no	—
259	Q₃₂	C₄×D₁₇	1	1	yes	characteristic
260	SD₃₂	—	17	2	no	—
261	SD₃₂	—	17	2	no	—
262	SD₃₂	—	17	2	no	—
263	SD₃₂	C₄×D₁₇	1	2	yes	—
264	D₁₆	—	17	2	no	—
265	D₁₆	—	17	1	no	—
266	D₁₆	C₄×D₁₇	1	1	yes	characteristic
267	M₃₂	—	17	2	no	—
268	M₃₂	—	17	2	no	—
269	C₂×C₁₆	—	17	1	no	—
270	C₂×C₁₆	—	17	1	no	—
271	C₂×C₁₆	—	34	2	no	—
272	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
273	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
274	#108	—	17	2	no	—
275	#108	—	34	2	no	—
276	#108	C₂²×D₁₇	1	2	yes	—
277	#104	—	2	2	no	—
278	#104	—	34	2	no	—
279	#104	—	34	2	no	—
280	#104	—	34	2	no	—
281	#97	—	17	1	no	—
282	#97	—	17	1	no	—
283	C₄×C₈	—	17	1	no	—
284	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
285	C₁₇	#33464	1	1	yes	characteristic, complemented, Sylow
286	#41	—	2	2	no	—
287	#41	—	34	2	no	—
288	#41	—	34	2	no	—
289	#41	—	34	2	no	—
290	#41	—	68	2	no	—
291	#41	—	68	2	no	—
292	C₂×Q₈	—	34	2	no	—
293	C₂×D₄	—	34	2	no	—
294	C₂×D₄	—	68	2	no	—
295	C₂²×C₄	—	17	1	no	—
296	C₂²×C₄	—	68	2	no	—
297	Q₁₆	—	17	2	no	—
298	Q₁₆	—	34	2	no	—
299	Q₁₆	C₂×C₄×D₁₇	1	2	yes	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	34	2	no	—
302	SD₁₆	—	34	2	no	—
303	SD₁₆	—	34	2	no	—
304	D₈	—	17	2	no	—
305	D₈	—	34	2	no	—
306	D₈	C₂×C₄×D₁₇	1	2	yes	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	4	2	no	—
309	M₁₆	—	17	2	no	—
310	M₁₆	—	17	2	no	—
311	M₁₆	—	34	2	no	—
312	M₁₆	—	68	2	no	—
313	M₁₆	—	68	2	no	—
314	M₁₆	—	68	2	no	—
315	M₁₆	—	68	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	17	1	no	—
318	C₂×C₈	—	17	1	no	—
319	C₂×C₈	—	17	2	no	—
320	C₂×C₈	—	17	2	no	—
321	C₂×C₈	—	68	2	no	—
322	C₂×C₈	—	68	2	no	—
323	C₂×C₈	—	68	2	no	—
324	C₂×C₈	—	68	2	no	—
325	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
326	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
327	C₄⋊C₄	—	34	1	no	—
328	C₂²⋊C₄	—	34	1	no	—
329	C₄²	—	17	1	no	—
330	C₄²	D₄×D₁₇	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	17	2	no	—
333	C₁₆	—	34	1	no	—
334	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
336	C₂³	—	17	1	no	—
337	C₂³	—	68	2	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	34	2	no	—
340	Q₈	—	68	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	34	2	no	—
344	D₄	—	68	2	no	—
345	D₄	—	68	2	no	—
346	D₄	—	68	2	no	—
347	D₄	—	68	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	17	1	no	—
351	C₂×C₄	—	17	2	no	—
352	C₂×C₄	—	17	2	no	—
353	C₂×C₄	—	34	1	no	—
354	C₂×C₄	—	68	2	no	—
355	C₂×C₄	—	68	2	no	—
356	C₂×C₄	—	68	2	no	—
357	C₂×C₄	—	68	2	no	—
358	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	17	2	no	—
361	C₈	—	17	2	no	—
362	C₈	—	68	2	no	—
363	C₈	D₈×D₁₇	1	1	yes	characteristic
364	C₈	D₈×D₁₇	1	1	yes	characteristic
365	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
366	C₈	#41×D₁₇	1	1	yes	characteristic
367	C₂²	—	4	2	no	—
368	C₂²	—	272	2	no	—
369	C₂²	—	17	2	no	—
370	C₂²	—	68	1	no	—
371	C₂²	—	68	2	no	—
372	C₂²	#135×D₁₇	1	1	yes	characteristic
373	C₄	—	4	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	17	2	no	—
376	C₄	—	68	1	no	—
377	C₄	—	68	2	no	—
378	C₄	C₄×D₄×D₁₇	1	1	yes	characteristic
379	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
380	C₂	—	2	1	no	—
381	C₂	—	8	2	no	—
382	C₂	—	34	2	no	—
383	C₂	—	136	2	no	—
384	C₂	C₄×D₈×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2178	C₂
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	#14541917	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2178	D₁₇
hypocenter	C₁₇	#33464
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #14537696	C₂	semidirect
3. #14530670	C₂	semidirect
4. #14530669	C₂	semidirect
5. #14530668	C₂	semidirect
6. #14530666	C₂	semidirect
7. #14530662	C₂	semidirect
8. #14530577	C₂	semidirect
9. D₁₇×#359	C₂	semidirect
10. #14530379	C₂	semidirect
11. C₁₇×#2178	C₂	semidirect
14. #11763251	C₄	semidirect
15. #11763248	C₄	semidirect
16. #11763246	C₄	semidirect
17. #11763243	C₄	semidirect
24. #11762927	C₂²	semidirect
25. #11762926	C₂²	semidirect
26. #11762925	C₂²	semidirect
30. #11762639	C₂²	semidirect
31. #11762638	C₂²	semidirect
32. #11762617	C₂²	semidirect
33. #11762616	C₂²	semidirect
34. #11762615	C₂²	semidirect
35. #11762585	C₂²	semidirect
36. #11762584	C₂²	semidirect
37. C₁₆×Dic₁₇	C₂²	semidirect
39. C₁₇×#443	C₂²	semidirect
40. C₁₇×#368	C₂²	semidirect
41. C₁₇×#359	C₂²	semidirect
42. C₄×C₁₆×C₁₇	C₂²	semidirect
182. #2178	D₁₇	semidirect
215. #443	C₂×D₁₇	semidirect
221. #368	C₂×D₁₇	semidirect
225. #359	C₂×D₁₇	semidirect
229. C₄×C₁₆	C₂×D₁₇	semidirect
285. C₁₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#14541917	2
#14537696	2
#14530670	2
#14530669	2
#14530668	2
#14530666	2
#14530662	2
#14530577	2
D₁₇×#359	2
#14530379	2
C₁₇×#2178	2
#33464	17