Subgroups — #16950782

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762588	C₂	1	1	yes	maximal, characteristic
4	C₈×#63227	C₂	1	1	yes	maximal, characteristic
5	#14526862	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526861	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₁₇	—	2	2	no	—
12	C₂×C₁₆×D₁₇	—	2	2	no	—
13	C₂×C₁₆×D₁₇	—	2	2	no	—
14	C₂×#10587491	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₄×#63227	C₄	1	1	yes	characteristic
17	#11762627	—	2	2	no	—
18	#11762588	—	2	2	no	—
19	#11762564	C₄	1	1	yes	characteristic
20	#11762563	C₄	1	1	yes	characteristic
21	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×C₄⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#63227	C₈	1	1	yes	characteristic
30	C₂×#63227	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₇	—	2	1	no	—
35	C₂×C₈×D₁₇	—	2	1	no	—
36	C₂×C₈×D₁₇	—	2	1	no	—
37	C₂×C₈×D₁₇	—	2	1	no	—
38	C₂×C₈×D₁₇	D₄	1	2	yes	—
39	C₂×C₈×D₁₇	Q₈	1	2	yes	—
40	C₂×C₁₆×C₁₇	—	2	1	no	—
41	C₂×C₁₆×C₁₇	—	2	1	no	—
42	C₂×C₁₆×C₁₇	—	2	1	no	—
43	C₂×C₁₆×C₁₇	—	2	1	no	—
44	C₂×C₁₆×C₁₇	—	2	1	no	—
45	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
46	C₄×C₈×C₁₇	—	2	1	no	—
47	C₄×C₈×C₁₇	—	2	1	no	—
48	#10587502	C₈	1	1	yes	characteristic
49	#10587502	C₈	1	1	yes	characteristic
50	#10587491	—	2	2	no	—
51	C₈×Dic₁₇	—	2	1	no	—
52	C₈×Dic₁₇	—	2	1	no	—
53	C₂×#63214	—	2	1	no	—
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	—	2	1	no	—
56	C₂×#63214	—	2	1	no	—
57	C₂×#63214	—	2	1	no	—
58	C₂×#63214	D₄	1	2	yes	—
59	C₁₆×D₁₇	—	2	8	no	—
60	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₁₇	M₁₆	1	2	yes	—
73	C₂×C₈×C₁₇	—	2	1	no	—
74	C₂×C₈×C₁₇	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	1	no	—
76	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
82	C₄²×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	—	2	1	no	—
84	#63227	—	2	2	no	—
85	#63227	C₁₆	1	4	yes	complemented
86	C₄×Dic₁₇	—	2	1	no	—
87	C₄×Dic₁₇	—	2	1	no	—
88	C₂×#3622	—	2	1	no	—
89	C₂×#3622	—	2	1	no	—
90	C₂×#3622	—	2	1	no	—
91	C₂×#3622	—	2	1	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	C₄⋊C₄	1	2	yes	—
94	C₈×D₁₇	—	2	2	no	—
95	C₈×D₁₇	—	2	2	no	—
96	C₈×D₁₇	—	2	2	no	—
97	C₈×D₁₇	—	2	2	no	—
98	C₁₆×C₁₇	—	2	2	no	—
99	C₁₆×C₁₇	—	4	1	no	—
100	#63214	—	2	2	no	—
101	#63214	—	4	1	no	—
102	#12912	—	17	1	no	maximal, Sylow
103	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₁₇	—	2	1	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	C₂²×D₁₇	C₄⋊C₈	1	2	yes	—
109	C₂²×D₁₇	C₂×C₁₆	1	2	yes	—
110	C₂²×D₁₇	M₃₂	1	2	yes	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	complemented
117	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
131	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	complemented
132	C₄×D₁₇	—	2	2	no	—
133	C₄×D₁₇	—	2	2	no	—
134	C₄×D₁₇	—	2	2	no	—
135	C₄×D₁₇	—	2	2	no	—
136	C₈×C₁₇	—	4	1	no	—
137	C₈×C₁₇	#120	1	1	yes	characteristic
138	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₈×C₁₇	#123	1	1	yes	characteristic
141	#3622	—	2	2	no	—
142	#3622	—	4	1	no	—
143	C₂³×C₁₆	—	17	1	no	—
144	C₂×C₄⋊C₁₆	—	17	1	no	—
145	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic, complemented
146	C₂×C₂²⋊C₁₆	—	17	1	no	—
147	C₈×C₂²⋊C₄	—	17	1	no	—
148	#1374	—	17	1	no	—
149	#1374	—	17	1	no	—
150	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
152	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
153	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
154	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
155	C₂²×C₁₇	#504	1	1	yes	characteristic
156	C₂²×C₁₇	#504	1	1	yes	characteristic
157	C₂×D₁₇	—	2	2	no	—
158	C₂×D₁₇	—	2	2	no	—
159	C₂×D₁₇	—	2	2	no	—
160	C₂×D₁₇	—	2	2	no	—
161	C₂×D₁₇	—	2	2	no	—
162	C₂×D₁₇	—	2	2	no	—
163	C₂×D₁₇	C₄⋊C₁₆	1	4	yes	complemented
164	C₄×C₁₇	—	2	2	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	#385	1	1	yes	characteristic
167	C₄×C₁₇	#424	1	1	yes	characteristic
168	C₄×C₁₇	#435	1	1	yes	characteristic
169	C₄×C₁₇	#435	1	1	yes	characteristic
170	Dic₁₇	—	2	2	no	—
171	Dic₁₇	—	2	2	no	—
172	Dic₁₇	—	4	1	no	—
173	C₂³×C₈	—	17	1	no	—
174	C₂²×C₁₆	—	17	1	no	—
175	C₂²×C₁₆	—	17	1	no	—
176	C₂²×C₁₆	—	34	2	no	—
177	C₂²×C₁₆	—	34	2	no	—
178	C₂²×C₁₆	—	34	2	no	—
179	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
180	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
181	C₂×C₂²⋊C₈	—	17	1	no	—
182	C₂×C₄×C₈	—	17	1	no	—
183	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
184	C₄×C₂²⋊C₄	—	17	1	no	—
185	C₄⋊C₁₆	—	2	2	no	—
186	C₄⋊C₁₆	—	34	2	no	—
187	C₂²⋊C₁₆	—	34	2	no	—
188	#336	—	17	1	no	—
189	#336	—	17	1	no	—
190	C₂×C₁₇	#1850	1	1	yes	characteristic
191	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
192	C₂×C₁₇	#2151	1	1	yes	characteristic
193	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
194	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
195	C₂×C₁₇	#2168	1	1	yes	characteristic
196	C₂×C₁₇	#2168	1	1	yes	characteristic
197	D₁₇	—	2	4	no	—
198	C₂³×C₄	—	17	1	no	—
199	C₂²×C₈	—	17	1	no	—
200	C₂²×C₈	—	17	2	no	—
201	C₂²×C₈	—	17	2	no	—
202	C₂²×C₈	—	17	1	no	—
203	C₂²×C₈	—	34	1	no	—
204	C₂²×C₈	—	34	1	no	—
205	C₂²×C₈	—	34	1	no	—
206	C₂²×C₈	—	34	1	no	—
207	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	17	1	no	—
210	C₂×C₂²⋊C₄	—	17	1	no	—
211	C₂×C₄²	—	17	1	no	—
212	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	17	2	no	—
219	C₂×C₁₆	—	34	8	no	—
220	C₂×C₁₆	—	34	1	no	—
221	C₂×C₁₆	—	34	1	no	—
222	C₂×C₁₆	—	34	1	no	—
223	C₂×C₁₆	—	34	1	no	—
224	C₂×C₁₆	—	34	1	no	—
225	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
226	C₂²⋊C₈	—	34	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	34	1	no	—
230	C₄×C₈	—	34	1	no	—
231	#95	—	17	1	no	—
232	#95	—	17	1	no	—
233	C₁₇	#12912	1	1	yes	characteristic, complemented, Sylow
234	C₂⁴	—	17	1	no	—
235	C₂²×C₄	—	17	1	no	—
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	17	2	no	—
238	C₂²×C₄	—	17	2	no	—
239	C₂²×C₄	—	17	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	—	34	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	C₂²×C₄	—	34	1	no	—
244	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	17	2	no	—
251	C₂×C₈	—	34	1	no	—
252	C₂×C₈	—	34	1	no	—
253	C₂×C₈	—	34	2	no	—
254	C₂×C₈	—	34	2	no	—
255	C₂×C₈	—	34	2	no	—
256	C₂×C₈	—	34	2	no	—
257	C₂×C₈	—	34	1	no	—
258	C₂×C₈	—	34	1	no	—
259	C₂×C₈	—	34	1	no	—
260	C₂×C₈	#63252	1	1	yes	characteristic
261	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
262	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
264	C₂×C₈	#63256	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
266	C₂²⋊C₄	—	17	4	no	—
267	C₂²⋊C₄	—	34	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	—	34	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₆	—	34	2	no	—
275	C₁₆	—	68	1	no	—
276	C₂³	—	17	2	no	—
277	C₂³	—	17	2	no	—
278	C₂³	—	17	2	no	—
279	C₂³	—	34	1	no	—
280	C₂³	—	34	1	no	—
281	C₂³	—	34	1	no	—
282	C₂³	—	34	1	no	—
283	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	17	2	no	—
290	C₂×C₄	—	17	2	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	2	no	—
296	C₂×C₄	—	34	2	no	—
297	C₂×C₄	—	34	2	no	—
298	C₂×C₄	—	34	2	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	C₁₆×D₁₇	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	#10587579	1	1	yes	characteristic
305	C₂×C₄	#10587587	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
307	C₂×C₄	#10587593	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	34	2	no	—
311	C₈	—	68	1	no	—
312	C₈	#10587580	1	1	yes	characteristic
313	C₈	#10587582	1	1	yes	characteristic
314	C₈	#10587623	1	1	yes	characteristic
315	C₈	#10587632	1	1	yes	characteristic
316	C₂²	—	17	4	no	—
317	C₂²	—	34	2	no	—
318	C₂²	—	34	2	no	—
319	C₂²	—	34	2	no	—
320	C₂²	—	34	2	no	—
321	C₂²	—	34	2	no	—
322	C₂²	—	34	2	no	—
323	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
324	C₂²	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
325	C₂²	#11762874	1	1	yes	characteristic
326	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
327	C₂²	M₃₂×D₁₇	1	1	yes	characteristic
328	C₂²	#11763225	1	1	yes	characteristic
329	C₂²	#11763227	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	34	2	no	—
333	C₄	—	34	2	no	—
334	C₄	—	68	1	no	—
335	C₄	#11762680	1	1	yes	characteristic
336	C₄	#11762834	1	1	yes	characteristic
337	C₄	#11762878	1	1	yes	characteristic
338	C₄	#11762879	1	1	yes	characteristic
339	C₂	—	34	4	no	—
340	C₂	#14528960	1	1	yes	characteristic
341	C₂	D₁₇×C₄⋊C₁₆	1	1	yes	characteristic
342	C₂	#14530541	1	1	yes	characteristic
343	C₂	C₁₆×C₂²⋊D₁₇	1	1	yes	characteristic
344	C₂	#14530603	1	1	yes	characteristic
345	C₂	#14530607	1	1	yes	characteristic
346	C₂	#14530608	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	#12912
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11762627	C₂	semidirect
5. #14526862	C₂	semidirect
6. #14526861	C₂	semidirect
7. C₂×C₁₇×C₄⋊C₁₆	C₂	semidirect
85. #63227	C₁₆	semidirect
116. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
131. C₂×Dic₁₇	C₂×C₁₆	semidirect
145. C₂×C₄⋊C₁₆	D₁₇	semidirect
163. C₂×D₁₇	C₄⋊C₁₆	semidirect
233. C₁₇	#12912	semidirect
302. C₂×C₄	C₁₆×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×#11762627	2
C₂×#11762588	2
C₈×#63227	2
#14526862	2
#14526861	2
C₂×C₁₇×C₄⋊C₁₆	2
#12912	17