Subgroups — #17187419

Order 4352.

582 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587587	C₂	1	1	yes	maximal, characteristic
2	#14530590	C₂	1	2	yes	maximal, complemented
3	#14530588	C₂	1	2	yes	maximal, complemented
4	#14530587	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530585	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530585	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530541	C₂	1	2	yes	maximal, complemented
8	#14530396	C₂	1	2	yes	maximal
9	#14530377	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587587	C₄	1	1	yes	characteristic
13	C₂×#10587587	C₂²	1	1	yes	characteristic
14	C₄×#63252	C₄	1	1	yes	characteristic
15	C₂×#10587469	C₂²	1	2	yes	—
16	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
17	#11762764	C₄	1	1	yes	characteristic
18	#11762764	C₂²	1	1	yes	characteristic
19	#11762763	C₄	1	1	yes	characteristic
20	#11762763	C₄	1	1	yes	characteristic
21	#11762763	C₂²	1	1	yes	characteristic
22	#11762763	C₂²	1	1	yes	characteristic
23	C₄×#63223	C₄	1	1	yes	characteristic
24	C₄×#63218	C₄	1	1	yes	characteristic
25	C₄×#63218	C₄	1	1	yes	characteristic
26	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	C₄⋊#63214	C₂²	1	2	yes	—
29	#11762627	C₂²	1	2	yes	—
30	#11762627	C₂²	1	2	yes	—
31	C₂²⋊#63214	C₂²	1	2	yes	—
32	#11762588	C₂²	1	2	yes	—
33	#11762588	C₂²	1	2	yes	—
34	#11762585	C₂²	1	1	yes	characteristic
35	#11762585	C₂²	1	1	yes	characteristic
36	#11762584	C₂²	1	2	yes	—
37	C₁₆×Dic₁₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
40	C₁₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₁₇×#346	C₂²	1	1	yes	characteristic
42	C₁₇×#346	C₂²	1	1	yes	characteristic
43	C₂×#63252	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₁₇	C₈	1	2	yes	—
48	C₄×C₂²⋊D₁₇	C₈	1	2	yes	—
49	#10587612	C₂×C₄	1	1	yes	characteristic
50	C₂×#63223	C₂×C₄	1	1	yes	characteristic
51	C₂×#63223	C₂×C₄	1	1	yes	characteristic
52	#10587587	—	2	4	no	—
53	#10587587	—	2	4	no	—
54	C₂×#63218	C₂×C₄	1	1	yes	characteristic
55	C₂×#63218	C₂×C₄	1	1	yes	characteristic
56	C₂×#63218	C₂×C₄	1	1	yes	characteristic
57	C₂×#63218	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
62	C₄×D₆₈	C₈	1	2	yes	—
63	#10587558	C₂×C₄	1	1	yes	characteristic
64	#10587558	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₃₄	C₈	1	2	yes	—
68	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
70	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	#10587486	C₂×C₄	1	1	yes	characteristic
75	#10587486	C₂×C₄	1	1	yes	characteristic
76	#10587486	C₂×C₄	1	1	yes	characteristic
77	#10587486	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
81	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
82	C₂×#63214	C₂³	1	2	yes	—
83	C₂×#63214	C₂³	1	2	yes	—
84	#10587474	C₂×C₄	1	1	yes	characteristic
85	#10587474	C₂×C₄	1	1	yes	characteristic
86	C₄×#3622	C₂×C₄	1	1	yes	characteristic
87	C₄×#3622	C₂×C₄	1	1	yes	characteristic
88	#10587469	—	2	4	no	—
89	C₁₆×D₁₇	—	2	4	no	—
90	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
91	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₇	C₂×C₈	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₇	C₂×C₈	1	1	yes	characteristic
94	#63252	—	2	2	no	—
95	#63252	—	2	2	no	—
96	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
102	C₂×C₈×C₁₇	—	2	2	no	—
103	C₂×C₈×C₁₇	—	2	2	no	—
104	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
108	C₄²×C₁₇	C₂×C₈	1	2	yes	—
109	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²⋊Dic₁₇	C₂×C₈	1	2	yes	—
112	#63227	C₂×C₈	1	2	yes	—
113	#63227	C₂×C₈	1	2	yes	—
114	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
115	#63225	C₂×C₈	1	2	yes	—
116	#63225	C₂×C₈	1	2	yes	—
117	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
118	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
119	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
121	#63223	—	2	2	no	—
122	#63223	—	2	2	no	—
123	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
124	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
125	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
126	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
127	#63218	—	2	2	no	—
128	#63218	—	2	2	no	—
129	#63218	—	2	2	no	—
130	#63218	—	2	2	no	—
131	C₈×D₁₇	—	2	2	no	—
132	C₈×D₁₇	—	2	2	no	—
133	C₈×D₁₇	M₁₆	1	4	yes	—
134	C₈×D₁₇	#41	1	4	yes	—
135	C₁₆×C₁₇	—	2	2	no	—
136	C₁₆×C₁₇	—	2	2	no	—
137	#63214	—	2	2	no	—
138	#63214	—	2	2	no	—
139	#33421	—	17	1	no	maximal, Sylow
140	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
141	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
142	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₁₇	—	2	1	no	—
144	C₂×C₄×C₁₇	—	2	1	no	—
145	C₂×C₄×C₁₇	—	4	2	no	—
146	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
150	C₂²⋊D₁₇	—	2	2	no	—
151	C₂²⋊D₁₇	—	2	2	no	—
152	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
153	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
154	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
155	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
156	D₆₈	—	2	2	no	—
157	C₄×D₁₇	—	2	1	no	—
158	C₄×D₁₇	—	2	1	no	—
159	C₄×D₁₇	—	4	2	no	—
160	C₄×D₁₇	—	4	2	no	—
161	C₄×D₁₇	#117	1	2	yes	—
162	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
163	Dic₃₄	—	2	2	no	—
164	C₈×C₁₇	—	2	1	no	—
165	C₈×C₁₇	—	2	1	no	—
166	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
167	C₈×C₁₇	C₂×#41	1	2	yes	—
168	#3622	—	2	1	no	—
169	#3622	—	2	1	no	—
170	#3622	#117	1	2	yes	—
171	#3622	C₂×M₁₆	1	2	yes	—
172	C₄×#131	—	17	1	no	—
173	#2164	—	17	2	no	—
174	#2164	—	17	2	no	—
175	#2163	—	17	1	no	—
176	#2163	—	17	1	no	—
177	#2163	—	17	1	no	—
178	#2163	D₁₇	1	1	yes	characteristic, complemented
179	#2151	—	17	2	no	—
180	#2113	—	17	2	no	—
181	C₄×M₃₂	—	17	1	no	—
182	C₂×#346	—	17	1	no	—
183	C₂²×C₁₇	—	2	2	no	—
184	C₂²×C₁₇	—	2	1	no	—
185	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
186	C₂×D₁₇	—	2	2	no	—
187	C₂×D₁₇	—	2	2	no	—
188	C₂×D₁₇	—	2	1	no	—
189	C₂×D₁₇	#431	1	2	yes	—
190	C₄×C₁₇	—	2	2	no	—
191	C₄×C₁₇	—	2	2	no	—
192	C₄×C₁₇	—	2	1	no	—
193	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
194	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
195	Dic₁₇	—	2	2	no	—
196	Dic₁₇	—	2	2	no	—
197	Dic₁₇	—	2	1	no	—
198	Dic₁₇	#431	1	2	yes	—
199	C₂×#131	—	17	1	no	—
200	C₂×#131	—	17	1	no	—
201	C₄×#41	—	17	1	no	—
202	C₂×M₃₂	—	17	2	no	—
203	C₂²×C₁₆	—	17	2	no	—
204	#405	—	17	1	no	—
205	#405	—	17	1	no	—
206	#405	—	17	1	no	—
207	#405	—	17	1	no	—
208	#405	—	17	1	no	—
209	#405	—	17	1	no	—
210	C₄×M₁₆	—	17	1	no	—
211	C₄×M₁₆	—	17	1	no	—
212	C₄×M₁₆	—	17	1	no	—
213	C₂×C₄×C₈	—	17	1	no	—
214	C₂×C₄×C₈	—	17	1	no	—
215	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
216	C₄⋊C₁₆	—	17	2	no	—
217	C₄⋊C₁₆	—	17	2	no	—
218	C₄⋊C₁₆	—	17	2	no	—
219	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
220	C₂²⋊C₁₆	—	17	2	no	—
221	C₂²⋊C₁₆	—	17	2	no	—
222	C₂²⋊C₁₆	—	17	2	no	—
223	C₂²⋊C₁₆	C₂×D₁₇	1	2	yes	—
224	#346	—	17	2	no	—
225	#346	—	17	1	no	—
226	#346	—	17	1	no	—
227	#346	C₂×D₁₇	1	1	yes	characteristic
228	#346	C₂×D₁₇	1	1	yes	characteristic
229	C₄×C₁₆	—	17	2	no	—
230	C₂×C₁₇	—	4	1	no	—
231	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
232	C₂×C₁₇	#3407	1	2	yes	—
233	D₁₇	—	2	2	no	—
234	D₁₇	—	4	1	no	—
235	C₂×#41	—	17	1	no	—
236	#131	—	34	4	no	—
237	#131	—	34	4	no	—
238	C₂×M₁₆	—	17	1	no	—
239	C₂×M₁₆	—	17	1	no	—
240	C₂×M₁₆	—	17	1	no	—
241	C₂×M₁₆	—	17	1	no	—
242	C₂×M₁₆	—	17	1	no	—
243	C₂×M₁₆	—	17	1	no	—
244	C₂²×C₈	—	17	1	no	—
245	C₂²×C₈	—	17	1	no	—
246	C₂²×C₈	—	17	1	no	—
247	C₂²×C₈	—	17	1	no	—
248	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₄×Q₈	—	17	2	no	—
251	C₄×D₄	—	17	2	no	—
252	C₄×D₄	—	17	2	no	—
253	C₄×D₄	—	17	2	no	—
254	#117	—	17	1	no	—
255	#117	—	17	1	no	—
256	#117	—	17	1	no	—
257	C₂×C₄²	—	17	1	no	—
258	C₂×C₄²	—	17	1	no	—
259	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
260	M₃₂	—	34	4	no	—
261	C₂×C₁₆	—	17	2	no	—
262	C₂×C₁₆	—	17	2	no	—
263	C₂×C₁₆	—	34	4	no	—
264	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
265	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
266	#97	—	17	1	no	—
267	#97	—	17	1	no	—
268	#97	—	17	1	no	—
269	#97	—	17	1	no	—
270	#97	—	17	1	no	—
271	#97	—	17	1	no	—
272	C₄×C₈	—	17	1	no	—
273	C₄×C₈	—	17	1	no	—
274	C₄×C₈	—	17	1	no	—
275	C₄×C₈	—	17	1	no	—
276	C₄×C₈	—	17	1	no	—
277	C₄×C₈	—	17	1	no	—
278	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
279	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
282	C₁₇	#33421	1	1	yes	characteristic, complemented, Sylow
283	#41	—	34	2	no	—
284	#41	—	34	2	no	—
285	C₂×Q₈	—	17	1	no	—
286	C₂×D₄	—	17	1	no	—
287	C₂×D₄	—	17	1	no	—
288	C₂×D₄	—	17	1	no	—
289	C₂²×C₄	—	17	1	no	—
290	C₂²×C₄	—	17	1	no	—
291	C₂²×C₄	—	17	2	no	—
292	C₂²×C₄	—	17	2	no	—
293	C₂²×C₄	C₈×D₁₇	1	2	yes	—
294	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
295	M₁₆	—	34	2	no	—
296	M₁₆	—	34	2	no	—
297	M₁₆	—	34	2	no	—
298	M₁₆	—	34	2	no	—
299	M₁₆	—	34	2	no	—
300	M₁₆	—	34	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	17	1	no	—
304	C₂×C₈	—	17	1	no	—
305	C₂×C₈	—	17	4	no	—
306	C₂×C₈	—	17	4	no	—
307	C₂×C₈	—	17	1	no	—
308	C₂×C₈	—	17	1	no	—
309	C₂×C₈	—	34	2	no	—
310	C₂×C₈	—	34	2	no	—
311	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
315	C₄⋊C₄	—	17	2	no	—
316	C₄⋊C₄	—	17	2	no	—
317	C₄⋊C₄	—	17	2	no	—
318	C₂²⋊C₄	—	17	2	no	—
319	C₂²⋊C₄	—	17	2	no	—
320	C₂²⋊C₄	—	17	2	no	—
321	C₄²	—	17	1	no	—
322	C₄²	—	17	1	no	—
323	C₄²	—	17	2	no	—
324	C₄²	—	17	2	no	—
325	C₄²	C₈×D₁₇	1	2	yes	—
326	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
328	C₁₆	—	2	2	no	—
329	C₁₆	—	2	2	no	—
330	C₁₆	—	34	2	no	—
331	C₁₆	—	34	2	no	—
332	C₂³	—	17	1	no	—
333	C₂³	—	17	1	no	—
334	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
335	Q₈	—	34	2	no	—
336	D₄	—	34	2	no	—
337	D₄	—	34	2	no	—
338	D₄	—	34	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	17	1	no	—
343	C₂×C₄	—	17	2	no	—
344	C₂×C₄	—	17	2	no	—
345	C₂×C₄	—	17	2	no	—
346	C₂×C₄	—	17	2	no	—
347	C₂×C₄	—	17	1	no	—
348	C₂×C₄	—	34	1	no	—
349	C₂×C₄	—	34	1	no	—
350	C₂×C₄	—	68	2	no	—
351	C₂×C₄	—	68	2	no	—
352	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
353	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
354	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	17	2	no	—
359	C₈	—	17	2	no	—
360	C₈	—	34	1	no	—
361	C₈	—	34	1	no	—
362	C₈	M₁₆×D₁₇	1	2	yes	—
363	C₈	#41×D₁₇	1	2	yes	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	17	2	no	—
367	C₂²	—	34	2	no	—
368	C₂²	—	34	2	no	—
369	C₂²	—	34	1	no	—
370	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	1	no	—
374	C₄	—	17	2	no	—
375	C₄	—	34	2	no	—
376	C₄	—	34	2	no	—
377	C₄	—	34	1	no	—
378	C₄	#117×D₁₇	1	1	yes	characteristic
379	C₄	C₂×M₁₆×D₁₇	1	1	yes	characteristic
380	C₂	—	4	1	no	—
381	C₂	—	34	2	no	—
382	C₂	—	68	1	no	—
383	C₂	D₁₇×#431	1	1	yes	characteristic
384	C₂	#14541886	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#2163	C₂
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2163	D₁₇
hypocenter	C₁₇	#33421
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #14530590	C₂	semidirect
3. #14530588	C₂	semidirect
4. #14530587	C₂	semidirect
5. #14530585	C₂	semidirect
6. #14530585	C₂	semidirect
7. #14530541	C₂	semidirect
9. #14530377	C₂	semidirect
10. D₁₇×#346	C₂	semidirect
11. C₁₇×#2163	C₂	semidirect
178. #2163	D₁₇	semidirect
282. C₁₇	#33421	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587587	2
#14530590	2
#14530588	2
#14530587	2
#14530585	2
#14530541	2
#14530396	2
#14530377	2
D₁₇×#346	2
C₁₇×#2163	2
#33421	17