Subgroups — #16950792

Order 4352.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530394	C₂	1	1	yes	maximal, characteristic
4	C₈×#63227	C₂	1	1	yes	maximal, characteristic
5	#14526866	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526865	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×#10587483	C₂²	1	1	yes	characteristic
10	C₂×#10587483	C₂²	1	1	yes	characteristic
11	C₂×#10587469	—	2	1	no	—
12	M₃₂×D₁₇	—	2	2	no	—
13	M₃₂×D₁₇	—	2	2	no	—
14	C₂×C₁₆×D₁₇	—	2	1	no	—
15	C₂×#10587491	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₄×#63227	C₄	1	1	yes	characteristic
18	#11762629	—	2	2	no	—
19	#11762588	—	2	1	no	—
20	#11762588	—	2	1	no	—
21	#11762564	C₄	1	1	yes	characteristic
22	#11762563	C₄	1	1	yes	characteristic
23	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₁₇×#364	—	2	2	no	—
27	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#63227	—	2	1	no	—
32	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
39	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
40	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
41	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
43	C₁₇×M₃₂	—	2	1	no	—
44	C₁₇×M₃₂	—	2	1	no	—
45	C₁₇×M₃₂	—	2	1	no	—
46	C₁₇×M₃₂	D₄	1	2	yes	—
47	C₂×C₁₆×C₁₇	—	2	1	no	—
48	C₂×C₁₆×C₁₇	—	2	1	no	—
49	C₄×C₈×C₁₇	—	2	1	no	—
50	C₄×C₈×C₁₇	—	2	1	no	—
51	#10587502	—	2	1	no	—
52	#10587491	C₈	1	2	yes	—
53	#10587491	C₈	1	2	yes	—
54	C₈×Dic₁₇	—	2	1	no	—
55	C₈×Dic₁₇	—	2	1	no	—
56	#10587483	—	2	1	no	—
57	#10587483	—	2	1	no	—
58	#10587483	—	2	1	no	—
59	#10587483	D₄	1	2	yes	—
60	C₂×#63214	—	2	1	no	—
61	C₂×#63214	—	2	1	no	—
62	#10587469	—	2	4	no	—
63	C₁₆×D₁₇	—	2	4	no	—
64	C₂³×D₁₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	2	1	no	—
68	C₂²×Dic₁₇	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₁₇	—	2	1	no	—
78	C₂×C₈×C₁₇	—	2	1	no	—
79	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	—	2	1	no	—
88	#63227	—	2	4	no	—
89	C₄×Dic₁₇	—	2	1	no	—
90	C₄×Dic₁₇	—	2	1	no	—
91	C₂×#3622	—	2	1	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	1	no	—
94	C₂×#3622	—	2	1	no	—
95	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
96	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
97	C₂×#3622	C₂×C₈	1	2	yes	—
98	C₈×D₁₇	—	2	1	no	—
99	C₈×D₁₇	—	2	1	no	—
100	C₈×D₁₇	—	2	1	no	—
101	C₈×D₁₇	—	2	1	no	—
102	C₈×D₁₇	—	2	1	no	—
103	C₈×D₁₇	—	2	1	no	—
104	C₈×D₁₇	—	2	1	no	—
105	C₈×D₁₇	—	2	1	no	—
106	C₁₆×C₁₇	—	2	2	no	—
107	C₁₆×C₁₇	—	4	1	no	—
108	#63214	—	2	2	no	—
109	#63214	—	4	1	no	—
110	#12914	—	17	1	no	maximal, Sylow
111	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	2	1	no	—
114	C₂²×D₁₇	—	2	2	no	—
115	C₂²×D₁₇	—	4	1	no	—
116	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
117	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	4	1	no	—
120	C₂×C₄×C₁₇	—	4	1	no	—
121	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
126	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	2	no	—
130	C₂×Dic₁₇	—	4	1	no	—
131	C₂×Dic₁₇	—	4	1	no	—
132	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
133	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	2	1	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	2	1	no	—
139	C₄×D₁₇	—	2	1	no	—
140	C₄×D₁₇	—	2	1	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	2	no	—
143	C₈×C₁₇	#120	1	1	yes	characteristic
144	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₇	#123	1	1	yes	characteristic
147	#3622	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3622	—	2	2	no	—
150	C₂²×M₃₂	—	17	1	no	—
151	C₂×#364	—	17	1	no	—
152	C₂×#364	D₁₇	1	1	yes	characteristic, complemented
153	#2112	—	17	1	no	—
154	C₈×C₂²⋊C₄	—	17	1	no	—
155	#1376	—	17	1	no	—
156	#1376	—	17	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₁₇	#435	1	1	yes	characteristic
161	C₂²×C₁₇	#435	1	1	yes	characteristic
162	C₂×D₁₇	—	2	1	no	—
163	C₂×D₁₇	—	2	1	no	—
164	C₂×D₁₇	—	2	1	no	—
165	C₂×D₁₇	—	2	1	no	—
166	C₂×D₁₇	—	2	1	no	—
167	C₂×D₁₇	—	4	2	no	—
168	C₂×D₁₇	—	4	2	no	—
169	C₂×D₁₇	#364	1	2	yes	complemented
170	C₄×C₁₇	—	4	2	no	—
171	C₄×C₁₇	#385	1	1	yes	characteristic
172	C₄×C₁₇	#424	1	1	yes	characteristic
173	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
174	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	—	2	1	no	—
177	Dic₁₇	—	4	2	no	—
178	C₂³×C₈	—	17	1	no	—
179	C₂×M₃₂	—	17	1	no	—
180	C₂×M₃₂	—	17	1	no	—
181	C₂×M₃₂	—	34	1	no	—
182	C₂×M₃₂	—	34	2	no	—
183	C₂×M₃₂	—	34	2	no	—
184	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
185	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
186	C₂²×C₁₆	—	34	1	no	—
187	C₂×C₂²⋊C₈	—	17	1	no	—
188	C₂×C₄×C₈	—	17	1	no	—
189	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
190	C₄×C₂²⋊C₄	—	17	1	no	—
191	#364	—	2	2	no	—
192	#364	—	34	2	no	—
193	C₂²⋊C₁₆	—	34	1	no	—
194	C₂²⋊C₁₆	—	34	1	no	—
195	#336	—	17	1	no	—
196	#336	—	17	1	no	—
197	C₂×C₁₇	—	2	1	no	—
198	C₂×C₁₇	—	2	1	no	—
199	C₂×C₁₇	#1850	1	1	yes	characteristic
200	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
201	C₂×C₁₇	#2153	1	1	yes	characteristic
202	D₁₇	—	2	2	no	—
203	D₁₇	—	4	1	no	—
204	C₂³×C₄	—	17	1	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₂²×C₈	—	17	1	no	—
208	C₂²×C₈	—	17	1	no	—
209	C₂²×C₈	—	17	1	no	—
210	C₂²×C₈	—	17	1	no	—
211	C₂²×C₈	—	34	1	no	—
212	C₂²×C₈	—	34	1	no	—
213	C₂²×C₈	—	34	1	no	—
214	C₂²×C₈	—	34	1	no	—
215	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	34	1	no	—
218	C₂×C₄²	—	17	1	no	—
219	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	17	2	no	—
224	M₃₂	—	34	4	no	—
225	M₃₂	—	34	1	no	—
226	M₃₂	—	34	1	no	—
227	M₃₂	—	34	1	no	—
228	M₃₂	C₂²⋊D₁₇	1	2	yes	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	—	2	1	no	—
231	C₂×C₁₆	—	34	4	no	—
232	C₂×C₁₆	—	34	1	no	—
233	C₂×C₁₆	—	34	1	no	—
234	C₂²⋊C₈	—	17	2	no	—
235	C₂²⋊C₈	—	17	2	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	34	1	no	—
239	C₄×C₈	—	34	1	no	—
240	#95	—	34	1	no	—
241	C₁₇	#12914	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	17	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	17	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	—	34	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	2	no	—
260	C₂×C₈	—	34	1	no	—
261	C₂×C₈	—	34	1	no	—
262	C₂×C₈	—	34	1	no	—
263	C₂×C₈	—	34	1	no	—
264	C₂×C₈	—	34	1	no	—
265	C₂×C₈	—	34	1	no	—
266	C₂×C₈	—	34	1	no	—
267	C₂×C₈	—	34	1	no	—
268	C₂×C₈	—	34	1	no	—
269	C₂×C₈	—	34	1	no	—
270	C₂×C₈	—	34	1	no	—
271	C₂×C₈	—	34	1	no	—
272	C₂×C₈	C₈×D₁₇	1	2	yes	—
273	C₂×C₈	#63252	1	1	yes	characteristic
274	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
275	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
277	C₂×C₈	#63256	1	1	yes	characteristic
278	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	C₂²⋊C₄	—	34	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	34	1	no	—
283	C₄²	—	34	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	34	2	no	—
287	C₁₆	—	68	1	no	—
288	C₂³	—	17	1	no	—
289	C₂³	—	17	1	no	—
290	C₂³	—	34	2	no	—
291	C₂³	—	34	1	no	—
292	C₂³	—	34	1	no	—
293	C₂³	—	68	1	no	—
294	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	17	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	2	no	—
311	C₂×C₄	—	68	1	no	—
312	C₂×C₄	—	68	1	no	—
313	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
314	C₂×C₄	#10587579	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
316	C₂×C₄	#10587587	1	1	yes	characteristic
317	C₂×C₄	#10587593	1	1	yes	characteristic
318	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	34	2	no	—
321	C₈	—	34	1	no	—
322	C₈	—	34	1	no	—
323	C₈	#10587580	1	1	yes	characteristic
324	C₈	#10587582	1	1	yes	characteristic
325	C₈	#10587623	1	1	yes	characteristic
326	C₈	#10587632	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	17	2	no	—
330	C₂²	—	34	1	no	—
331	C₂²	—	34	1	no	—
332	C₂²	—	34	1	no	—
333	C₂²	—	34	1	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	68	2	no	—
336	C₂²	—	68	2	no	—
337	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
338	C₂²	#11762878	1	1	yes	characteristic
339	C₂²	#11762879	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	34	1	no	—
342	C₄	—	34	1	no	—
343	C₄	—	68	2	no	—
344	C₄	#11762680	1	1	yes	characteristic
345	C₄	#11762834	1	1	yes	characteristic
346	C₄	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
347	C₄	#11762874	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	34	2	no	—
351	C₂	—	68	1	no	—
352	C₂	#14528960	1	1	yes	characteristic
353	C₂	D₁₇×#364	1	1	yes	characteristic
354	C₂	#14530548	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#364	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×D₁₇	M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#364	D₁₇
hypocenter	C₁₇	#12914
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#11762629	C₂	semidirect
5. #14526866	C₂	semidirect
6. #14526865	C₂	semidirect
7. C₂×C₁₇×#364	C₂	semidirect
152. C₂×#364	D₁₇	semidirect
169. C₂×D₁₇	#364	semidirect
241. C₁₇	#12914	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₁₇	2
C₂×#11762629	2
#14530394	2
C₈×#63227	2
#14526866	2
#14526865	2
C₂×C₁₇×#364	2
#12914	17