Subgroups — #17005202

Order 4352.

576 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537933	C₂	1	2	yes	maximal, complemented
2	#14527656	C₂	1	2	yes	maximal, complemented
3	#14527649	C₂	1	2	yes	maximal, complemented
4	#14527628	C₂	1	2	yes	maximal, complemented
5	#14527621	C₂	1	2	yes	maximal, complemented
6	#14527540	C₂	1	1	yes	maximal, characteristic, complemented
7	#14527249	C₂	1	2	yes	maximal, complemented
8	#14527076	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#3168	C₂	1	1	yes	maximal, characteristic
10	#11762940	C₂²	1	2	yes	—
11	Q₈×#3622	C₂²	1	2	yes	—
12	#11762887	—	2	2	no	—
13	#11762887	C₂²	1	2	yes	—
14	#11762881	—	2	2	no	—
15	D₄×#3622	C₂²	1	2	yes	—
16	#11762870	—	2	2	no	—
17	#11762862	C₂²	1	1	yes	characteristic
18	#11762862	C₂²	1	1	yes	characteristic
19	#11762836	—	2	2	no	—
20	#11762833	C₂²	1	2	yes	—
21	#11762560	C₂²	1	1	yes	characteristic
22	#11762560	C₂²	1	1	yes	characteristic
23	#11762560	C₂²	1	2	yes	—
24	Q₈⋊#3622	C₂²	1	2	yes	—
25	Q₈⋊#3622	C₂²	1	2	yes	—
26	D₄⋊#3622	C₂²	1	2	yes	—
27	D₄⋊#3622	C₂²	1	2	yes	—
28	#11762531	C₂²	1	2	yes	—
29	#11762531	C₂²	1	2	yes	—
30	C₁₇×#529	C₂²	1	2	yes	—
31	C₁₇×#499	C₄	1	1	yes	characteristic
32	C₁₇×#495	C₄	1	1	yes	characteristic
33	C₁₇×#488	C₄	1	2	yes	—
34	C₁₇×#467	C₄	1	1	yes	characteristic
35	C₁₇×#466	C₄	1	1	yes	characteristic
36	C₁₇×#465	C₄	1	2	yes	—
37	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₁₇	C₂²	1	2	yes	—
39	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#126	—	2	2	no	—
47	C₁₇×#125	C₂×C₄	1	2	yes	—
48	C₁₇×#124	C₂×C₄	1	2	yes	—
49	C₁₇×#123	—	2	2	no	—
50	C₁₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
51	C₁₇×#121	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
53	C₄×D₄×C₁₇	—	2	2	no	—
54	C₄×D₄×C₁₇	C₂³	1	2	yes	—
55	C₁₇×#117	—	2	2	no	—
56	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
57	C₂×#63223	—	2	2	no	—
58	C₂×#63223	—	2	2	no	—
59	C₂²×#3622	—	2	2	no	—
60	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
61	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₇×#106	C₂×C₄	1	2	yes	—
63	C₁₇×#103	C₂×C₄	1	2	yes	—
64	C₁₇×#103	C₂×C₄	1	2	yes	—
65	C₁₇×#102	C₂×C₄	1	2	yes	—
66	C₁₇×#102	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂²⋊#3622	—	2	2	no	—
72	C₂²⋊#3622	—	2	2	no	—
73	C₂²⋊#3622	—	2	1	no	—
74	C₂²⋊#3622	D₄	1	1	yes	characteristic
75	C₂²⋊#3622	D₄	1	1	yes	characteristic
76	C₄⋊#3622	—	2	2	no	—
77	C₄⋊#3622	—	2	2	no	—
78	C₄⋊#3622	—	2	2	no	—
79	C₄⋊#3622	—	2	1	no	—
80	C₄⋊#3622	D₄	1	2	yes	—
81	C₄⋊#3622	C₂³	1	2	yes	—
82	C₄⋊#3622	C₂³	1	2	yes	—
83	#10587474	—	2	2	no	—
84	C₄×#3622	—	2	2	no	—
85	C₄×#3622	C₂³	1	2	yes	—
86	C₄×#3622	C₂³	1	2	yes	—
87	C₂×Q₈×C₁₇	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₁₇	—	2	2	no	—
89	C₂×D₄×C₁₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₁₇	—	2	2	no	—
91	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
94	Q₁₆×C₁₇	—	4	1	no	—
95	SD₁₆×C₁₇	—	4	2	no	—
96	D₈×C₁₇	—	4	1	no	—
97	C₂×C₈×C₁₇	—	4	1	no	—
98	C₂×C₈×C₁₇	—	4	1	no	—
99	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₇	—	2	2	no	—
104	C₄⋊C₄×C₁₇	—	2	2	no	—
105	C₄⋊C₄×C₁₇	—	2	2	no	—
106	C₄⋊C₄×C₁₇	—	2	2	no	—
107	C₄⋊C₄×C₁₇	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₁₇	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₁₇	—	2	2	no	—
110	C₂²⋊C₄×C₁₇	—	2	2	no	—
111	C₂²⋊C₄×C₁₇	—	2	2	no	—
112	C₂²⋊C₄×C₁₇	—	2	2	no	—
113	C₂²⋊C₄×C₁₇	—	2	2	no	—
114	C₄²×C₁₇	—	2	2	no	—
115	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₇	#41	1	1	yes	characteristic
118	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
119	#63223	—	8	2	no	—
120	#63223	—	8	2	no	—
121	C₂×#3622	—	2	2	no	—
122	C₂×#3622	—	2	2	no	—
123	C₂×#3622	—	2	2	no	—
124	C₂×#3622	—	8	2	no	—
125	C₂×#3622	C₂×D₄	1	2	yes	—
126	C₂×#3622	C₂×D₄	1	2	yes	—
127	#17993	—	17	1	no	maximal, Sylow
128	C₂³×C₁₇	—	2	2	no	—
129	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₁₇	—	2	2	no	—
131	Q₈×C₁₇	—	2	2	no	—
132	D₄×C₁₇	—	2	2	no	—
133	D₄×C₁₇	—	2	2	no	—
134	D₄×C₁₇	—	8	2	no	—
135	C₂×C₄×C₁₇	—	2	1	no	—
136	C₂×C₄×C₁₇	—	2	1	no	—
137	C₂×C₄×C₁₇	—	2	2	no	—
138	C₂×C₄×C₁₇	—	2	2	no	—
139	C₂×C₄×C₁₇	—	2	2	no	—
140	C₂×C₄×C₁₇	—	2	2	no	—
141	C₂×C₄×C₁₇	—	4	1	no	—
142	C₂×C₄×C₁₇	—	4	2	no	—
143	C₂×C₄×C₁₇	—	4	2	no	—
144	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
150	C₈×C₁₇	—	2	1	no	—
151	C₈×C₁₇	—	2	1	no	—
152	C₈×C₁₇	—	2	1	no	—
153	C₈×C₁₇	—	2	1	no	—
154	#3622	—	2	2	no	—
155	#3622	—	2	2	no	—
156	#3622	—	4	2	no	—
157	#3622	—	4	2	no	—
158	#3622	—	4	2	no	—
159	#3168	D₁₇	1	1	yes	characteristic
160	#2993	—	17	2	no	—
161	#1587	—	17	2	no	—
162	#1586	—	17	2	no	—
163	#1583	—	17	2	no	—
164	#1582	—	17	2	no	—
165	#1566	—	17	1	no	—
166	#1495	—	17	2	no	—
167	#1452	—	17	1	no	—
168	C₂²×C₁₇	—	2	1	no	—
169	C₂²×C₁₇	—	2	1	no	—
170	C₂²×C₁₇	—	2	1	no	—
171	C₂²×C₁₇	—	4	2	no	—
172	C₂²×C₁₇	—	4	2	no	—
173	C₂²×C₁₇	—	4	2	no	—
174	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₇	—	2	1	no	—
176	C₄×C₁₇	—	2	1	no	—
177	C₄×C₁₇	—	2	1	no	—
178	C₄×C₁₇	—	2	1	no	—
179	C₄×C₁₇	—	4	2	no	—
180	C₄×C₁₇	—	4	2	no	—
181	C₄×C₁₇	—	4	2	no	—
182	C₄×C₁₇	#568	1	1	yes	characteristic
183	C₄×C₁₇	#568	1	1	yes	characteristic
184	C₄×C₁₇	#577	1	2	yes	—
185	#529	C₂×D₁₇	1	2	yes	—
186	#499	Dic₁₇	1	1	yes	characteristic
187	#495	Dic₁₇	1	1	yes	characteristic
188	#488	Dic₁₇	1	2	yes	—
189	#467	Dic₁₇	1	1	yes	characteristic
190	#466	Dic₁₇	1	1	yes	characteristic
191	#465	Dic₁₇	1	2	yes	—
192	#446	—	17	2	no	—
193	C₈×Q₈	—	17	2	no	—
194	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
195	C₄×SD₁₆	C₂×D₁₇	1	2	yes	—
196	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
197	#436	—	17	2	no	—
198	#436	—	34	2	no	—
199	#435	—	34	2	no	—
200	C₈×D₄	—	17	2	no	—
201	#433	—	34	2	no	—
202	#431	—	17	1	no	—
203	#431	—	17	1	no	—
204	#424	—	34	2	no	—
205	#423	—	17	2	no	—
206	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
207	#334	—	17	1	no	—
208	#334	—	17	1	no	—
209	#334	—	17	2	no	—
210	Q₈⋊C₈	—	17	2	no	—
211	Q₈⋊C₈	—	17	2	no	—
212	D₄⋊C₈	—	17	2	no	—
213	D₄⋊C₈	—	17	2	no	—
214	#322	—	17	2	no	—
215	#322	—	17	2	no	—
216	C₂×C₁₇	—	4	1	no	—
217	C₂×C₁₇	—	8	2	no	—
218	C₂×C₁₇	#2932	1	1	yes	characteristic
219	C₂×C₁₇	#2952	1	1	yes	characteristic
220	C₂×C₁₇	#2957	1	1	yes	characteristic
221	C₂×Q₁₆	C₂×Dic₁₇	1	1	yes	characteristic
222	C₂×SD₁₆	C₂×Dic₁₇	1	2	yes	—
223	C₂×D₈	C₂×Dic₁₇	1	1	yes	characteristic
224	C₂×M₁₆	—	34	2	no	—
225	C₂×M₁₆	—	34	2	no	—
226	C₂²×C₈	—	34	2	no	—
227	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
228	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
229	#126	—	2	2	no	—
230	#125	C₂×Dic₁₇	1	2	yes	—
231	#124	C₂×Dic₁₇	1	2	yes	—
232	#123	—	2	2	no	—
233	C₂²⋊Q₈	C₂×Dic₁₇	1	2	yes	—
234	#121	C₂×Dic₁₇	1	2	yes	—
235	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	C₂²×D₁₇	1	2	yes	—
238	#117	—	2	2	no	—
239	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
240	#107	C₂×Dic₁₇	1	1	yes	characteristic
241	#107	C₂×Dic₁₇	1	1	yes	characteristic
242	#106	C₂×Dic₁₇	1	2	yes	—
243	C₄⋊C₈	—	17	2	no	—
244	C₄⋊C₈	—	17	2	no	—
245	C₄⋊C₈	—	17	2	no	—
246	C₄⋊C₈	—	34	2	no	—
247	C₄⋊C₈	—	34	2	no	—
248	C₄⋊C₈	—	34	2	no	—
249	C₄⋊C₈	—	34	1	no	—
250	#103	C₂×Dic₁₇	1	2	yes	—
251	#103	C₂×Dic₁₇	1	2	yes	—
252	#102	C₂×Dic₁₇	1	2	yes	—
253	#102	C₂×Dic₁₇	1	2	yes	—
254	C₂²⋊C₈	—	17	1	no	—
255	C₂²⋊C₈	—	17	1	no	—
256	C₂²⋊C₈	—	34	2	no	—
257	C₂²⋊C₈	—	34	2	no	—
258	C₂²⋊C₈	—	34	1	no	—
259	#97	—	34	2	no	—
260	C₄×C₈	—	17	2	no	—
261	C₄×C₈	—	17	2	no	—
262	C₄×C₈	—	34	2	no	—
263	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
264	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
267	C₁₇	#17993	1	1	yes	characteristic, complemented, Sylow
268	C₂×Q₈	C₂²×Dic₁₇	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₂²×Dic₁₇	1	2	yes	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
273	C₂²×C₄	#63254	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
275	Q₁₆	—	4	1	no	—
276	SD₁₆	—	4	2	no	—
277	D₈	—	4	1	no	—
278	M₁₆	—	136	2	no	—
279	M₁₆	—	136	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	17	2	no	—
283	C₂×C₈	—	17	2	no	—
284	C₂×C₈	—	34	2	no	—
285	C₂×C₈	—	34	2	no	—
286	C₂×C₈	—	34	2	no	—
287	C₂×C₈	—	136	2	no	—
288	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
289	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂²×Dic₁₇	1	2	yes	—
297	C₄⋊C₄	C₂²×Dic₁₇	1	2	yes	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	D₄×D₁₇	1	1	yes	characteristic
305	C₄²	#63254	1	1	yes	characteristic
306	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
307	C₄²	C₂³×D₁₇	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	D₄×Dic₁₇	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	Q₈	—	2	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	8	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
325	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
326	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
328	C₂×C₄	C₂×#63254	1	1	yes	characteristic
329	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	34	2	no	—
335	C₈	—	34	2	no	—
336	C₈	—	68	2	no	—
337	C₈	—	68	2	no	—
338	C₈	—	68	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	#11763576	1	1	yes	characteristic
354	C₄	#11763576	1	1	yes	characteristic
355	C₄	#11763616	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	8	2	no	—
358	C₂	#14537502	1	1	yes	characteristic
359	C₂	#14537668	1	1	yes	characteristic
360	C₂	#14537701	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#3168	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₁₇×#466	C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#3168	D₁₇
hypocenter	C₁₇	#17993
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #14537933	C₂	semidirect
2. #14527656	C₂	semidirect
3. #14527649	C₂	semidirect
4. #14527628	C₂	semidirect
5. #14527621	C₂	semidirect
6. #14527540	C₂	semidirect
7. #14527249	C₂	semidirect
8. #14527076	C₂	semidirect
267. C₁₇	#17993	semidirect

Maximal subgroups

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

subgroup	index
#14537933	2
#14527656	2
#14527649	2
#14527628	2
#14527621	2
#14527540	2
#14527249	2
#14527076	2
C₁₇×#3168	2
#17993	17