Subgroups — #17960784

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545749	C₂	1	1	yes	maximal, characteristic
2	#14544996	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11763696	C₂	1	1	yes	maximal, characteristic, complemented
4	#14544558	C₂	1	1	yes	maximal, characteristic
5	#14543837	C₂	1	1	yes	maximal, characteristic, complemented
6	#14543827	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₇×#350	C₂	1	1	yes	maximal, characteristic
8	#11763868	—	2	2	no	—
9	#11763868	C₄	1	2	yes	complemented
10	#11763867	C₄	1	2	yes	complemented
11	#11763865	—	2	1	no	—
12	#11763865	C₂²	1	1	yes	characteristic
13	C₂×#10587529	—	2	1	no	—
14	C₂×#10587529	C₂²	1	1	yes	characteristic
15	D₁₇⋊#104	C₄	1	1	yes	characteristic
16	#11763799	C₄	1	1	yes	characteristic
17	#11763772	C₂²	1	1	yes	characteristic
18	C₂×C₈×#595	C₂²	1	1	yes	characteristic
19	#11763696	—	2	2	no	—
20	#11763696	—	2	1	no	—
21	C₁₆×#595	—	2	1	no	—
22	#11763684	C₄	1	1	yes	characteristic
23	#11763681	C₄	1	1	yes	characteristic
24	#131×D₁₇	C₂²	1	1	yes	characteristic
25	M₃₂×D₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
27	#11762594	C₄	1	1	yes	characteristic, complemented
28	#11762593	C₄	1	1	yes	characteristic, complemented
29	#11762592	C₄	1	1	yes	characteristic, complemented
30	C₁₇×#350	C₄	1	1	yes	characteristic, complemented
31	#41×D₁₇	C₂×C₄	1	1	yes	characteristic
32	#10587677	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
34	D₁₇⋊M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×#63242	C₂×C₄	1	1	yes	characteristic
36	C₁₇×#131	C₂×C₄	1	1	yes	characteristic
37	#10587634	C₂×C₄	1	1	yes	characteristic
38	#10587593	—	2	1	no	—
39	#10587593	C₂×C₄	1	1	yes	characteristic
40	M₁₆×D₁₇	—	2	1	no	—
41	M₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
42	#10587587	—	2	1	no	—
43	#10587587	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₇	—	2	1	no	—
45	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
46	#10587550	C₈	1	2	yes	—
47	#10587549	C₈	1	2	yes	—
48	#10587548	C₈	1	2	yes	—
49	#10587547	C₈	1	2	yes	—
50	#10587538	—	2	2	no	—
51	#10587538	C₂×C₄	1	2	yes	—
52	C₂×#63216	—	2	2	no	—
53	C₂×#63216	C₂×C₄	1	2	yes	—
54	#10587532	—	2	1	no	—
55	C₈×#595	—	2	1	no	—
56	C₈×#595	—	2	1	no	—
57	C₈×#595	—	2	1	no	—
58	#10587530	—	2	1	no	—
59	#10587530	—	2	2	no	—
60	#10587529	—	2	1	no	—
61	#10587529	—	2	2	no	—
62	#10587529	D₄	1	1	yes	characteristic
63	#10587529	D₄	1	1	yes	characteristic
64	#10587529	D₄	1	2	yes	—
65	C₁₇×M₃₂	C₂×C₄	1	1	yes	characteristic
66	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
67	#10587483	C₂×C₄	1	1	yes	characteristic
68	C₂×#63214	C₂×C₄	1	1	yes	characteristic
69	#10587469	—	2	1	no	—
70	C₁₆×D₁₇	—	2	1	no	—
71	C₁₆×D₁₇	—	2	1	no	—
72	C₁₆×D₁₇	—	2	1	no	—
73	C₂²×#595	—	2	1	no	—
74	#41×C₁₇	C₄²	1	1	yes	characteristic
75	#63256	C₂×C₈	1	1	yes	characteristic
76	Q₈×D₁₇	C₂×C₈	1	1	yes	characteristic
77	#63254	—	2	1	no	—
78	#63254	C₂×C₈	1	1	yes	characteristic
79	D₄×D₁₇	—	2	1	no	—
80	D₄×D₁₇	C₂×C₈	1	1	yes	characteristic
81	#63252	—	2	1	no	—
82	#63252	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₁₇	—	2	1	no	—
84	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊#595	—	2	1	no	—
86	#63247	—	2	1	no	—
87	C₂×#3624	—	2	1	no	—
88	C₄⋊#595	C₂×C₈	1	2	yes	—
89	C₄×#595	—	2	1	no	—
90	C₄×#595	—	2	1	no	—
91	C₄×#595	C₂×C₈	1	2	yes	—
92	#63243	C₂×C₈	1	2	yes	—
93	#63242	—	2	1	no	—
94	#63242	—	2	1	no	—
95	#63242	C₂×C₈	1	2	yes	—
96	M₁₆×C₁₇	—	2	1	no	—
97	M₁₆×C₁₇	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₁₇	—	2	1	no	—
99	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	#63223	—	2	1	no	—
101	#63223	C₄²	1	1	yes	characteristic
102	C₂×#3622	—	2	1	no	—
103	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
104	#63218	—	2	1	no	—
105	#63218	—	2	1	no	—
106	#63218	—	2	1	no	—
107	C₈×D₁₇	—	2	1	no	—
108	C₈×D₁₇	—	2	1	no	—
109	C₈×D₁₇	—	2	1	no	—
110	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
111	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
112	C₈×D₁₇	#41	1	1	yes	characteristic
113	C₈×D₁₇	#41	1	1	yes	characteristic
114	#63216	—	2	2	no	—
115	#63216	C₂²⋊C₄	1	2	yes	—
116	#63216	C₂²⋊C₄	1	2	yes	—
117	C₁₆×C₁₇	—	2	1	no	—
118	C₁₆×C₁₇	—	2	1	no	—
119	#63214	—	2	1	no	—
120	#63214	—	2	1	no	—
121	#12615	—	17	1	no	maximal, Sylow
122	C₂²×D₁₇	—	2	1	no	—
123	C₂²×D₁₇	#131	1	1	yes	characteristic
124	C₂×#595	—	2	2	no	—
125	C₂×#595	—	2	2	no	—
126	C₂×#595	—	4	2	no	—
127	Q₈×C₁₇	C₄×C₈	1	1	yes	characteristic
128	D₄×C₁₇	—	2	1	no	—
129	D₄×C₁₇	C₄×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₇	—	2	1	no	—
131	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
132	C₂²⋊D₁₇	—	2	1	no	—
133	C₂²⋊D₁₇	—	2	1	no	—
134	C₂²⋊D₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	#131	1	1	yes	characteristic
137	D₆₈	—	2	1	no	—
138	D₆₈	C₄×C₈	1	1	yes	characteristic
139	C₄×D₁₇	—	2	1	no	—
140	C₄×D₁₇	—	2	1	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
143	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
144	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
145	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
146	Dic₃₄	—	2	1	no	—
147	Dic₃₄	C₄×C₈	1	1	yes	characteristic
148	#3624	—	2	2	no	—
149	#3624	—	2	2	no	—
150	C₈×C₁₇	—	2	1	no	—
151	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
152	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
153	#3622	—	2	1	no	—
154	#3622	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	#3622	#117	1	1	yes	characteristic
156	C₂×#504	—	17	1	no	—
157	C₂×#350	—	17	1	no	—
158	#2108	—	17	1	no	—
159	C₂×#346	—	17	1	no	—
160	#1764	—	17	1	no	—
161	#1379	—	17	1	no	—
162	#1376	—	17	1	no	—
163	C₂²×C₁₇	—	2	1	no	—
164	C₂²×C₁₇	#405	1	1	yes	characteristic
165	C₂×D₁₇	—	2	2	no	—
166	C₂×D₁₇	—	2	1	no	—
167	C₂×D₁₇	—	4	2	no	—
168	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
169	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
170	#595	—	4	2	no	—
171	#595	—	4	2	no	—
172	C₄×C₁₇	—	2	1	no	—
173	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	C₈×D₄	1	1	yes	characteristic
177	Dic₁₇	C₈×D₄	1	1	yes	characteristic
178	C₂×#131	—	17	1	no	—
179	#504	—	17	2	no	—
180	#504	—	17	2	no	—
181	#504	—	34	2	no	—
182	C₂×M₃₂	—	17	1	no	—
183	C₂×M₃₂	—	17	1	no	—
184	C₂×M₃₂	—	34	1	no	—
185	C₂²×C₁₆	—	17	1	no	—
186	C₂²×C₁₆	—	17	1	no	—
187	C₂²×C₁₆	—	34	1	no	—
188	C₂×#104	—	17	1	no	—
189	#416	—	17	1	no	—
190	#405	—	17	1	no	—
191	C₂×C₄×C₈	—	17	1	no	—
192	#350	—	17	1	no	—
193	#350	—	17	1	no	—
194	#350	—	17	1	no	—
195	#350	#595	1	1	yes	characteristic, complemented
196	#346	—	34	2	no	—
197	#346	—	34	1	no	—
198	C₄×C₁₆	—	34	1	no	—
199	#341	—	17	1	no	—
200	#339	—	17	1	no	—
201	C₂×C₁₇	—	2	1	no	—
202	C₂×C₁₇	—	4	1	no	—
203	C₂×C₁₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
204	D₁₇	—	2	1	no	—
205	D₁₇	—	4	1	no	—
206	D₁₇	#2171	1	2	yes	—
207	C₂×#41	—	17	1	no	—
208	#131	—	17	1	no	—
209	#131	—	17	1	no	—
210	#131	—	17	1	no	—
211	#131	—	34	1	no	—
212	#131	—	34	1	no	—
213	#131	C₂×#595	1	1	yes	characteristic
214	C₂×M₁₆	—	17	1	no	—
215	C₂×M₁₆	—	17	1	no	—
216	C₂×M₁₆	—	34	1	no	—
217	C₂²×C₈	—	17	1	no	—
218	C₂²×C₈	—	17	1	no	—
219	C₂²×C₈	—	34	1	no	—
220	#117	—	17	1	no	—
221	C₂×C₄²	—	17	1	no	—
222	M₃₂	—	17	2	no	—
223	M₃₂	—	17	1	no	—
224	M₃₂	—	34	1	no	—
225	M₃₂	—	34	2	no	—
226	M₃₂	—	34	2	no	—
227	M₃₂	—	34	1	no	—
228	M₃₂	C₂×#595	1	1	yes	characteristic
229	C₂×C₁₆	—	17	1	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	2	no	—
232	C₂×C₁₆	—	17	2	no	—
233	C₂×C₁₆	—	17	1	no	—
234	C₂×C₁₆	—	34	1	no	—
235	C₂×C₁₆	—	34	1	no	—
236	C₂×C₁₆	—	34	1	no	—
237	C₂×C₁₆	—	34	2	no	—
238	C₂×C₁₆	—	34	2	no	—
239	C₂×C₁₆	—	34	1	no	—
240	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
241	#104	—	17	2	no	—
242	#104	—	17	2	no	—
243	#103	—	17	2	no	—
244	#102	—	17	2	no	—
245	#97	—	34	1	no	—
246	C₄×C₈	—	34	1	no	—
247	C₄×C₈	—	34	1	no	—
248	C₄×C₈	—	34	1	no	—
249	C₁₇	#12615	1	1	yes	characteristic, complemented, Sylow
250	#41	—	17	1	no	—
251	#41	—	17	1	no	—
252	#41	—	17	1	no	—
253	#41	—	34	1	no	—
254	#41	—	34	1	no	—
255	#41	C₄×#595	1	1	yes	characteristic
256	C₂×Q₈	—	17	1	no	—
257	C₂×D₄	—	17	1	no	—
258	C₂×D₄	—	34	1	no	—
259	C₂²×C₄	—	17	1	no	—
260	C₂²×C₄	—	34	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	17	2	no	—
264	M₁₆	—	17	1	no	—
265	M₁₆	—	34	1	no	—
266	M₁₆	—	34	1	no	—
267	M₁₆	—	34	1	no	—
268	M₁₆	—	34	1	no	—
269	M₁₆	—	34	1	no	—
270	M₁₆	C₄×#595	1	1	yes	characteristic
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	17	1	no	—
273	C₂×C₈	—	17	1	no	—
274	C₂×C₈	—	17	1	no	—
275	C₂×C₈	—	17	1	no	—
276	C₂×C₈	—	17	1	no	—
277	C₂×C₈	—	17	2	no	—
278	C₂×C₈	—	34	1	no	—
279	C₂×C₈	—	34	1	no	—
280	C₂×C₈	—	34	1	no	—
281	C₂×C₈	—	34	1	no	—
282	C₂×C₈	—	34	1	no	—
283	C₂×C₈	—	34	1	no	—
284	C₂×C₈	—	34	1	no	—
285	C₂×C₈	C₂²×#595	1	1	yes	characteristic
286	C₄⋊C₄	—	17	2	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₄²	—	17	2	no	—
289	C₄²	—	34	1	no	—
290	C₄²	—	34	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	17	2	no	—
294	C₁₆	—	17	2	no	—
295	C₁₆	—	34	2	no	—
296	C₁₆	—	34	1	no	—
297	C₁₆	—	34	1	no	—
298	C₂³	—	17	1	no	—
299	C₂³	—	34	1	no	—
300	Q₈	—	17	1	no	—
301	Q₈	—	34	1	no	—
302	Q₈	C₈×#595	1	1	yes	characteristic
303	D₄	—	2	1	no	—
304	D₄	—	17	1	no	—
305	D₄	—	34	1	no	—
306	D₄	—	34	1	no	—
307	D₄	—	34	1	no	—
308	D₄	—	34	1	no	—
309	D₄	C₈×#595	1	1	yes	characteristic
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	2	no	—
321	C₂×C₄	—	34	2	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	17	1	no	—
326	C₈	—	17	1	no	—
327	C₈	—	34	2	no	—
328	C₈	—	34	2	no	—
329	C₈	—	34	1	no	—
330	C₈	D₄×#595	1	1	yes	characteristic
331	C₈	D₄×#595	1	1	yes	characteristic
332	C₂²	—	2	1	no	—
333	C₂²	—	17	1	no	—
334	C₂²	—	17	1	no	—
335	C₂²	—	34	2	no	—
336	C₂²	—	34	1	no	—
337	C₂²	—	68	2	no	—
338	C₂²	#11763775	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	17	1	no	—
341	C₄	—	17	1	no	—
342	C₄	—	34	1	no	—
343	C₄	—	68	2	no	—
344	C₄	—	68	2	no	—
345	C₄	C₂²⋊C₄×#595	1	1	yes	characteristic
346	C₄	C₂×C₈×#595	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	4	1	no	—
349	C₂	—	17	2	no	—
350	C₂	—	34	1	no	—
351	C₂	—	68	1	no	—
352	C₂	#14544519	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#350	C₄
socle	C₂×C₁₇	C₈×C₂²⋊C₄
prime radical	#41×D₁₇	C₂×C₄
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#350	#595
hypocenter	C₁₇	#12615
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

Complements

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

N	H ≅ G/N	type
2. #14544996	C₂	semidirect
3. C₂×#11763696	C₂	semidirect
5. #14543837	C₂	semidirect
6. #14543827	C₂	semidirect
9. #11763868	C₄	semidirect
10. #11763867	C₄	semidirect
27. #11762594	C₄	semidirect
28. #11762593	C₄	semidirect
29. #11762592	C₄	semidirect
30. C₁₇×#350	C₄	semidirect
195. #350	#595	semidirect
249. C₁₇	#12615	semidirect

Maximal subgroups

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

subgroup	index
#14545749	2
#14544996	2
C₂×#11763696	2
#14544558	2
#14543837	2
#14543827	2
D₁₇×#350	2
#12615	17