Subgroups — #17017469

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539905	C₂	1	1	yes	maximal, characteristic
2	#14539709	C₂	1	1	yes	maximal, characteristic
3	#14537497	C₂	1	1	yes	maximal, characteristic
4	#14528328	C₂	1	4	yes	maximal, complemented
5	#14528204	C₂	1	4	yes	maximal, complemented
6	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
7	#14527469	C₂	1	1	yes	maximal, characteristic, complemented
8	#14527456	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587587	—	2	1	no	—
11	#11763359	C₂²	1	1	yes	characteristic
12	#11763212	—	2	1	no	—
13	#11763183	—	2	1	no	—
14	#11763159	—	2	1	no	—
15	#11763157	—	2	1	no	—
16	#11763122	—	2	2	no	—
17	#11763086	C₂²	1	4	yes	—
18	#11763011	C₂²	1	4	yes	—
19	#11762948	—	2	2	no	—
20	#11762888	—	2	1	no	—
21	#11762888	C₂²	1	2	yes	—
22	C₈×D₆₈	—	2	1	no	—
23	C₈×D₆₈	C₂²	1	2	yes	—
24	#11762862	C₂²	1	1	yes	characteristic
25	#11762862	C₂²	1	1	yes	characteristic
26	#11762833	C₂²	1	1	yes	characteristic
27	#11762775	—	2	1	no	—
28	#11762562	C₂²	1	2	yes	—
29	#11762560	C₂²	1	2	yes	—
30	#11762555	C₂²	1	4	yes	—
31	#11762555	C₂²	1	4	yes	—
32	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
34	C₁₇×#487	—	2	1	no	—
35	C₁₇×#487	—	2	1	no	—
36	C₁₇×#487	C₂²	1	2	yes	—
37	C₁₇×#487	C₂²	1	2	yes	—
38	C₂×C₁₇×#103	—	2	1	no	—
39	C₂×C₁₇×#103	—	2	1	no	—
40	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₇	—	2	1	no	—
42	C₂²×Q₈×C₁₇	—	2	1	no	—
43	C₂×#63252	—	2	1	no	—
44	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₁₇×C₄⋊Q₈	—	2	1	no	—
47	C₁₇×C₄⋊Q₈	—	2	1	no	—
48	C₁₇×C₄⋊Q₈	—	2	1	no	—
49	C₁₇×C₄⋊Q₈	—	2	1	no	—
50	C₁₇×C₄⋊Q₈	—	2	1	no	—
51	C₁₇×C₄⋊Q₈	—	2	1	no	—
52	C₁₇×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₁₇×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
59	C₂×#63230	—	2	4	no	—
60	#10587616	—	2	1	no	—
61	#10587616	D₄	1	2	yes	—
62	C₂×#63223	—	2	1	no	—
63	#10587587	—	4	4	no	—
64	C₂×#63218	—	2	2	no	—
65	C₂×C₈×D₁₇	—	2	2	no	—
66	#10587561	—	2	1	no	—
67	#10587560	D₄	1	1	yes	characteristic
68	C₄×D₆₈	—	2	1	no	—
69	C₄×D₆₈	C₂³	1	2	yes	—
70	#10587555	D₄	1	1	yes	characteristic
71	C₁₇×#103	—	2	2	no	—
72	C₁₇×#103	—	2	2	no	—
73	C₁₇×#103	—	2	2	no	—
74	C₁₇×#103	—	2	2	no	—
75	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	#10587506	—	2	4	no	—
80	C₂²⋊#3622	—	2	1	no	—
81	C₂²⋊#3622	—	2	1	no	—
82	#10587491	—	2	2	no	—
83	#10587491	—	2	2	no	—
84	#10587480	—	2	4	no	—
85	#10587478	—	2	4	no	—
86	C₄⋊#3622	—	2	1	no	—
87	C₄⋊#3622	C₂³	1	2	yes	—
88	C₄⋊#3622	C₂³	1	2	yes	—
89	C₄⋊#3622	C₂³	1	2	yes	—
90	C₄×#3622	—	2	1	no	—
91	C₄×#3622	—	2	1	no	—
92	C₂×Q₈×C₁₇	—	2	2	no	—
93	C₂×Q₈×C₁₇	—	2	2	no	—
94	C₂×Q₈×C₁₇	—	4	1	no	—
95	C₂×Q₈×C₁₇	—	4	1	no	—
96	C₂×Q₈×C₁₇	—	4	2	no	—
97	C₂×Q₈×C₁₇	—	4	2	no	—
98	C₂×Q₈×C₁₇	—	8	2	no	—
99	C₂²×C₄×C₁₇	—	4	1	no	—
100	C₂²×C₄×C₁₇	—	4	1	no	—
101	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
104	C₂×C₂²⋊D₁₇	—	2	2	no	—
105	#63252	—	4	2	no	—
106	#63252	—	4	2	no	—
107	C₂×D₆₈	—	2	1	no	—
108	C₂×D₆₈	—	2	1	no	—
109	C₂×D₆₈	C₂×D₄	1	2	yes	—
110	C₂×C₄×D₁₇	—	2	2	no	—
111	C₂×Dic₃₄	—	2	1	no	—
112	C₂×C₈×C₁₇	—	2	2	no	—
113	C₂×C₈×C₁₇	—	2	2	no	—
114	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₁₇	—	2	2	no	—
119	C₄⋊C₄×C₁₇	—	2	2	no	—
120	C₄⋊C₄×C₁₇	—	2	2	no	—
121	C₄⋊C₄×C₁₇	—	2	2	no	—
122	C₄⋊C₄×C₁₇	—	4	1	no	—
123	C₄⋊C₄×C₁₇	—	4	1	no	—
124	C₄⋊C₄×C₁₇	—	4	1	no	—
125	C₄⋊C₄×C₁₇	—	4	1	no	—
126	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₁₇	#41	1	1	yes	characteristic
129	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
130	#63230	—	8	4	no	—
131	#63227	—	2	2	no	—
132	#63227	—	2	2	no	—
133	C₄⋊Dic₁₇	—	2	1	no	—
134	C₄⋊Dic₁₇	C₂×D₄	1	2	yes	—
135	#63223	—	4	2	no	—
136	C₂×#3622	—	2	2	no	—
137	C₂×#3622	—	2	2	no	—
138	C₂×#3622	—	2	2	no	—
139	#63218	—	4	4	no	—
140	C₈×D₁₇	—	4	4	no	—
141	#19197	—	17	1	no	maximal, Sylow
142	C₂³×C₁₇	#121	1	1	yes	characteristic
143	C₂²×D₁₇	—	2	2	no	—
144	Q₈×C₁₇	—	4	2	no	—
145	Q₈×C₁₇	—	4	2	no	—
146	Q₈×C₁₇	—	4	4	no	—
147	Q₈×C₁₇	—	8	2	no	—
148	Q₈×C₁₇	—	8	2	no	—
149	C₂×C₄×C₁₇	—	2	2	no	—
150	C₂×C₄×C₁₇	—	2	2	no	—
151	C₂×C₄×C₁₇	—	4	2	no	—
152	C₂×C₄×C₁₇	—	4	2	no	—
153	C₂×C₄×C₁₇	—	8	2	no	—
154	C₂×C₄×C₁₇	—	8	2	no	—
155	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₇	#135	1	2	yes	—
159	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₁₇	—	4	4	no	—
164	C₂×Dic₁₇	—	2	2	no	—
165	D₆₈	—	2	4	no	—
166	D₆₈	—	4	2	no	—
167	D₆₈	—	4	2	no	—
168	C₄×D₁₇	—	4	2	no	—
169	C₄×D₁₇	—	4	2	no	—
170	Dic₃₄	—	4	2	no	—
171	C₈×C₁₇	—	2	1	no	—
172	C₈×C₁₇	—	2	1	no	—
173	C₈×C₁₇	—	2	1	no	—
174	C₈×C₁₇	—	2	1	no	—
175	#3622	—	4	2	no	—
176	#3622	—	4	2	no	—
177	#3622	—	4	2	no	—
178	#3226	—	17	1	no	—
179	#3206	—	17	1	no	—
180	C₂×#487	D₁₇	1	1	yes	characteristic, complemented
181	#2931	—	17	1	no	—
182	#1699	—	17	4	no	—
183	#1679	—	17	4	no	—
184	#1564	—	17	1	no	—
185	#1549	—	17	1	no	—
186	#1546	—	17	1	no	—
187	C₂²×C₁₇	—	2	1	no	—
188	C₂²×C₁₇	—	2	1	no	—
189	C₂²×C₁₇	#463	1	2	yes	—
190	C₂²×C₁₇	C₂×#121	1	1	yes	characteristic
191	C₂×D₁₇	—	4	2	no	—
192	C₂×D₁₇	—	4	2	no	—
193	C₂×D₁₇	—	4	2	no	—
194	C₄×C₁₇	—	2	1	no	—
195	C₄×C₁₇	—	2	1	no	—
196	C₄×C₁₇	—	8	2	no	—
197	C₄×C₁₇	—	8	2	no	—
198	C₄×C₁₇	#463	1	2	yes	—
199	C₄×C₁₇	#473	1	2	yes	—
200	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
201	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₄×C₁₇	#577	1	2	yes	—
203	Dic₁₇	—	4	2	no	—
204	C₂×#131	—	34	1	no	—
205	#532	—	17	1	no	—
206	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
207	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
208	#500	—	34	1	no	—
209	#492	—	34	1	no	—
210	#487	—	2	1	no	—
211	#487	—	2	1	no	—
212	#487	—	34	1	no	—
213	#487	C₂×D₁₇	1	2	yes	—
214	#487	C₂×D₁₇	1	2	yes	—
215	#486	—	34	1	no	—
216	#480	—	34	2	no	—
217	#474	—	17	4	no	—
218	#461	—	17	4	no	—
219	#448	—	34	2	no	—
220	#436	—	17	2	no	—
221	#436	—	34	1	no	—
222	C₈×D₄	—	17	2	no	—
223	C₈×D₄	—	34	1	no	—
224	#431	—	17	1	no	—
225	#431	—	17	1	no	—
226	#423	—	17	1	no	—
227	C₂×#103	—	2	1	no	—
228	C₂×#103	—	2	1	no	—
229	#408	—	34	1	no	—
230	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
231	#335	—	17	2	no	—
232	#334	—	17	2	no	—
233	#332	—	17	4	no	—
234	#332	—	17	4	no	—
235	C₂×C₁₇	—	2	2	no	—
236	C₂×C₁₇	C₂×#463	1	1	yes	characteristic
237	C₂×C₁₇	#3057	1	1	yes	characteristic
238	C₂×C₁₇	C₂×#473	1	1	yes	characteristic
239	D₁₇	—	8	2	no	—
240	C₂×#41	—	34	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂×SD₁₆	—	34	4	no	—
244	#131	—	68	4	no	—
245	C₂×M₁₆	—	34	1	no	—
246	C₂×M₁₆	—	34	2	no	—
247	C₂²×C₈	—	34	2	no	—
248	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
249	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	17	1	no	—
257	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
258	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
259	#127	—	17	1	no	—
260	#124	—	34	1	no	—
261	#121	—	17	2	no	—
262	#121	—	34	1	no	—
263	C₄×D₄	—	17	2	no	—
264	C₄×D₄	—	34	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
270	#107	—	34	4	no	—
271	C₄⋊C₈	—	17	2	no	—
272	C₄⋊C₈	—	17	2	no	—
273	C₄⋊C₈	—	17	2	no	—
274	C₄⋊C₈	—	34	1	no	—
275	#103	—	2	2	no	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#103	—	2	2	no	—
279	#103	—	34	4	no	—
280	#102	—	34	4	no	—
281	C₂²⋊C₈	—	34	1	no	—
282	C₂²⋊C₈	—	34	1	no	—
283	C₂²⋊C₈	—	34	2	no	—
284	C₂²⋊C₈	—	34	2	no	—
285	C₄×C₈	—	34	1	no	—
286	C₄×C₈	—	34	1	no	—
287	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
288	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
291	C₁₇	#19197	1	1	yes	characteristic, complemented, Sylow
292	#41	—	68	2	no	—
293	#41	—	68	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	8	2	no	—
301	C₂×Q₈	—	34	1	no	—
302	C₂×D₄	—	17	2	no	—
303	C₂×D₄	—	34	2	no	—
304	C₂×D₄	—	34	1	no	—
305	C₂×D₄	—	34	1	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	34	2	no	—
309	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
310	C₂²×C₄	#63254	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
312	SD₁₆	—	136	4	no	—
313	M₁₆	—	68	4	no	—
314	M₁₆	—	68	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	34	2	no	—
318	C₂×C₈	—	34	2	no	—
319	C₂×C₈	—	34	2	no	—
320	C₂×C₈	—	68	4	no	—
321	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₄⋊C₄	—	17	2	no	—
334	C₄⋊C₄	—	34	1	no	—
335	C₂²⋊C₄	—	34	2	no	—
336	C₂²⋊C₄	—	34	2	no	—
337	C₄²	D₄×D₁₇	1	1	yes	characteristic
338	C₄²	#63254	1	1	yes	characteristic
339	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
340	C₄²	C₂³×D₁₇	1	1	yes	characteristic
341	C₂³	—	34	2	no	—
342	C₂³	#10587624	1	1	yes	characteristic
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	4	no	—
346	Q₈	—	8	2	no	—
347	Q₈	—	8	2	no	—
348	Q₈	—	68	2	no	—
349	D₄	—	34	4	no	—
350	D₄	—	68	4	no	—
351	D₄	—	68	2	no	—
352	D₄	—	68	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	34	2	no	—
360	C₂×C₄	—	68	2	no	—
361	C₂×C₄	—	68	2	no	—
362	C₂×C₄	#10587624	1	1	yes	characteristic
363	C₂×C₄	#10587624	1	1	yes	characteristic
364	C₂×C₄	#10587624	1	1	yes	characteristic
365	C₂×C₄	#10587635	1	2	yes	—
366	C₂×C₄	#10587636	1	2	yes	—
367	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#63254	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	68	2	no	—
375	C₈	—	68	2	no	—
376	C₈	—	68	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	68	2	no	—
380	C₂²	—	68	2	no	—
381	C₂²	—	68	2	no	—
382	C₂²	#11763021	1	2	yes	—
383	C₂²	C₂×#10587624	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	68	2	no	—
389	C₄	#11763021	1	2	yes	—
390	C₄	#11763081	1	2	yes	—
391	C₄	C₂×#10587636	1	1	yes	characteristic
392	C₄	C₂×#10587635	1	1	yes	characteristic
393	C₄	#11763615	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	136	2	no	—
396	C₂	C₂×#11763021	1	1	yes	characteristic
397	C₂	#14538419	1	1	yes	characteristic
398	C₂	C₂×#11763081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587624
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#487	C₂
socle	C₂²×C₁₇	C₂×#121
prime radical	#11763359	C₂²
hypercenter	C₂×#487	D₁₇
hypocenter	C₁₇	#19197
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
4. #14528328	C₂	semidirect
5. #14528204	C₂	semidirect
6. #14527532	C₂	semidirect
7. #14527469	C₂	semidirect
8. #14527456	C₂	semidirect
9. C₂×C₁₇×#487	C₂	semidirect
180. C₂×#487	D₁₇	semidirect
291. C₁₇	#19197	semidirect

Maximal subgroups

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

subgroup	index
#14539905	2
#14539709	2
#14537497	2
#14528328	2
#14528204	2
#14527532	2
#14527469	2
#14527456	2
C₂×C₁₇×#487	2
#19197	17