Subgroups — #17034694

Order 4352.

608 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	#14537211	C₂	1	1	yes	maximal, characteristic
2	#14528932	C₂	1	2	yes	maximal, complemented
3	#14528908	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528886	C₂	1	2	yes	maximal
5	#14528831	C₂	1	1	yes	maximal, characteristic
6	#14528827	C₂	1	1	yes	maximal, characteristic
7	#14528671	C₂	1	1	yes	maximal, characteristic, complemented
8	#14528670	C₂	1	1	yes	maximal, characteristic, complemented
9	#14528583	C₂	1	2	yes	maximal
10	#14528538	C₂	1	2	yes	maximal, complemented
11	C₁₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587593	C₂²	1	1	yes	characteristic
13	C₂×#10587587	C₂²	1	1	yes	characteristic
14	#11763277	C₂²	1	1	yes	characteristic
15	C₂×#10587490	C₂²	1	1	yes	characteristic
16	#11762847	C₂²	1	1	yes	characteristic
17	#11762791	C₂²	1	2	yes	—
18	#11762782	C₂²	1	2	yes	—
19	#11762767	C₂²	1	2	yes	—
20	#11762766	C₄	1	2	yes	complemented
21	#11762765	C₂²	1	2	yes	—
22	#11762765	C₂²	1	1	yes	characteristic
23	#11762763	C₄	1	2	yes	complemented
24	M₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
25	C₄×#63218	C₄	1	2	yes	complemented
26	C₂×#10587486	C₂²	1	1	yes	characteristic
27	#97×D₁₇	C₄	1	2	yes	complemented
28	#11762580	C₂²	1	2	yes	—
29	#11762570	C₂²	1	2	yes	—
30	#11762568	C₂²	1	2	yes	—
31	#11762567	C₂²	1	2	yes	—
32	#11762565	C₂²	1	2	yes	—
33	#11762565	C₂²	1	2	yes	—
34	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
35	C₁₇×#427	C₂²	1	1	yes	characteristic
36	C₁₇×#405	C₂²	1	1	yes	characteristic
37	C₁₇×#344	C₂²	1	2	yes	—
38	C₁₇×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#63252	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
41	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₁₇×#117	C₂³	1	2	yes	—
44	C₁₇×#117	C₂³	1	1	yes	characteristic
45	#10587612	C₂³	1	2	yes	—
46	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
47	C₂×#63223	C₂³	1	2	yes	—
48	C₂×#63223	C₂³	1	1	yes	characteristic
49	C₂²×#3622	C₂³	1	1	yes	characteristic
50	#10587593	—	2	2	no	—
51	#10587593	—	2	2	no	—
52	#10587587	—	2	1	no	—
53	#10587587	—	2	1	no	—
54	#10587587	—	2	1	no	—
55	#10587587	—	2	1	no	—
56	C₂×#63218	C₂×C₄	1	2	yes	—
57	C₂×#63218	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
59	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
60	#10587579	—	2	1	no	—
61	#10587573	—	2	1	no	—
62	#10587567	—	2	2	no	—
63	#10587558	C₂×C₄	1	2	yes	—
64	C₄²×D₁₇	C₂×C₄	1	2	yes	—
65	C₁₇×#108	—	2	1	no	—
66	C₁₇×#108	—	2	1	no	—
67	C₁₇×#107	—	2	1	no	—
68	C₁₇×#106	—	2	1	no	—
69	C₁₇×#97	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
71	#10587495	—	2	2	no	—
72	#10587494	—	2	2	no	—
73	#10587490	—	2	1	no	—
74	#10587490	—	2	1	no	—
75	#10587489	—	2	1	no	—
76	#10587488	—	2	1	no	—
77	#10587486	—	2	2	no	—
78	#10587486	C₂×C₄	1	1	yes	characteristic
79	#10587486	C₂×C₄	1	2	yes	—
80	#10587486	C₂×C₄	1	1	yes	characteristic
81	#10587486	C₂×C₄	1	1	yes	characteristic
82	#10587486	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₇	—	2	2	no	—
84	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
85	#10587477	—	2	4	no	—
86	#10587474	—	2	2	no	—
87	#10587474	C₂×C₄	1	2	yes	—
88	C₄×#3622	—	2	2	no	—
89	C₄×#3622	C₂×C₄	1	2	yes	—
90	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₇	#41	1	2	yes	—
92	C₂²×Dic₁₇	—	2	1	no	—
93	#63252	—	2	1	no	—
94	#63252	—	2	1	no	—
95	#63252	—	2	1	no	—
96	#63252	—	2	1	no	—
97	C₂×D₆₈	#41	1	1	yes	characteristic
98	C₂×C₄×D₁₇	—	2	2	no	—
99	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
100	C₂×Dic₃₄	#41	1	1	yes	characteristic
101	M₁₆×C₁₇	—	2	1	no	—
102	M₁₆×C₁₇	—	2	1	no	—
103	M₁₆×C₁₇	—	2	2	no	—
104	M₁₆×C₁₇	—	2	2	no	—
105	C₂×C₈×C₁₇	—	2	2	no	—
106	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
113	C₄⋊C₄×C₁₇	—	2	1	no	—
114	C₄⋊C₄×C₁₇	—	2	2	no	—
115	C₂²⋊C₄×C₁₇	—	2	1	no	—
116	C₂²⋊C₄×C₁₇	—	2	2	no	—
117	C₄²×C₁₇	—	2	2	no	—
118	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
119	C₂²⋊Dic₁₇	—	2	2	no	—
120	#63227	—	2	2	no	—
121	C₄⋊Dic₁₇	—	2	2	no	—
122	#63225	—	2	2	no	—
123	C₄×Dic₁₇	—	2	1	no	—
124	C₄×Dic₁₇	—	2	2	no	—
125	C₄×Dic₁₇	—	2	2	no	—
126	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
128	#63223	—	2	1	no	—
129	#63223	—	2	1	no	—
130	#63223	—	2	2	no	—
131	#63223	—	2	2	no	—
132	C₂×#3622	—	2	2	no	—
133	C₂×#3622	C₂²×C₄	1	2	yes	—
134	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
135	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
136	C₂×#3622	#41	1	2	yes	—
137	C₂×#3622	#41	1	2	yes	—
138	#63218	—	2	2	no	—
139	#63218	—	2	2	no	—
140	#63218	—	2	4	no	—
141	C₈×D₁₇	—	2	2	no	—
142	C₈×D₁₇	—	2	2	no	—
143	C₈×D₁₇	—	2	4	no	—
144	#20788	—	17	1	no	maximal, Sylow
145	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
146	C₂²×D₁₇	#117	1	2	yes	—
147	C₂×C₄×C₁₇	—	2	2	no	—
148	C₂×C₄×C₁₇	—	4	2	no	—
149	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₁₇	—	2	2	no	—
156	C₂²⋊D₁₇	—	2	2	no	—
157	C₂×Dic₁₇	—	2	1	no	—
158	C₂×Dic₁₇	—	2	1	no	—
159	C₂×Dic₁₇	—	4	2	no	—
160	C₂×Dic₁₇	—	8	1	no	—
161	C₂×Dic₁₇	#117	1	2	yes	—
162	D₆₈	—	2	1	no	—
163	D₆₈	—	2	1	no	—
164	C₄×D₁₇	—	2	2	no	—
165	C₄×D₁₇	—	2	2	no	—
166	C₄×D₁₇	—	8	2	no	—
167	Dic₃₄	—	2	1	no	—
168	Dic₃₄	—	2	1	no	—
169	C₈×C₁₇	—	2	2	no	—
170	C₈×C₁₇	—	4	2	no	—
171	C₈×C₁₇	#117	1	1	yes	characteristic
172	C₈×C₁₇	#117	1	1	yes	characteristic
173	C₈×C₁₇	#117	1	1	yes	characteristic
174	C₈×C₁₇	#117	1	1	yes	characteristic
175	#3622	—	2	2	no	—
176	#3622	—	4	2	no	—
177	#3622	#117	1	2	yes	—
178	#3622	#117	1	2	yes	—
179	#2875	—	17	1	no	—
180	#1841	—	17	2	no	—
181	#1833	—	17	1	no	—
182	#1833	D₁₇	1	1	yes	characteristic, complemented
183	#1827	—	17	2	no	—
184	#1813	—	17	1	no	—
185	#1812	—	17	1	no	—
186	#1773	—	17	1	no	—
187	#1773	—	17	1	no	—
188	#1757	—	17	2	no	—
189	#1747	—	17	2	no	—
190	C₂²×C₁₇	—	4	1	no	—
191	C₂²×C₁₇	C₂×#117	1	1	yes	characteristic
192	C₂²×C₁₇	#529	1	1	yes	characteristic
193	C₂²×C₁₇	#529	1	1	yes	characteristic
194	C₂×D₁₇	—	2	2	no	—
195	C₂×D₁₇	—	4	2	no	—
196	C₄×C₁₇	—	4	2	no	—
197	C₄×C₁₇	—	8	2	no	—
198	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
199	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
200	C₄×C₁₇	#529	1	1	yes	characteristic
201	C₄×C₁₇	#529	1	1	yes	characteristic
202	Dic₁₇	—	2	2	no	—
203	Dic₁₇	—	4	1	no	—
204	Dic₁₇	—	4	1	no	—
205	Dic₁₇	—	8	2	no	—
206	C₂×#131	—	17	1	no	—
207	C₂×#131	—	17	1	no	—
208	C₄×#41	—	17	1	no	—
209	C₂×#108	—	17	1	no	—
210	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
211	#427	—	17	1	no	—
212	#427	C₂×D₁₇	1	1	yes	characteristic
213	#413	—	17	2	no	—
214	#410	—	17	2	no	—
215	#405	—	17	1	no	—
216	#405	—	17	2	no	—
217	#405	—	17	2	no	—
218	#405	—	17	2	no	—
219	#405	—	17	2	no	—
220	#405	C₂×D₁₇	1	1	yes	characteristic
221	C₄×M₁₆	—	17	1	no	—
222	C₄×M₁₆	—	17	2	no	—
223	C₂×#97	—	17	1	no	—
224	C₂×#97	—	17	2	no	—
225	#344	—	17	2	no	—
226	#344	C₂×D₁₇	1	2	yes	—
227	#339	—	17	2	no	—
228	#339	—	17	2	no	—
229	#338	—	17	2	no	—
230	H(3,ℤ/4)	—	17	2	no	—
231	H(3,ℤ/4)	—	17	2	no	—
232	H(3,ℤ/4)	C₂×D₁₇	1	2	yes	—
233	C₂×C₁₇	—	2	1	no	—
234	C₂×C₁₇	—	2	1	no	—
235	C₂×C₁₇	—	2	1	no	—
236	C₂×C₁₇	#2283	1	1	yes	characteristic
237	D₁₇	—	4	2	no	—
238	C₂×#41	—	17	1	no	—
239	#131	—	34	2	no	—
240	#131	—	34	2	no	—
241	#131	—	34	1	no	—
242	#131	—	34	1	no	—
243	#131	—	34	1	no	—
244	#131	—	34	1	no	—
245	C₂×M₁₆	—	17	1	no	—
246	C₂×M₁₆	—	17	2	no	—
247	C₂×M₁₆	—	17	2	no	—
248	C₂×M₁₆	—	17	2	no	—
249	C₂×M₁₆	C₂²×D₁₇	1	2	yes	—
250	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
251	C₂²×C₈	—	17	1	no	—
252	C₂²×C₈	—	17	2	no	—
253	C₂²×C₈	—	17	2	no	—
254	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₄×Q₈	—	34	1	no	—
256	C₄×D₄	—	34	2	no	—
257	C₄×D₄	—	34	1	no	—
258	#117	—	17	2	no	—
259	#117	—	17	2	no	—
260	#117	C₂²×D₁₇	1	2	yes	—
261	#117	C₂²×D₁₇	1	1	yes	characteristic
262	C₂×C₄²	—	17	1	no	—
263	C₂×C₄²	—	17	2	no	—
264	#108	—	2	1	no	—
265	#108	—	2	1	no	—
266	#108	—	34	1	no	—
267	#108	—	34	1	no	—
268	#107	—	2	1	no	—
269	#107	—	34	1	no	—
270	#106	—	2	1	no	—
271	#106	—	34	1	no	—
272	#101	—	34	2	no	—
273	#100	—	34	2	no	—
274	#99	—	34	4	no	—
275	#97	—	17	1	no	—
276	#97	—	17	1	no	—
277	#97	—	17	2	no	—
278	#97	—	17	1	no	—
279	#97	—	17	1	no	—
280	#97	—	17	2	no	—
281	#97	—	34	2	no	—
282	#97	—	34	2	no	—
283	#97	C₄×D₁₇	1	2	yes	—
284	C₄×C₈	—	17	2	no	—
285	C₄×C₈	—	17	2	no	—
286	C₄×C₈	—	34	2	no	—
287	C₄×C₈	—	34	2	no	—
288	C₄×C₈	C₄×D₁₇	1	2	yes	—
289	C₁₇	#20788	1	1	yes	characteristic, complemented, Sylow
290	#41	—	34	1	no	—
291	#41	—	34	1	no	—
292	#41	—	34	1	no	—
293	#41	—	34	1	no	—
294	C₂×Q₈	—	17	1	no	—
295	C₂×D₄	—	17	2	no	—
296	C₂×D₄	—	17	1	no	—
297	C₂²×C₄	—	17	2	no	—
298	C₂²×C₄	—	34	2	no	—
299	C₂²×C₄	—	34	1	no	—
300	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	34	4	no	—
306	M₁₆	—	34	1	no	—
307	M₁₆	—	34	1	no	—
308	M₁₆	—	34	2	no	—
309	M₁₆	—	34	2	no	—
310	M₁₆	—	34	2	no	—
311	M₁₆	—	34	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	17	1	no	—
314	C₂×C₈	—	17	1	no	—
315	C₂×C₈	—	17	2	no	—
316	C₂×C₈	—	17	2	no	—
317	C₂×C₈	—	17	2	no	—
318	C₂×C₈	—	34	4	no	—
319	C₂×C₈	—	34	2	no	—
320	C₂×C₈	—	34	2	no	—
321	C₂×C₈	—	34	2	no	—
322	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
325	C₂×C₈	#63254	1	1	yes	characteristic
326	C₂×C₈	#63254	1	1	yes	characteristic
327	C₂×C₈	#63256	1	1	yes	characteristic
328	C₂×C₈	#63256	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	34	2	no	—
332	C₄⋊C₄	—	34	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	34	2	no	—
336	C₂²⋊C₄	—	34	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	17	1	no	—
339	C₄²	—	17	1	no	—
340	C₄²	—	34	2	no	—
341	C₄²	—	34	1	no	—
342	C₄²	—	34	2	no	—
343	C₄²	C₂×C₄×D₁₇	1	2	yes	—
344	C₂³	—	17	2	no	—
345	C₂³	#41×D₁₇	1	1	yes	characteristic
346	Q₈	—	34	1	no	—
347	Q₈	—	34	1	no	—
348	D₄	—	34	2	no	—
349	D₄	—	34	2	no	—
350	D₄	—	34	1	no	—
351	D₄	—	34	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	17	2	no	—
355	C₂×C₄	—	34	2	no	—
356	C₂×C₄	—	34	2	no	—
357	C₂×C₄	—	34	1	no	—
358	C₂×C₄	—	34	1	no	—
359	C₂×C₄	—	68	2	no	—
360	C₂×C₄	—	136	2	no	—
361	C₂×C₄	—	136	1	no	—
362	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
363	C₂×C₄	C₂×#63254	1	1	yes	characteristic
364	C₂×C₄	C₂×#63256	1	1	yes	characteristic
365	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
366	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
367	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	17	2	no	—
371	C₈	—	17	2	no	—
372	C₈	—	34	2	no	—
373	C₈	—	68	2	no	—
374	C₈	#10587578	1	1	yes	characteristic
375	C₈	#10587578	1	1	yes	characteristic
376	C₈	#10587578	1	1	yes	characteristic
377	C₈	#10587578	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	34	2	no	—
380	C₂²	—	68	2	no	—
381	C₂²	#117×D₁₇	1	1	yes	characteristic
382	C₂²	#11763338	1	1	yes	characteristic
383	C₂²	#11763344	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	34	2	no	—
387	C₄	—	68	1	no	—
388	C₄	—	68	1	no	—
389	C₄	—	136	2	no	—
390	C₄	#117×D₁₇	1	1	yes	characteristic
391	C₄	C₂×#10587578	1	1	yes	characteristic
392	C₄	#11763338	1	1	yes	characteristic
393	C₄	#11763344	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	68	2	no	—
398	C₂	#14531177	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#1833	C₂
socle	C₂×C₁₇	#2283
prime radical	C₂×#63252	C₂³
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#1833	D₁₇
hypocenter	C₁₇	#20788
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #14528932	C₂	semidirect
3. #14528908	C₂	semidirect
7. #14528671	C₂	semidirect
8. #14528670	C₂	semidirect
10. #14528538	C₂	semidirect
11. C₁₇×#1833	C₂	semidirect
20. #11762766	C₄	semidirect
23. #11762763	C₄	semidirect
25. C₄×#63218	C₄	semidirect
27. #97×D₁₇	C₄	semidirect
182. #1833	D₁₇	semidirect
289. C₁₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#14537211	2
#14528932	2
#14528908	2
#14528886	2
#14528831	2
#14528827	2
#14528671	2
#14528670	2
#14528583	2
#14528538	2
C₁₇×#1833	2
#20788	17