Subgroups — #17017487

Order 4352.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539912	C₂	1	1	yes	maximal, characteristic
2	#14539905	C₂	1	1	yes	maximal, characteristic
3	#14537497	C₂	1	1	yes	maximal, characteristic
4	#14528356	C₂	1	4	yes	maximal, complemented
5	#14528232	C₂	1	4	yes	maximal, complemented
6	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
7	#14527465	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	#11763205	—	2	1	no	—
14	#11763157	—	2	1	no	—
15	#11763157	—	2	1	no	—
16	#11763127	—	2	2	no	—
17	#11763122	—	2	2	no	—
18	#11763096	C₂²	1	4	yes	—
19	#11763086	C₂²	1	4	yes	—
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	#11762558	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	#10587616	—	2	1	no	—
60	#10587616	D₄	1	2	yes	—
61	C₂×#63223	—	2	1	no	—
62	#10587587	—	4	4	no	—
63	C₂×#63218	—	2	2	no	—
64	C₂×C₈×D₁₇	—	2	2	no	—
65	#10587561	—	2	1	no	—
66	#10587560	D₄	1	1	yes	characteristic
67	C₄×D₆₈	—	2	1	no	—
68	C₄×D₆₈	C₂³	1	2	yes	—
69	#10587555	D₄	1	1	yes	characteristic
70	C₁₇×#103	—	2	2	no	—
71	C₁₇×#103	—	2	2	no	—
72	C₁₇×#103	—	2	2	no	—
73	C₁₇×#103	—	2	2	no	—
74	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
75	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊#3622	—	2	1	no	—
79	C₂²⋊#3622	—	2	1	no	—
80	#10587491	—	2	2	no	—
81	#10587491	—	2	2	no	—
82	#10587480	—	2	4	no	—
83	#10587480	—	2	4	no	—
84	#10587479	—	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	#63227	—	2	2	no	—
131	#63227	—	2	2	no	—
132	C₄⋊Dic₁₇	—	2	1	no	—
133	C₄⋊Dic₁₇	C₂×D₄	1	2	yes	—
134	#63223	—	4	2	no	—
135	C₂×#3622	—	2	2	no	—
136	C₂×#3622	—	2	2	no	—
137	C₂×#3622	—	2	2	no	—
138	#63218	—	4	4	no	—
139	C₈×D₁₇	—	4	4	no	—
140	#19199	—	17	1	no	maximal, Sylow
141	C₂³×C₁₇	#121	1	1	yes	characteristic
142	C₂²×D₁₇	—	2	2	no	—
143	Q₈×C₁₇	—	4	2	no	—
144	Q₈×C₁₇	—	4	2	no	—
145	Q₈×C₁₇	—	8	2	no	—
146	Q₈×C₁₇	—	8	2	no	—
147	Q₈×C₁₇	—	8	2	no	—
148	C₂×C₄×C₁₇	—	2	2	no	—
149	C₂×C₄×C₁₇	—	2	2	no	—
150	C₂×C₄×C₁₇	—	4	2	no	—
151	C₂×C₄×C₁₇	—	4	2	no	—
152	C₂×C₄×C₁₇	—	8	2	no	—
153	C₂×C₄×C₁₇	—	8	2	no	—
154	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
155	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₇	#135	1	2	yes	—
158	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
162	C₂²⋊D₁₇	—	4	4	no	—
163	C₂×Dic₁₇	—	2	2	no	—
164	D₆₈	—	2	4	no	—
165	D₆₈	—	4	2	no	—
166	D₆₈	—	4	2	no	—
167	C₄×D₁₇	—	4	2	no	—
168	C₄×D₁₇	—	4	2	no	—
169	Dic₃₄	—	4	2	no	—
170	C₈×C₁₇	—	2	1	no	—
171	C₈×C₁₇	—	2	1	no	—
172	C₈×C₁₇	—	2	1	no	—
173	C₈×C₁₇	—	2	1	no	—
174	#3622	—	4	2	no	—
175	#3622	—	4	2	no	—
176	#3622	—	4	2	no	—
177	#3227	—	17	1	no	—
178	#3226	—	17	1	no	—
179	C₂×#487	D₁₇	1	1	yes	characteristic, complemented
180	#2931	—	17	1	no	—
181	#1703	—	17	4	no	—
182	#1683	—	17	4	no	—
183	#1564	—	17	1	no	—
184	#1548	—	17	1	no	—
185	#1546	—	17	1	no	—
186	C₂²×C₁₇	—	2	1	no	—
187	C₂²×C₁₇	—	2	1	no	—
188	C₂²×C₁₇	#463	1	2	yes	—
189	C₂²×C₁₇	C₂×#121	1	1	yes	characteristic
190	C₂×D₁₇	—	4	2	no	—
191	C₂×D₁₇	—	4	2	no	—
192	C₂×D₁₇	—	4	2	no	—
193	C₄×C₁₇	—	2	1	no	—
194	C₄×C₁₇	—	2	1	no	—
195	C₄×C₁₇	—	8	2	no	—
196	C₄×C₁₇	—	8	2	no	—
197	C₄×C₁₇	#463	1	2	yes	—
198	C₄×C₁₇	#471	1	2	yes	—
199	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
200	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₄×C₁₇	#578	1	2	yes	—
202	Dic₁₇	—	4	2	no	—
203	C₂×#131	—	34	1	no	—
204	#532	—	17	1	no	—
205	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
206	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
207	#500	—	34	1	no	—
208	#498	—	34	1	no	—
209	#487	—	2	1	no	—
210	#487	—	2	1	no	—
211	#487	C₂×D₁₇	1	2	yes	—
212	#487	C₂×D₁₇	1	2	yes	—
213	#486	—	34	1	no	—
214	#486	—	34	1	no	—
215	#481	—	34	2	no	—
216	#480	—	34	2	no	—
217	#476	—	17	4	no	—
218	#474	—	17	4	no	—
219	#436	—	17	2	no	—
220	#436	—	34	1	no	—
221	C₈×D₄	—	17	2	no	—
222	C₈×D₄	—	34	1	no	—
223	#431	—	17	1	no	—
224	#431	—	17	1	no	—
225	#423	—	17	1	no	—
226	C₂×#103	—	2	1	no	—
227	C₂×#103	—	2	1	no	—
228	#408	—	34	1	no	—
229	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
230	#335	—	17	2	no	—
231	#334	—	17	2	no	—
232	#333	—	17	4	no	—
233	#332	—	17	4	no	—
234	C₂×C₁₇	—	2	2	no	—
235	C₂×C₁₇	C₂×#463	1	1	yes	characteristic
236	C₂×C₁₇	#3059	1	1	yes	characteristic
237	C₂×C₁₇	C₂×#471	1	1	yes	characteristic
238	D₁₇	—	8	2	no	—
239	C₂×#41	—	34	1	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	#131	—	68	4	no	—
243	C₂×M₁₆	—	34	1	no	—
244	C₂×M₁₆	—	34	2	no	—
245	C₂²×C₈	—	34	2	no	—
246	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
247	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
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	—	17	1	no	—
255	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
256	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
257	#127	—	17	1	no	—
258	#124	—	34	1	no	—
259	#121	—	17	2	no	—
260	#121	—	34	1	no	—
261	C₄×D₄	—	17	2	no	—
262	C₄×D₄	—	34	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	#107	—	34	4	no	—
269	#106	—	34	4	no	—
270	C₄⋊C₈	—	17	2	no	—
271	C₄⋊C₈	—	17	2	no	—
272	C₄⋊C₈	—	17	2	no	—
273	C₄⋊C₈	—	34	1	no	—
274	#103	—	2	2	no	—
275	#103	—	2	2	no	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#102	—	34	4	no	—
279	#102	—	34	4	no	—
280	C₂²⋊C₈	—	34	1	no	—
281	C₂²⋊C₈	—	34	1	no	—
282	C₂²⋊C₈	—	34	2	no	—
283	C₂²⋊C₈	—	34	2	no	—
284	C₄×C₈	—	34	1	no	—
285	C₄×C₈	—	34	1	no	—
286	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
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	C₁₇	#19199	1	1	yes	characteristic, complemented, Sylow
291	#41	—	68	2	no	—
292	#41	—	68	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	8	2	no	—
300	C₂×Q₈	—	34	1	no	—
301	C₂×D₄	—	17	2	no	—
302	C₂×D₄	—	34	2	no	—
303	C₂×D₄	—	34	1	no	—
304	C₂×D₄	—	34	1	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	34	2	no	—
308	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
309	C₂²×C₄	#63254	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
311	M₁₆	—	68	4	no	—
312	M₁₆	—	68	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	34	2	no	—
316	C₂×C₈	—	34	2	no	—
317	C₂×C₈	—	34	2	no	—
318	C₂×C₈	—	68	4	no	—
319	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
321	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	17	2	no	—
332	C₄⋊C₄	—	34	1	no	—
333	C₂²⋊C₄	—	34	2	no	—
334	C₂²⋊C₄	—	34	2	no	—
335	C₄²	D₄×D₁₇	1	1	yes	characteristic
336	C₄²	#63254	1	1	yes	characteristic
337	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
338	C₄²	C₂³×D₁₇	1	1	yes	characteristic
339	C₂³	—	34	2	no	—
340	C₂³	#10587624	1	1	yes	characteristic
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	8	2	no	—
346	Q₈	—	68	2	no	—
347	D₄	—	34	4	no	—
348	D₄	—	68	4	no	—
349	D₄	—	68	2	no	—
350	D₄	—	68	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	2	no	—
356	C₂×C₄	—	8	2	no	—
357	C₂×C₄	—	34	2	no	—
358	C₂×C₄	—	68	2	no	—
359	C₂×C₄	—	68	2	no	—
360	C₂×C₄	#10587624	1	1	yes	characteristic
361	C₂×C₄	#10587624	1	1	yes	characteristic
362	C₂×C₄	#10587624	1	1	yes	characteristic
363	C₂×C₄	#10587635	1	2	yes	—
364	C₂×C₄	#10587636	1	2	yes	—
365	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
366	C₂×C₄	C₂×#63254	1	1	yes	characteristic
367	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	68	2	no	—
373	C₈	—	68	2	no	—
374	C₈	—	68	2	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	68	2	no	—
378	C₂²	—	68	2	no	—
379	C₂²	—	68	2	no	—
380	C₂²	#11763021	1	2	yes	—
381	C₂²	C₂×#10587624	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	68	2	no	—
387	C₄	#11763021	1	2	yes	—
388	C₄	#11763069	1	2	yes	—
389	C₄	C₂×#10587636	1	1	yes	characteristic
390	C₄	C₂×#10587635	1	1	yes	characteristic
391	C₄	#11763622	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	136	2	no	—
394	C₂	C₂×#11763021	1	1	yes	characteristic
395	C₂	#14538435	1	1	yes	characteristic
396	C₂	C₂×#11763069	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	#19199
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. #14528356	C₂	semidirect
5. #14528232	C₂	semidirect
6. #14527532	C₂	semidirect
7. #14527465	C₂	semidirect
8. #14527456	C₂	semidirect
9. C₂×C₁₇×#487	C₂	semidirect
179. C₂×#487	D₁₇	semidirect
290. C₁₇	#19199	semidirect

Maximal subgroups

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

subgroup	index
#14539912	2
#14539905	2
#14537497	2
#14528356	2
#14528232	2
#14527532	2
#14527465	2
#14527456	2
C₂×C₁₇×#487	2
#19199	17