Subgroups — #17192061

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541758	C₂	1	1	yes	maximal, characteristic
2	#14541673	C₂	1	1	yes	maximal, characteristic, complemented
3	#14531024	C₂	1	2	yes	maximal, complemented
4	#14531008	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530940	C₂	1	2	yes	maximal, complemented
6	#14530921	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530824	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530822	C₂	1	2	yes	maximal, complemented
9	#14530692	C₂	1	1	yes	maximal, characteristic, complemented
10	#14530651	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₇×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763538	—	2	1	no	—
13	#11763480	—	2	1	no	—
14	C₂×#10587499	C₂²	1	2	yes	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763210	C₂²	1	2	yes	—
17	#11763186	C₂²	1	1	yes	characteristic
18	#11763184	C₂²	1	2	yes	—
19	#11763118	—	2	2	no	—
20	#11763094	—	2	1	no	—
21	#11763004	—	2	1	no	—
22	#11763003	—	2	2	no	—
23	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₁₇	C₂²	1	2	yes	—
25	#11762891	C₂²	1	2	yes	—
26	#11762634	C₂²	1	2	yes	—
27	C₄⋊#63214	C₂²	1	1	yes	characteristic
28	#11762627	C₂²	1	2	yes	—
29	#11762611	C₂²	1	2	yes	—
30	#11762611	C₂²	1	2	yes	—
31	#11762610	C₂²	1	2	yes	—
32	#11762609	C₂²	1	2	yes	—
33	#11762584	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	Q₈×Dic₁₇	—	2	2	no	—
47	#10587626	—	2	2	no	—
48	C₂×#63228	—	2	2	no	—
49	#10587616	—	4	1	no	—
50	C₂×D₁₃₆	—	2	1	no	—
51	C₂×D₁₃₆	C₂³	1	2	yes	—
52	#10587579	—	2	2	no	—
53	#10587576	—	2	2	no	—
54	#10587576	—	4	2	no	—
55	#10587570	—	4	2	no	—
56	#10587560	—	2	1	no	—
57	#10587555	—	2	1	no	—
58	#10587555	—	4	1	no	—
59	C₄×Dic₃₄	—	4	1	no	—
60	C₄×Dic₃₄	D₄	1	1	yes	characteristic
61	C₁₇×SD₃₂	—	2	2	no	—
62	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
64	C₁₇×#107	C₂³	1	1	yes	characteristic
65	C₁₇×#107	C₂³	1	1	yes	characteristic
66	C₁₇×#103	—	2	1	no	—
67	C₁₇×#102	—	2	1	no	—
68	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	#10587506	—	2	2	no	—
70	#10587499	—	2	4	no	—
71	#10587489	—	2	1	no	—
72	#10587489	C₂³	1	2	yes	—
73	#10587485	D₄	1	2	yes	—
74	C₈×Dic₁₇	C₂³	1	2	yes	—
75	C₂×#63214	C₂³	1	2	yes	—
76	#10587480	—	2	2	no	—
77	#10587479	—	2	2	no	—
78	C₄⋊#3622	D₄	1	1	yes	characteristic
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	C₂×C₂²⋊D₁₇	—	4	2	no	—
84	#63252	—	8	4	no	—
85	C₂×D₆₈	—	2	2	no	—
86	C₂×D₆₈	—	4	1	no	—
87	C₂×D₆₈	—	4	1	no	—
88	C₂×C₄×D₁₇	—	4	2	no	—
89	C₂×Dic₃₄	—	4	1	no	—
90	C₂×Dic₃₄	D₈	1	1	yes	characteristic
91	Q₁₆×C₁₇	—	2	1	no	—
92	Q₁₆×C₁₇	#41	1	2	yes	complemented
93	D₈×C₁₇	—	2	1	no	—
94	D₈×C₁₇	—	2	1	no	—
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₁₇	—	2	1	no	—
98	C₄⋊C₄×C₁₇	—	2	1	no	—
99	C₄⋊C₄×C₁₇	—	4	1	no	—
100	C₂²⋊C₄×C₁₇	—	4	1	no	—
101	C₄²×C₁₇	—	4	1	no	—
102	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
103	#63228	—	8	2	no	—
104	#63227	—	4	2	no	—
105	C₄⋊Dic₁₇	—	2	2	no	—
106	C₄⋊Dic₁₇	—	4	1	no	—
107	C₄⋊Dic₁₇	—	4	2	no	—
108	C₄⋊Dic₁₇	—	4	2	no	—
109	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
110	#63225	—	4	2	no	—
111	#63225	D₈	1	2	yes	—
112	C₄×Dic₁₇	—	4	2	no	—
113	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
114	C₂×#3622	C₂×D₄	1	2	yes	—
115	D₁₃₆	—	2	2	no	—
116	D₁₃₆	—	2	2	no	—
117	D₁₃₆	—	8	1	no	—
118	C₁₆×C₁₇	—	2	1	no	—
119	C₁₆×C₁₇	—	2	1	no	—
120	#63214	—	2	2	no	—
121	#33935	—	17	1	no	maximal, Sylow
122	C₂³×C₁₇	—	4	1	no	—
123	C₂²×D₁₇	—	4	2	no	—
124	Q₈×C₁₇	—	2	2	no	—
125	Q₈×C₁₇	—	4	1	no	—
126	D₄×C₁₇	—	4	1	no	—
127	D₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	1	no	—
131	C₂×C₄×C₁₇	—	8	2	no	—
132	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₁₇	—	8	4	no	—
136	C₂²⋊D₁₇	—	16	2	no	—
137	C₂×Dic₁₇	—	4	2	no	—
138	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
139	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
140	D₆₈	—	4	2	no	—
141	D₆₈	—	4	2	no	—
142	D₆₈	—	8	2	no	—
143	D₆₈	—	16	1	no	—
144	C₄×D₁₇	—	8	4	no	—
145	Dic₃₄	—	2	2	no	—
146	Dic₃₄	—	8	2	no	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
150	#3622	—	2	2	no	—
151	#3391	—	17	1	no	—
152	#3383	—	17	1	no	—
153	#2244	—	17	2	no	—
154	#2240	—	17	1	no	—
155	#2228	—	17	2	no	—
156	#2224	—	17	1	no	—
157	#2207	—	17	2	no	—
158	#2207	—	17	1	no	—
159	#2183	—	17	1	no	—
160	#2175	—	17	2	no	—
161	C₄×SD₃₂	D₁₇	1	1	yes	characteristic, complemented
162	C₂²×C₁₇	—	8	1	no	—
163	C₂²×C₁₇	—	8	1	no	—
164	C₂²×C₁₇	—	8	1	no	—
165	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₁₇	—	8	2	no	—
167	C₂×D₁₇	—	8	2	no	—
168	C₂×D₁₇	—	8	2	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	4	2	no	—
171	C₄×C₁₇	—	8	1	no	—
172	C₄×C₁₇	—	8	1	no	—
173	C₄×C₁₇	#509	1	2	yes	—
174	C₄×C₁₇	#548	1	1	yes	characteristic
175	C₄×C₁₇	#577	1	1	yes	characteristic
176	Dic₁₇	—	2	2	no	—
177	Dic₁₇	—	2	2	no	—
178	Dic₁₇	—	8	2	no	—
179	#557	—	34	1	no	—
180	#550	—	34	1	no	—
181	C₂×SD₃₂	—	17	2	no	—
182	C₂×SD₃₂	C₂×D₁₇	1	1	yes	characteristic
183	#500	—	17	1	no	—
184	#499	—	17	2	no	—
185	#493	—	17	2	no	—
186	#493	—	17	1	no	—
187	#479	—	34	2	no	—
188	#475	—	34	1	no	—
189	#459	—	34	2	no	—
190	#459	—	34	1	no	—
191	C₈×Q₈	—	17	1	no	—
192	C₄×Q₁₆	—	17	2	no	—
193	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
194	C₄×D₈	—	17	2	no	—
195	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
196	#367	C₂×D₁₇	1	1	yes	characteristic
197	#366	—	17	2	no	—
198	C₄⋊C₁₆	—	17	2	no	—
199	C₄⋊C₁₆	—	17	1	no	—
200	#358	C₂×D₁₇	1	1	yes	characteristic
201	#357	—	17	2	no	—
202	#357	—	17	2	no	—
203	#357	—	17	2	no	—
204	#357	—	17	2	no	—
205	#357	C₂×D₁₇	1	1	yes	characteristic
206	#346	—	17	2	no	—
207	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
208	C₂×C₁₇	—	16	1	no	—
209	C₂×C₁₇	#3358	1	1	yes	characteristic
210	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
211	C₂×C₁₇	#3416	1	1	yes	characteristic
212	D₁₇	—	16	2	no	—
213	C₂×#41	—	68	1	no	—
214	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
215	C₂×D₈	—	17	2	no	—
216	C₂×D₈	—	34	2	no	—
217	C₂×D₈	—	34	1	no	—
218	C₂×D₈	C₂²×D₁₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	34	1	no	—
220	C₄⋊Q₈	—	68	1	no	—
221	#127	—	34	2	no	—
222	#127	—	34	1	no	—
223	#125	—	34	2	no	—
224	#125	—	68	2	no	—
225	#121	—	68	2	no	—
226	#121	—	68	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	17	1	no	—
229	C₄×Q₈	—	34	2	no	—
230	C₄×Q₈	—	68	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	34	2	no	—
233	SD₃₂	—	2	2	no	—
234	SD₃₂	—	34	4	no	—
235	C₂×C₁₆	—	17	2	no	—
236	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
238	#107	—	17	2	no	—
239	#107	—	34	1	no	—
240	#107	C₂²×D₁₇	1	1	yes	characteristic
241	#107	C₂²×D₁₇	1	1	yes	characteristic
242	#106	—	34	2	no	—
243	C₄⋊C₈	—	17	2	no	—
244	C₄⋊C₈	—	17	1	no	—
245	#103	—	2	1	no	—
246	#103	—	34	2	no	—
247	#102	—	2	1	no	—
248	#102	—	34	2	no	—
249	C₄×C₈	—	17	2	no	—
250	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
251	C₁₇	#33935	1	1	yes	characteristic, complemented, Sylow
252	#41	—	136	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	17	1	no	—
255	C₂×Q₈	—	68	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	34	2	no	—
258	C₂×D₄	—	68	2	no	—
259	C₂×D₄	—	68	2	no	—
260	C₂×D₄	—	68	1	no	—
261	C₂×D₄	—	68	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	68	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	#63252	1	2	yes	complemented
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	34	2	no	—
269	D₈	—	34	2	no	—
270	D₈	—	136	1	no	—
271	D₈	—	136	2	no	—
272	C₂×C₈	—	17	2	no	—
273	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	17	1	no	—
279	C₄⋊C₄	—	17	2	no	—
280	C₄⋊C₄	—	34	2	no	—
281	C₄⋊C₄	—	68	2	no	—
282	C₄⋊C₄	—	68	2	no	—
283	C₄⋊C₄	—	68	2	no	—
284	C₄⋊C₄	—	68	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	68	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	17	2	no	—
289	C₄²	—	68	2	no	—
290	C₄²	D₄×D₁₇	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	34	2	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	68	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	34	2	no	—
299	Q₈	—	136	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	272	1	no	—
303	D₄	—	272	2	no	—
304	D₄	—	68	2	no	—
305	D₄	—	68	2	no	—
306	D₄	—	136	4	no	—
307	D₄	—	136	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	17	2	no	—
313	C₂×C₄	—	17	2	no	—
314	C₂×C₄	—	68	2	no	—
315	C₂×C₄	—	136	4	no	—
316	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
317	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	34	2	no	—
321	C₈	C₂×#63252	1	1	yes	characteristic
322	C₈	#10587697	1	1	yes	characteristic
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	136	2	no	—
327	C₂²	—	136	2	no	—
328	C₂²	—	136	2	no	—
329	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	34	2	no	—
335	C₄	—	34	2	no	—
336	C₄	—	136	2	no	—
337	C₄	#11763246	1	2	yes	—
338	C₄	#11763468	1	1	yes	characteristic
339	C₄	#11763616	1	1	yes	characteristic
340	C₂	—	16	1	no	—
341	C₂	—	272	2	no	—
342	C₂	#14541417	1	1	yes	characteristic
343	C₂	C₂×#11763246	1	1	yes	characteristic
344	C₂	#14541927	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	#14541758	C₂
hypercenter	C₄×SD₃₂	D₁₇
hypocenter	C₁₇	#33935
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
2. #14541673	C₂	semidirect
3. #14531024	C₂	semidirect
4. #14531008	C₂	semidirect
5. #14530940	C₂	semidirect
6. #14530921	C₂	semidirect
7. #14530824	C₂	semidirect
8. #14530822	C₂	semidirect
9. #14530692	C₂	semidirect
10. #14530651	C₂	semidirect
11. C₄×C₁₇×SD₃₂	C₂	semidirect
92. Q₁₆×C₁₇	#41	semidirect
161. C₄×SD₃₂	D₁₇	semidirect
251. C₁₇	#33935	semidirect
265. Q₁₆	#63252	semidirect

Maximal subgroups

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

subgroup	index
#14541758	2
#14541673	2
#14531024	2
#14531008	2
#14530940	2
#14530921	2
#14530824	2
#14530822	2
#14530692	2
#14530651	2
C₄×C₁₇×SD₃₂	2
#33935	17