Subgroups — #17187747

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763239	C₂	1	1	yes	maximal, characteristic
2	#14537701	C₂	1	1	yes	maximal, characteristic, complemented
3	#14537700	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530667	C₂	1	4	yes	maximal, complemented
5	#14530662	C₂	1	4	yes	maximal, complemented
6	C₂×#11762638	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530513	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530383	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#2143	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587590	C₂²	1	1	yes	characteristic
11	#11763576	—	2	1	no	—
12	#11763575	—	2	1	no	—
13	#11763239	C₄	1	2	yes	complemented
14	#11763239	C₄	1	2	yes	complemented
15	#11763239	C₂²	1	2	yes	—
16	#11763239	C₂²	1	2	yes	—
17	C₂×#10587500	C₄	1	1	yes	characteristic, complemented
18	C₂×#10587498	C₄	1	1	yes	characteristic, complemented
19	C₂×#10587499	C₄	1	1	yes	characteristic, complemented
20	C₂×#10587497	C₄	1	1	yes	characteristic, complemented
21	C₂²×#63214	C₂²	1	1	yes	characteristic
22	#11762931	—	2	2	no	—
23	#11762930	—	2	2	no	—
24	#11762926	C₂²	1	4	yes	complemented
25	#11762925	C₂²	1	4	yes	complemented
26	C₂×#10587490	C₂²	1	1	yes	characteristic
27	#11762826	—	2	1	no	—
28	#11762825	—	2	1	no	—
29	#11762764	C₂²	1	1	yes	characteristic
30	#11762638	C₂²	1	2	yes	complemented
31	#11762638	C₂²	1	2	yes	complemented
32	#11762617	C₂²	1	4	yes	complemented
33	#11762584	C₂²	1	2	yes	complemented
34	C₁₆×Dic₁₇	C₂²	1	2	yes	complemented
35	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
36	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
37	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
38	C₁₇×#359	C₂²	1	4	yes	complemented
39	C₂×#41×C₁₇	—	2	1	no	—
40	C₂×#63252	—	2	1	no	—
41	C₁₇×#135	—	2	1	no	—
42	C₁₇×#135	—	2	1	no	—
43	C₁₇×#135	C₂×C₄	1	2	yes	—
44	C₁₇×#135	C₂³	1	2	yes	—
45	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₁₇	—	2	1	no	—
47	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₁₇	—	2	1	no	—
49	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	#10587634	—	2	4	no	—
51	#10587634	—	4	2	no	—
52	#10587612	D₄	1	1	yes	characteristic
53	C₂×#63223	—	2	1	no	—
54	C₂×#63223	—	4	2	no	—
55	C₂×#63223	D₄	1	1	yes	characteristic
56	#10587593	—	2	4	no	—
57	M₁₆×D₁₇	—	4	2	no	—
58	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
59	#10587590	—	2	1	no	—
60	#10587590	—	2	1	no	—
61	#10587590	C₂×C₄	1	2	yes	—
62	#10587590	C₂³	1	2	yes	—
63	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
64	C₂×#63219	—	2	1	no	—
65	#10587587	—	4	2	no	—
66	C₁₇×M₃₂	—	2	1	no	—
67	C₁₇×M₃₂	—	2	1	no	—
68	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
69	C₁₇×#108	—	2	1	no	—
70	C₁₇×#108	C₂³	1	2	yes	—
71	#10587508	—	2	4	no	—
72	#10587500	C₂×C₄	1	2	yes	—
73	#10587500	C₂×C₄	1	2	yes	—
74	#10587499	C₂×C₄	1	2	yes	—
75	#10587499	C₂×C₄	1	2	yes	—
76	#10587498	C₂×C₄	1	2	yes	—
77	#10587498	C₂×C₄	1	2	yes	—
78	#10587497	C₂×C₄	1	2	yes	—
79	#10587497	C₂×C₄	1	2	yes	—
80	#10587496	—	2	4	no	—
81	#10587490	—	2	1	no	—
82	#10587490	C₂³	1	2	yes	—
83	#10587486	D₄	1	2	yes	—
84	C₈×Dic₁₇	C₂³	1	2	yes	—
85	C₂×#63214	C₂×C₄	1	1	yes	characteristic
86	C₂×#63214	C₂×C₄	1	1	yes	characteristic
87	C₂×#63214	C₂×C₄	1	1	yes	characteristic
88	C₂×#63214	C₂×C₄	1	1	yes	characteristic
89	C₂×#63214	C₂³	1	1	yes	characteristic
90	C₂×#63214	C₂³	1	1	yes	characteristic
91	#41×C₁₇	—	2	2	no	—
92	#41×C₁₇	—	2	2	no	—
93	#41×C₁₇	—	4	1	no	—
94	#41×C₁₇	—	4	1	no	—
95	C₂×Q₈×C₁₇	—	2	1	no	—
96	C₂×D₄×C₁₇	—	2	1	no	—
97	C₂×D₄×C₁₇	—	4	1	no	—
98	C₂²×C₄×C₁₇	—	4	1	no	—
99	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₂²⋊D₁₇	—	4	1	no	—
101	#63252	—	2	2	no	—
102	#63252	—	2	2	no	—
103	#63252	—	4	1	no	—
104	#63252	—	4	1	no	—
105	C₂×D₆₈	—	2	1	no	—
106	C₂×C₄×D₁₇	—	4	1	no	—
107	C₂×Dic₃₄	—	2	1	no	—
108	Q₁₆×C₁₇	—	2	1	no	—
109	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
110	SD₁₆×C₁₇	—	2	2	no	—
111	SD₁₆×C₁₇	—	2	2	no	—
112	D₈×C₁₇	—	2	1	no	—
113	D₈×C₁₇	C₂²×C₄	1	2	yes	—
114	M₁₆×C₁₇	—	2	2	no	—
115	M₁₆×C₁₇	—	4	1	no	—
116	C₂×C₈×C₁₇	—	4	1	no	—
117	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
121	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
122	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₁₇	—	2	1	no	—
124	C₄⋊Dic₁₇	—	2	1	no	—
125	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
126	#63223	—	2	2	no	—
127	#63223	—	4	1	no	—
128	#63223	—	4	4	no	—
129	#63223	D₈	1	2	yes	—
130	#63223	D₈	1	2	yes	—
131	C₂×#3622	—	4	1	no	—
132	C₂×#3622	—	4	4	no	—
133	C₂×#3622	C₂×D₄	1	2	yes	—
134	Dic₆₈	—	2	1	no	—
135	Dic₆₈	C₂²×C₄	1	2	yes	—
136	D₁₃₆	—	2	1	no	—
137	D₁₃₆	C₂²×C₄	1	2	yes	—
138	#63219	—	2	2	no	—
139	#63219	—	2	2	no	—
140	#63218	—	4	4	no	—
141	C₈×D₁₇	—	4	4	no	—
142	C₁₆×C₁₇	—	2	2	no	—
143	#63214	C₂²×C₄	1	1	yes	characteristic
144	#63214	C₂²×C₄	1	1	yes	characteristic
145	#63214	C₂²×C₄	1	1	yes	characteristic
146	#63214	C₂²×C₄	1	1	yes	characteristic
147	#33465	—	17	1	no	maximal, Sylow
148	C₂³×C₁₇	—	4	1	no	—
149	C₂³×C₁₇	#135	1	1	yes	characteristic
150	C₂²×D₁₇	—	4	1	no	—
151	Q₈×C₁₇	—	2	2	no	—
152	Q₈×C₁₇	—	4	1	no	—
153	D₄×C₁₇	—	2	2	no	—
154	D₄×C₁₇	—	4	2	no	—
155	D₄×C₁₇	—	4	2	no	—
156	D₄×C₁₇	—	4	1	no	—
157	C₂×C₄×C₁₇	—	4	1	no	—
158	C₂×C₄×C₁₇	—	4	2	no	—
159	C₂×C₄×C₁₇	—	4	2	no	—
160	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
165	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
166	C₂²⋊D₁₇	—	4	2	no	—
167	C₂²⋊D₁₇	—	4	2	no	—
168	C₂×Dic₁₇	—	2	2	no	—
169	C₂×Dic₁₇	—	4	1	no	—
170	D₆₈	—	2	2	no	—
171	D₆₈	—	4	1	no	—
172	C₄×D₁₇	—	4	2	no	—
173	C₄×D₁₇	—	4	2	no	—
174	Dic₃₄	—	2	2	no	—
175	Dic₃₄	—	4	1	no	—
176	C₈×C₁₇	—	4	2	no	—
177	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
178	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
179	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
180	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
181	#3622	—	4	2	no	—
182	#3622	C₂×D₈	1	4	yes	—
183	C₂×#508	—	17	1	no	—
184	#2957	—	17	1	no	—
185	#2957	—	17	1	no	—
186	#2179	—	17	4	no	—
187	#2178	—	17	4	no	—
188	C₂×#368	—	17	1	no	—
189	#2143	—	17	1	no	—
190	#2143	D₁₇	1	1	yes	characteristic, complemented
191	#2108	—	17	1	no	—
192	C₂²×C₁₇	—	4	1	no	—
193	C₂²×C₁₇	—	4	2	no	—
194	C₂²×C₁₇	—	16	1	no	—
195	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
196	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
197	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
198	C₂×D₁₇	—	4	2	no	—
199	C₂×D₁₇	—	16	1	no	—
200	C₄×C₁₇	—	4	2	no	—
201	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
202	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
203	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
205	Dic₁₇	—	4	2	no	—
206	Dic₁₇	—	4	2	no	—
207	C₂×#135	—	17	1	no	—
208	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
209	#568	—	34	1	no	—
210	#568	—	34	1	no	—
211	#508	—	17	2	no	—
212	#508	—	17	2	no	—
213	#508	—	17	2	no	—
214	#508	—	17	2	no	—
215	C₂×Q₃₂	—	17	1	no	—
216	C₂×SD₃₂	—	17	1	no	—
217	C₂×SD₃₂	—	17	1	no	—
218	C₂×D₁₆	—	17	1	no	—
219	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
220	C₂²×C₁₆	—	17	1	no	—
221	#444	—	34	2	no	—
222	#444	—	34	2	no	—
223	#443	—	17	4	no	—
224	#443	—	17	4	no	—
225	C₂×#108	—	17	1	no	—
226	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
227	#421	—	34	1	no	—
228	#421	—	34	1	no	—
229	#405	—	17	1	no	—
230	#368	—	17	2	no	—
231	#368	—	17	2	no	—
232	#359	—	17	4	no	—
233	#359	C₂×D₁₇	1	4	yes	complemented
234	#346	—	17	2	no	—
235	C₄×C₁₆	—	17	2	no	—
236	C₂×C₁₇	—	2	1	no	—
237	C₂×C₁₇	—	2	1	no	—
238	C₂×C₁₇	—	2	1	no	—
239	C₂×C₁₇	—	8	2	no	—
240	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
241	D₁₇	—	8	2	no	—
242	C₂×#41	—	2	1	no	—
243	C₂×#41	—	34	1	no	—
244	#135	—	2	1	no	—
245	#135	—	2	1	no	—
246	#135	—	17	2	no	—
247	#135	—	17	2	no	—
248	#135	—	34	1	no	—
249	#135	—	34	1	no	—
250	#135	C₄×D₁₇	1	2	yes	—
251	#135	C₂²×D₁₇	1	2	yes	—
252	C₂×Q₁₆	—	17	1	no	—
253	C₂×Q₁₆	C₄×D₁₇	1	1	yes	characteristic
254	C₂×SD₁₆	—	2	1	no	—
255	C₂×SD₁₆	—	34	1	no	—
256	C₂×D₈	—	17	1	no	—
257	C₂×D₈	C₄×D₁₇	1	1	yes	characteristic
258	#131	—	34	4	no	—
259	#131	—	34	4	no	—
260	#131	—	68	2	no	—
261	#131	—	68	2	no	—
262	C₂×M₁₆	—	2	1	no	—
263	C₂×M₁₆	—	17	1	no	—
264	C₂×M₁₆	—	34	1	no	—
265	C₂×M₁₆	—	68	2	no	—
266	C₂×M₁₆	—	68	2	no	—
267	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
268	#117	—	17	1	no	—
269	Q₃₂	—	17	2	no	—
270	Q₃₂	—	17	2	no	—
271	SD₃₂	—	17	2	no	—
272	SD₃₂	—	17	2	no	—
273	SD₃₂	—	17	2	no	—
274	SD₃₂	—	17	2	no	—
275	D₁₆	—	17	2	no	—
276	D₁₆	—	17	2	no	—
277	M₃₂	—	2	1	no	—
278	M₃₂	—	2	1	no	—
279	C₂×C₁₆	—	17	1	no	—
280	C₂×C₁₆	—	17	1	no	—
281	C₂×C₁₆	—	17	1	no	—
282	C₂×C₁₆	—	17	1	no	—
283	C₂×C₁₆	—	17	1	no	—
284	C₂×C₁₆	—	17	1	no	—
285	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
286	#108	—	2	1	no	—
287	#108	—	17	2	no	—
288	#108	—	34	1	no	—
289	#108	C₂²×D₁₇	1	2	yes	—
290	#104	—	34	4	no	—
291	#104	—	34	4	no	—
292	#97	—	17	2	no	—
293	C₄×C₈	—	17	2	no	—
294	C₁₇	#33465	1	1	yes	characteristic, complemented, Sylow
295	#41	—	2	2	no	—
296	#41	—	2	2	no	—
297	#41	—	4	1	no	—
298	#41	—	4	1	no	—
299	#41	—	34	2	no	—
300	#41	—	34	2	no	—
301	#41	—	68	1	no	—
302	#41	—	68	1	no	—
303	C₂×Q₈	—	2	1	no	—
304	C₂×Q₈	—	34	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	34	1	no	—
308	C₂×D₄	—	68	1	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	68	1	no	—
311	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
312	Q₁₆	—	2	1	no	—
313	Q₁₆	—	17	2	no	—
314	Q₁₆	—	34	1	no	—
315	Q₁₆	C₂×C₄×D₁₇	1	2	yes	—
316	SD₁₆	—	2	2	no	—
317	SD₁₆	—	2	2	no	—
318	SD₁₆	—	34	2	no	—
319	SD₁₆	—	34	2	no	—
320	D₈	—	2	1	no	—
321	D₈	—	17	2	no	—
322	D₈	—	34	1	no	—
323	D₈	C₂×C₄×D₁₇	1	2	yes	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	17	2	no	—
327	M₁₆	—	17	2	no	—
328	M₁₆	—	34	2	no	—
329	M₁₆	—	68	4	no	—
330	M₁₆	—	68	4	no	—
331	M₁₆	—	68	1	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	17	2	no	—
334	C₂×C₈	—	68	4	no	—
335	C₂×C₈	—	68	4	no	—
336	C₂×C₈	—	68	1	no	—
337	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
338	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
339	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
340	C₂×C₈	#63254	1	1	yes	characteristic
341	C₂×C₈	#63254	1	1	yes	characteristic
342	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
343	C₄⋊C₄	—	34	1	no	—
344	C₂²⋊C₄	—	34	1	no	—
345	C₄²	—	17	2	no	—
346	C₁₆	—	2	2	no	—
347	C₁₆	—	17	1	no	—
348	C₁₆	—	17	1	no	—
349	C₁₆	—	17	1	no	—
350	C₁₆	—	17	1	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	68	1	no	—
353	C₂³	#10587602	1	1	yes	characteristic
354	Q₈	—	2	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	34	2	no	—
357	Q₈	—	68	1	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	1	no	—
362	D₄	—	34	2	no	—
363	D₄	—	68	2	no	—
364	D₄	—	68	2	no	—
365	D₄	—	68	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	34	2	no	—
370	C₂×C₄	—	68	1	no	—
371	C₂×C₄	—	68	2	no	—
372	C₂×C₄	—	68	2	no	—
373	C₂×C₄	#10587567	1	1	yes	characteristic
374	C₂×C₄	#10587567	1	1	yes	characteristic
375	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
376	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
377	C₂×C₄	#10587602	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
379	C₈	—	4	2	no	—
380	C₈	—	17	4	no	—
381	C₈	—	68	2	no	—
382	C₈	#10587567	1	1	yes	characteristic
383	C₈	#10587567	1	1	yes	characteristic
384	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
385	C₈	C₂×#63254	1	1	yes	characteristic
386	C₂²	—	4	1	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	16	1	no	—
389	C₂²	—	272	1	no	—
390	C₂²	—	68	2	no	—
391	C₂²	#11762889	1	1	yes	characteristic
392	C₂²	#11762889	1	1	yes	characteristic
393	C₂²	C₂×#10587602	1	1	yes	characteristic
394	C₄	—	4	2	no	—
395	C₄	—	68	2	no	—
396	C₄	—	68	2	no	—
397	C₄	#11762889	1	1	yes	characteristic
398	C₄	#11762889	1	1	yes	characteristic
399	C₄	C₂×#10587567	1	1	yes	characteristic
400	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	—	8	2	no	—
405	C₂	—	136	2	no	—
406	C₂	C₂×#11762889	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2143	C₂
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	C₂×#11763239	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2143	D₁₇
hypocenter	C₁₇	#33465
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2143

Complements

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

N	H ≅ G/N	type
2. #14537701	C₂	semidirect
3. #14537700	C₂	semidirect
4. #14530667	C₂	semidirect
5. #14530662	C₂	semidirect
6. C₂×#11762638	C₂	semidirect
7. #14530513	C₂	semidirect
8. #14530383	C₂	semidirect
9. C₁₇×#2143	C₂	semidirect
13. #11763239	C₄	semidirect
14. #11763239	C₄	semidirect
17. C₂×#10587500	C₄	semidirect
18. C₂×#10587498	C₄	semidirect
19. C₂×#10587499	C₄	semidirect
20. C₂×#10587497	C₄	semidirect
24. #11762926	C₂²	semidirect
25. #11762925	C₂²	semidirect
30. #11762638	C₂²	semidirect
31. #11762638	C₂²	semidirect
32. #11762617	C₂²	semidirect
33. #11762584	C₂²	semidirect
34. C₁₆×Dic₁₇	C₂²	semidirect
38. C₁₇×#359	C₂²	semidirect
190. #2143	D₁₇	semidirect
233. #359	C₂×D₁₇	semidirect
294. C₁₇	#33465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763239	2
#14537701	2
#14537700	2
#14530667	2
#14530662	2
C₂×#11762638	2
#14530513	2
#14530383	2
C₁₇×#2143	2
#33465	17