Subgroups — #17191920

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541216	C₂	1	1	yes	maximal, characteristic, complemented
2	#14541133	C₂	1	1	yes	maximal, characteristic
3	#14531059	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530985	C₂	1	2	yes	maximal, complemented
5	#14530973	C₂	1	2	yes	maximal, complemented
6	#14530926	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530921	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530894	C₂	1	2	yes	maximal, complemented
9	#14530617	C₂	1	1	yes	maximal, characteristic, complemented
10	#14530571	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₇×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763532	—	2	1	no	—
13	#11763464	—	2	1	no	—
14	C₂×#10587469	C₂²	1	2	yes	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763214	C₂²	1	1	yes	characteristic
17	#11763149	—	2	2	no	—
18	#11763121	—	2	2	no	—
19	#11763094	—	2	1	no	—
20	#11763089	—	2	1	no	—
21	#11763047	C₂²	1	2	yes	—
22	#11763042	C₂²	1	2	yes	—
23	C₈×D₆₈	C₂²	1	1	yes	characteristic
24	#11762846	C₂²	1	2	yes	—
25	#107×D₁₇	C₂²	1	2	yes	—
26	#11762636	C₂²	1	2	yes	—
27	#11762635	C₂²	1	2	yes	—
28	#11762635	C₂²	1	2	yes	—
29	#11762634	C₂²	1	2	yes	—
30	C₄⋊#63214	C₂²	1	1	yes	characteristic
31	#11762613	C₂²	1	2	yes	—
32	#11762610	C₂²	1	2	yes	—
33	#11762588	C₂²	1	2	yes	—
34	C₂×C₁₇×SD₃₂	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
37	C₁₇×#367	C₂²	1	1	yes	characteristic
38	C₁₇×#358	C₂²	1	1	yes	characteristic
39	C₁₇×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×#63252	—	4	1	no	—
42	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₇	—	2	1	no	—
45	C₄×D₄×C₁₇	—	2	1	no	—
46	#10587632	—	2	2	no	—
47	#10587624	—	2	2	no	—
48	#10587610	—	4	1	no	—
49	C₂×C₈×D₁₇	C₂³	1	2	yes	—
50	#10587584	—	4	2	no	—
51	#10587578	—	2	2	no	—
52	C₄⋊C₄×D₁₇	—	2	2	no	—
53	#10587569	—	4	2	no	—
54	#10587561	—	4	1	no	—
55	C₄×D₆₈	D₄	1	1	yes	characteristic
56	#10587555	—	2	1	no	—
57	#10587555	—	2	1	no	—
58	C₄×Dic₃₄	—	4	1	no	—
59	C₁₇×SD₃₂	—	2	2	no	—
60	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
62	C₁₇×#107	C₂³	1	1	yes	characteristic
63	C₁₇×#107	C₂³	1	1	yes	characteristic
64	C₁₇×#103	—	2	1	no	—
65	C₁₇×#102	—	2	1	no	—
66	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	#10587506	—	2	2	no	—
68	#10587503	—	2	2	no	—
69	#10587491	D₄	1	2	yes	—
70	#10587489	—	2	1	no	—
71	#10587489	—	2	1	no	—
72	#10587489	C₂³	1	2	yes	—
73	#10587489	C₂³	1	2	yes	—
74	C₂×#63214	C₂³	1	2	yes	—
75	#10587479	—	2	2	no	—
76	#10587478	—	2	2	no	—
77	C₄⋊#3622	D₄	1	1	yes	characteristic
78	#10587469	—	2	4	no	—
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×D₄×C₁₇	—	2	1	no	—
81	C₂²×C₄×C₁₇	—	4	1	no	—
82	C₂×C₂²⋊D₁₇	—	4	2	no	—
83	#63252	—	8	4	no	—
84	C₂×D₆₈	D₈	1	1	yes	characteristic
85	C₂×C₄×D₁₇	—	4	2	no	—
86	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
87	C₂×Dic₃₄	—	4	1	no	—
88	C₂×Dic₃₄	—	4	1	no	—
89	Q₁₆×C₁₇	—	2	1	no	—
90	Q₁₆×C₁₇	—	2	1	no	—
91	D₈×C₁₇	—	2	1	no	—
92	D₈×C₁₇	—	2	1	no	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₁₇	—	2	1	no	—
96	C₄⋊C₄×C₁₇	—	2	1	no	—
97	C₄⋊C₄×C₁₇	—	4	1	no	—
98	C₂²⋊C₄×C₁₇	—	4	1	no	—
99	C₄²×C₁₇	—	4	1	no	—
100	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₇	—	4	2	no	—
102	#63227	—	4	2	no	—
103	#63227	—	4	2	no	—
104	#63227	D₈	1	2	yes	—
105	C₄⋊Dic₁₇	—	2	2	no	—
106	C₄⋊Dic₁₇	—	2	2	no	—
107	C₄⋊Dic₁₇	—	4	1	no	—
108	C₄⋊Dic₁₇	—	4	1	no	—
109	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
110	#63225	—	4	2	no	—
111	#63225	—	4	2	no	—
112	C₄×Dic₁₇	—	4	2	no	—
113	C₂×#3622	C₂×D₄	1	2	yes	—
114	C₈×D₁₇	—	2	2	no	—
115	C₈×D₁₇	—	2	2	no	—
116	C₁₆×C₁₇	—	2	1	no	—
117	C₁₆×C₁₇	#41	1	2	yes	complemented
118	#63214	—	2	2	no	—
119	#33920	—	17	1	no	maximal, Sylow
120	C₂³×C₁₇	—	4	1	no	—
121	C₂²×D₁₇	C₂×D₈	1	2	yes	—
122	Q₈×C₁₇	—	4	1	no	—
123	Q₈×C₁₇	—	4	1	no	—
124	D₄×C₁₇	—	4	1	no	—
125	D₄×C₁₇	—	4	1	no	—
126	C₂×C₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	8	2	no	—
130	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₁₇	—	8	4	no	—
134	C₂×Dic₁₇	—	4	2	no	—
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
137	D₆₈	—	2	2	no	—
138	C₄×D₁₇	—	2	2	no	—
139	C₄×D₁₇	—	2	2	no	—
140	C₄×D₁₇	—	8	4	no	—
141	Dic₃₄	—	8	2	no	—
142	Dic₃₄	—	8	2	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
146	#3622	—	2	2	no	—
147	#3342	—	17	1	no	—
148	#3334	—	17	1	no	—
149	#2252	—	17	1	no	—
150	#2236	—	17	2	no	—
151	#2234	—	17	2	no	—
152	#2225	—	17	1	no	—
153	#2224	—	17	1	no	—
154	#2220	—	17	2	no	—
155	C₄×SD₃₂	D₁₇	1	1	yes	characteristic, complemented
156	#2169	—	17	1	no	—
157	#2159	—	17	2	no	—
158	C₂²×C₁₇	—	8	1	no	—
159	C₂²×C₁₇	—	8	1	no	—
160	C₂²×C₁₇	—	8	1	no	—
161	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₁₇	—	2	2	no	—
163	C₂×D₁₇	—	2	2	no	—
164	C₂×D₁₇	—	2	2	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	8	1	no	—
167	C₄×C₁₇	—	8	1	no	—
168	C₄×C₁₇	—	8	1	no	—
169	C₄×C₁₇	#509	1	2	yes	—
170	C₄×C₁₇	#547	1	1	yes	characteristic
171	C₄×C₁₇	#578	1	1	yes	characteristic
172	Dic₁₇	—	2	2	no	—
173	Dic₁₇	—	8	2	no	—
174	Dic₁₇	—	8	2	no	—
175	#556	—	34	1	no	—
176	#547	—	34	1	no	—
177	C₂×SD₃₂	C₂×D₁₇	1	1	yes	characteristic
178	C₂×M₃₂	—	17	2	no	—
179	#500	—	17	1	no	—
180	#500	—	17	1	no	—
181	#485	—	34	2	no	—
182	#480	—	34	2	no	—
183	#475	—	34	1	no	—
184	#474	—	34	1	no	—
185	#467	—	17	2	no	—
186	#466	—	17	2	no	—
187	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
188	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
189	C₈×D₄	—	17	1	no	—
190	#427	—	17	2	no	—
191	C₂×#107	—	17	2	no	—
192	#367	—	17	2	no	—
193	#367	C₂×D₁₇	1	1	yes	characteristic
194	#366	—	17	2	no	—
195	#366	—	17	2	no	—
196	#366	—	17	2	no	—
197	C₄⋊C₁₆	—	17	1	no	—
198	#358	—	17	2	no	—
199	#358	C₂×D₁₇	1	1	yes	characteristic
200	#357	—	17	2	no	—
201	#357	C₂×D₁₇	1	1	yes	characteristic
202	C₂²⋊C₁₆	—	17	2	no	—
203	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
204	C₂×C₁₇	—	16	1	no	—
205	C₂×C₁₇	#3334	1	1	yes	characteristic
206	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
207	C₂×C₁₇	#3417	1	1	yes	characteristic
208	D₁₇	—	4	2	no	—
209	C₂×#41	—	68	1	no	—
210	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
211	C₂×D₈	C₂²×D₁₇	1	1	yes	characteristic
212	C₂²×C₈	—	17	2	no	—
213	C₄⋊Q₈	—	34	1	no	—
214	C₄⋊Q₈	—	34	1	no	—
215	#126	—	68	2	no	—
216	#124	—	68	1	no	—
217	#123	—	68	2	no	—
218	C₂²⋊Q₈	—	34	2	no	—
219	C₂²⋊Q₈	—	68	1	no	—
220	#121	—	34	2	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	68	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	—	17	1	no	—
225	#117	—	34	2	no	—
226	C₂×C₄⋊C₄	—	34	2	no	—
227	SD₃₂	—	2	2	no	—
228	M₃₂	—	34	4	no	—
229	C₂×C₁₆	—	17	2	no	—
230	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
232	#107	—	17	2	no	—
233	#107	—	17	2	no	—
234	#107	—	34	2	no	—
235	#107	—	34	1	no	—
236	#107	—	34	1	no	—
237	#107	C₂²×D₁₇	1	1	yes	characteristic
238	#107	C₂²×D₁₇	1	1	yes	characteristic
239	#106	—	34	2	no	—
240	C₄⋊C₈	—	17	1	no	—
241	#103	—	2	1	no	—
242	#103	—	34	2	no	—
243	#102	—	2	1	no	—
244	#102	—	34	2	no	—
245	C₂²⋊C₈	—	17	2	no	—
246	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₁₇	#33920	1	1	yes	characteristic, complemented, Sylow
248	#41	—	136	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	68	1	no	—
251	C₂×Q₈	—	68	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	17	1	no	—
254	C₂×D₄	—	68	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	17	2	no	—
257	C₂²×C₄	—	68	2	no	—
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	17	2	no	—
263	C₂×C₈	—	34	2	no	—
264	C₂×C₈	—	34	2	no	—
265	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	17	1	no	—
271	C₄⋊C₄	—	34	2	no	—
272	C₄⋊C₄	—	34	2	no	—
273	C₄⋊C₄	—	68	2	no	—
274	C₄⋊C₄	—	68	2	no	—
275	C₄⋊C₄	—	68	1	no	—
276	C₄⋊C₄	—	68	1	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	17	2	no	—
279	C₂²⋊C₄	—	68	2	no	—
280	C₂²⋊C₄	—	68	2	no	—
281	C₂²⋊C₄	—	68	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	68	2	no	—
284	C₄²	D₄×D₁₇	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	34	2	no	—
287	C₁₆	#63252	1	2	yes	complemented
288	C₂³	—	4	1	no	—
289	C₂³	—	17	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	136	2	no	—
293	Q₈	—	136	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	34	2	no	—
297	D₄	—	136	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	17	2	no	—
303	C₂×C₄	—	34	2	no	—
304	C₂×C₄	—	34	2	no	—
305	C₂×C₄	—	68	2	no	—
306	C₂×C₄	—	68	2	no	—
307	C₂×C₄	—	136	4	no	—
308	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
309	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	34	2	no	—
313	C₈	C₂×#63252	1	1	yes	characteristic
314	C₈	#10587697	1	1	yes	characteristic
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	34	2	no	—
319	C₂²	—	34	2	no	—
320	C₂²	—	34	2	no	—
321	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	34	2	no	—
327	C₄	—	136	2	no	—
328	C₄	—	136	2	no	—
329	C₄	#11763246	1	2	yes	—
330	C₄	#11763464	1	1	yes	characteristic
331	C₄	#11763623	1	1	yes	characteristic
332	C₂	—	16	1	no	—
333	C₂	—	68	2	no	—
334	C₂	#14541133	1	1	yes	characteristic
335	C₂	C₂×#11763246	1	1	yes	characteristic
336	C₂	#14541934	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₄×C₁₇×SD₃₂	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	#14541133	C₂
hypercenter	C₄×SD₃₂	D₁₇
hypocenter	C₁₇	#33920
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #14541216	C₂	semidirect
3. #14531059	C₂	semidirect
4. #14530985	C₂	semidirect
5. #14530973	C₂	semidirect
6. #14530926	C₂	semidirect
7. #14530921	C₂	semidirect
8. #14530894	C₂	semidirect
9. #14530617	C₂	semidirect
10. #14530571	C₂	semidirect
11. C₄×C₁₇×SD₃₂	C₂	semidirect
117. C₁₆×C₁₇	#41	semidirect
155. C₄×SD₃₂	D₁₇	semidirect
247. C₁₇	#33920	semidirect
287. C₁₆	#63252	semidirect

Maximal subgroups

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

subgroup	index
#14541216	2
#14541133	2
#14531059	2
#14530985	2
#14530973	2
#14530926	2
#14530921	2
#14530894	2
#14530617	2
#14530571	2
C₄×C₁₇×SD₃₂	2
#33920	17