Subgroups — #17978467

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545152	C₂	1	2	yes	maximal, complemented
2	#14545151	C₂	1	2	yes	maximal, complemented
3	#14545150	C₂	1	2	yes	maximal, complemented
4	#14545149	C₂	1	2	yes	maximal, complemented
5	#14545117	C₂	1	2	yes	maximal, complemented
6	#14545012	C₂	1	2	yes	maximal
7	#14544991	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10587530	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₇×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587530	C₂²	1	2	yes	—
11	C₂×#10587530	C₂²	1	2	yes	—
12	#11763730	C₂²	1	4	yes	complemented
13	C₄⋊#63216	C₂²	1	4	yes	complemented
14	#11763704	C₂²	1	4	yes	—
15	C₂²⋊#63216	C₂²	1	4	yes	—
16	#11763698	C₂²	1	2	yes	complemented
17	#11763697	C₂²	1	2	yes	complemented
18	#11763695	C₂²	1	2	yes	complemented
19	C₄×#63216	C₂²	1	2	yes	complemented
20	C₂×M₁₆×D₁₇	C₄	1	1	yes	characteristic
21	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	#117×D₁₇	C₄	1	1	yes	characteristic
23	#11762767	C₂²	1	1	yes	characteristic
24	#11762766	C₂²	1	1	yes	characteristic
25	#11762766	C₂²	1	1	yes	characteristic
26	#11762765	C₂²	1	1	yes	characteristic
27	#11762764	C₄	1	1	yes	characteristic
28	#11762763	C₄	1	1	yes	characteristic
29	#11762763	C₄	1	1	yes	characteristic
30	#97×D₁₇	C₄	1	1	yes	characteristic
31	#97×D₁₇	C₄	1	1	yes	characteristic
32	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
34	C₁₇×#405	C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
39	#10587612	C₂×C₄	1	1	yes	characteristic
40	C₂×#63223	C₂×C₄	1	1	yes	characteristic
41	C₂²×#3622	C₂³	1	1	yes	characteristic
42	M₁₆×D₁₇	—	2	4	no	—
43	C₂×#63218	C₂×C₄	1	1	yes	characteristic
44	C₂×#63218	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₇	—	2	4	no	—
46	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
52	#10587578	C₈	1	2	yes	—
53	C₄⋊C₄×D₁₇	C₈	1	2	yes	—
54	C₂²⋊C₄×D₁₇	C₈	1	2	yes	—
55	#10587563	C₈	1	2	yes	—
56	#10587558	C₂×C₄	1	1	yes	characteristic
57	#10587558	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×#63216	C₂³	1	4	yes	—
61	C₂×#63216	C₂³	1	4	yes	—
62	#10587530	—	2	4	no	—
63	#10587530	—	2	4	no	—
64	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
65	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
68	#10587486	C₂×C₄	1	1	yes	characteristic
69	#10587486	C₂×C₄	1	1	yes	characteristic
70	#10587486	C₂×C₄	1	1	yes	characteristic
71	#10587486	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
75	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
76	#10587474	C₂×C₄	1	1	yes	characteristic
77	#10587474	C₂×C₄	1	1	yes	characteristic
78	C₄×#3622	C₂³	1	1	yes	characteristic
79	C₄×#3622	C₂³	1	1	yes	characteristic
80	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₇	—	2	2	no	—
84	C₂×C₄×D₁₇	—	2	2	no	—
85	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
91	M₁₆×C₁₇	—	2	2	no	—
92	C₂×C₈×C₁₇	—	2	2	no	—
93	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
99	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₇	C₂×C₈	1	2	yes	—
102	#63227	C₂×C₈	1	2	yes	—
103	#63227	C₂×C₈	1	2	yes	—
104	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
105	#63225	C₂×C₈	1	2	yes	—
106	#63225	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
109	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
111	#63223	—	2	2	no	—
112	C₂×#3622	—	2	2	no	—
113	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
114	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
115	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
116	C₂×#3622	C₂⁴	1	1	yes	characteristic
117	#63218	—	2	2	no	—
118	#63218	—	2	2	no	—
119	C₈×D₁₇	—	2	2	no	—
120	C₈×D₁₇	—	2	2	no	—
121	C₈×D₁₇	—	2	2	no	—
122	C₈×D₁₇	—	2	2	no	—
123	C₈×D₁₇	M₁₆	1	4	yes	—
124	C₈×D₁₇	#41	1	4	yes	—
125	#63216	—	2	4	no	—
126	#63216	—	2	4	no	—
127	#33419	—	17	1	no	maximal, Sylow
128	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
129	C₂²×D₁₇	—	2	2	no	—
130	C₂²×D₁₇	—	4	2	no	—
131	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
132	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	—	2	1	no	—
134	C₂×C₄×C₁₇	—	2	1	no	—
135	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₇	—	2	2	no	—
140	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
141	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
142	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
144	C₄×D₁₇	—	2	2	no	—
145	C₄×D₁₇	—	2	1	no	—
146	C₄×D₁₇	—	2	1	no	—
147	C₄×D₁₇	—	2	2	no	—
148	C₄×D₁₇	—	4	2	no	—
149	C₄×D₁₇	—	4	2	no	—
150	C₄×D₁₇	#117	1	2	yes	—
151	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
152	C₈×C₁₇	—	2	1	no	—
153	C₈×C₁₇	—	2	1	no	—
154	C₈×C₁₇	#117	1	2	yes	—
155	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
156	#3622	—	2	1	no	—
157	#3622	—	2	1	no	—
158	#3622	C₂×M₁₆	1	2	yes	—
159	#3622	C₂×#41	1	2	yes	—
160	C₂×#405	—	17	1	no	—
161	#2164	—	17	2	no	—
162	#2164	—	17	2	no	—
163	#2164	—	17	2	no	—
164	#2164	—	17	2	no	—
165	#2150	—	17	2	no	—
166	#2112	—	17	2	no	—
167	C₄×M₃₂	—	17	1	no	—
168	C₄×M₃₂	—	17	1	no	—
169	C₂²×C₁₇	—	2	2	no	—
170	C₂²×C₁₇	—	2	1	no	—
171	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
172	C₂×D₁₇	—	2	2	no	—
173	C₂×D₁₇	—	2	2	no	—
174	C₂×D₁₇	—	2	1	no	—
175	C₂×D₁₇	—	4	4	no	—
176	C₂×D₁₇	#431	1	2	yes	—
177	C₄×C₁₇	—	2	2	no	—
178	C₄×C₁₇	—	2	2	no	—
179	C₄×C₁₇	—	2	1	no	—
180	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
181	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
182	Dic₁₇	—	2	2	no	—
183	Dic₁₇	—	2	2	no	—
184	Dic₁₇	—	2	1	no	—
185	Dic₁₇	#431	1	2	yes	—
186	C₂²×M₁₆	—	17	1	no	—
187	C₂³×C₈	—	17	1	no	—
188	C₂×#117	—	17	1	no	—
189	C₂×M₃₂	—	17	2	no	—
190	C₂×M₃₂	—	17	2	no	—
191	#405	—	17	1	no	—
192	#405	—	17	1	no	—
193	#405	—	17	1	no	—
194	#405	—	17	1	no	—
195	#405	—	17	1	no	—
196	#405	—	17	1	no	—
197	#405	—	17	1	no	—
198	#405	#595	1	1	yes	characteristic
199	C₂×#97	—	17	1	no	—
200	C₂×#97	—	17	1	no	—
201	C₂×C₄×C₈	—	17	1	no	—
202	C₂×C₄×C₈	—	17	1	no	—
203	C₄⋊C₁₆	—	17	4	no	—
204	C₄⋊C₁₆	—	17	4	no	—
205	C₂²⋊C₁₆	—	17	4	no	—
206	C₂²⋊C₁₆	—	17	4	no	—
207	#346	—	17	2	no	—
208	#346	—	17	2	no	—
209	C₄×C₁₆	—	17	2	no	—
210	C₄×C₁₆	—	17	2	no	—
211	C₂×C₁₇	—	4	1	no	—
212	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
213	C₂×C₁₇	#3407	1	2	yes	—
214	D₁₇	—	2	2	no	—
215	D₁₇	—	4	1	no	—
216	C₂³×C₄	—	17	1	no	—
217	C₂×M₁₆	—	17	1	no	—
218	C₂×M₁₆	—	17	1	no	—
219	C₂×M₁₆	—	17	1	no	—
220	C₂×M₁₆	—	34	4	no	—
221	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
222	C₂²×C₈	—	17	1	no	—
223	C₂²×C₈	—	17	1	no	—
224	C₂²×C₈	—	17	1	no	—
225	C₂²×C₈	—	17	1	no	—
226	C₂²×C₈	—	17	1	no	—
227	C₂²×C₈	—	17	1	no	—
228	C₂²×C₈	—	17	1	no	—
229	C₂²×C₈	—	34	4	no	—
230	C₂²×C₈	C₂×#595	1	1	yes	characteristic
231	#117	—	17	1	no	—
232	#117	—	17	2	no	—
233	#117	—	17	2	no	—
234	#117	—	17	1	no	—
235	#117	—	17	1	no	—
236	#117	C₂×#595	1	1	yes	characteristic
237	C₂×C₄⋊C₄	—	17	2	no	—
238	C₂×C₂²⋊C₄	—	17	2	no	—
239	C₂×C₄²	—	17	1	no	—
240	C₂×C₄²	—	17	1	no	—
241	M₃₂	—	34	4	no	—
242	M₃₂	—	34	4	no	—
243	C₂×C₁₆	—	17	4	no	—
244	C₂×C₁₆	—	17	4	no	—
245	#97	—	17	1	no	—
246	#97	—	17	1	no	—
247	#97	—	17	1	no	—
248	#97	—	17	1	no	—
249	#97	—	17	1	no	—
250	#97	—	17	1	no	—
251	#97	C₂×#595	1	1	yes	characteristic
252	#97	C₂×#595	1	1	yes	characteristic
253	C₄×C₈	—	17	1	no	—
254	C₄×C₈	—	17	1	no	—
255	C₄×C₈	—	17	1	no	—
256	C₄×C₈	—	17	1	no	—
257	C₄×C₈	—	17	1	no	—
258	C₄×C₈	—	17	1	no	—
259	C₄×C₈	C₂×#595	1	1	yes	characteristic
260	C₄×C₈	C₂×#595	1	1	yes	characteristic
261	C₁₇	#33419	1	1	yes	characteristic, complemented, Sylow
262	C₂⁴	—	17	1	no	—
263	C₂²×C₄	—	17	1	no	—
264	C₂²×C₄	—	17	1	no	—
265	C₂²×C₄	—	17	1	no	—
266	C₂²×C₄	—	17	1	no	—
267	C₂²×C₄	—	17	1	no	—
268	C₂²×C₄	—	17	2	no	—
269	C₂²×C₄	—	17	2	no	—
270	C₂²×C₄	—	34	2	no	—
271	C₂²×C₄	—	34	2	no	—
272	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	34	2	no	—
275	M₁₆	—	34	2	no	—
276	M₁₆	—	34	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	17	1	no	—
279	C₂×C₈	—	17	1	no	—
280	C₂×C₈	—	17	4	no	—
281	C₂×C₈	—	17	4	no	—
282	C₂×C₈	—	17	1	no	—
283	C₂×C₈	—	17	1	no	—
284	C₂×C₈	—	34	2	no	—
285	C₂×C₈	—	34	2	no	—
286	C₂×C₈	—	34	2	no	—
287	C₂×C₈	—	34	2	no	—
288	C₂×C₈	—	34	2	no	—
289	C₂×C₈	C₂²×#595	1	1	yes	characteristic
290	C₂×C₈	C₂²×#595	1	1	yes	characteristic
291	C₂×C₈	C₂²×#595	1	1	yes	characteristic
292	C₂×C₈	C₂²×#595	1	1	yes	characteristic
293	C₄⋊C₄	—	17	2	no	—
294	C₄⋊C₄	—	17	2	no	—
295	C₄⋊C₄	—	17	2	no	—
296	C₄⋊C₄	#63242	1	2	yes	—
297	C₂²⋊C₄	—	17	2	no	—
298	C₂²⋊C₄	—	17	2	no	—
299	C₂²⋊C₄	—	17	2	no	—
300	C₂²⋊C₄	#63242	1	2	yes	—
301	C₄²	—	17	2	no	—
302	C₄²	—	17	2	no	—
303	C₄²	—	17	1	no	—
304	C₄²	—	17	1	no	—
305	C₄²	C₂²×#595	1	1	yes	characteristic
306	C₄²	C₂²×#595	1	1	yes	characteristic
307	C₁₆	—	34	4	no	—
308	C₁₆	—	34	4	no	—
309	C₂³	—	17	1	no	—
310	C₂³	—	17	1	no	—
311	C₂³	—	34	2	no	—
312	C₂³	—	68	2	no	—
313	C₂³	C₂×#63242	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	17	1	no	—
317	C₂×C₄	—	17	2	no	—
318	C₂×C₄	—	17	2	no	—
319	C₂×C₄	—	17	2	no	—
320	C₂×C₄	—	17	2	no	—
321	C₂×C₄	—	17	1	no	—
322	C₂×C₄	—	34	2	no	—
323	C₂×C₄	—	34	2	no	—
324	C₂×C₄	—	34	2	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	C₂×#63242	1	2	yes	—
330	C₂×C₄	C₂×#63242	1	2	yes	—
331	C₂×C₄	C₂×#63242	1	1	yes	characteristic
332	C₂×C₄	C₂³×#595	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	17	2	no	—
336	C₈	—	17	2	no	—
337	C₈	—	34	1	no	—
338	C₈	—	34	1	no	—
339	C₈	D₁₇⋊M₁₆	1	2	yes	—
340	C₈	#10587677	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	17	2	no	—
344	C₂²	—	34	2	no	—
345	C₂²	—	34	1	no	—
346	C₂²	—	34	2	no	—
347	C₂²	—	68	4	no	—
348	C₂²	C₂²×#63242	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	17	2	no	—
353	C₄	—	34	1	no	—
354	C₄	—	34	2	no	—
355	C₄	—	34	2	no	—
356	C₄	C₂×#10587677	1	1	yes	characteristic
357	C₄	C₂×D₁₇⋊M₁₆	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	34	2	no	—
360	C₂	—	68	1	no	—
361	C₂	#14545448	1	1	yes	characteristic
362	C₂	#14545752	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#405	C₄
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#405	#595
hypocenter	C₁₇	#33419
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #14545152	C₂	semidirect
2. #14545151	C₂	semidirect
3. #14545150	C₂	semidirect
4. #14545149	C₂	semidirect
5. #14545117	C₂	semidirect
7. #14544991	C₂	semidirect
8. C₄×#10587530	C₂	semidirect
12. #11763730	C₂²	semidirect
13. C₄⋊#63216	C₂²	semidirect
16. #11763698	C₂²	semidirect
17. #11763697	C₂²	semidirect
18. #11763695	C₂²	semidirect
19. C₄×#63216	C₂²	semidirect
261. C₁₇	#33419	semidirect

Maximal subgroups

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

subgroup	index
#14545152	2
#14545151	2
#14545150	2
#14545149	2
#14545117	2
#14545012	2
#14544991	2
C₄×#10587530	2
D₁₇×#405	2
#33419	17