Subgroups — #17191960

Order 4352.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541207	C₂	1	1	yes	maximal, characteristic
2	#14531057	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530989	C₂	1	2	yes	maximal, complemented
4	#14530964	C₂	1	2	yes	maximal, complemented
5	#14530936	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530928	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530899	C₂	1	2	yes	maximal, complemented
8	C₄×#10587499	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530616	C₂	1	1	yes	maximal, characteristic, complemented
10	#14530570	C₂	1	2	yes	maximal, complemented
11	C₁₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763524	—	2	1	no	—
13	C₂×#10587499	C₂²	1	1	yes	characteristic
14	C₂×#10587483	C₂²	1	2	yes	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763134	—	2	2	no	—
17	#11763100	—	2	1	no	—
18	#11763045	C₂²	1	2	yes	—
19	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
20	#11762847	C₂²	1	2	yes	—
21	#11762636	C₂²	1	1	yes	characteristic
22	#11762636	C₂²	1	2	yes	—
23	#11762634	C₂²	1	2	yes	—
24	#11762634	C₂²	1	2	yes	—
25	#11762634	C₂²	1	2	yes	—
26	C₄⋊#63214	C₂²	1	1	yes	characteristic
27	#11762613	C₂²	1	1	yes	characteristic
28	#11762613	C₂²	1	2	yes	—
29	#11762609	C₂²	1	1	yes	characteristic
30	#11762609	C₂²	1	2	yes	—
31	C₂²⋊#63214	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	C₁₇×#554	—	2	1	no	—
34	C₁₇×#500	C₂²	1	1	yes	characteristic
35	C₁₇×#484	—	2	2	no	—
36	C₁₇×#477	—	2	1	no	—
37	C₁₇×#467	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
40	C₂×C₁₇×#107	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
43	C₁₇×C₄⋊Q₈	—	2	1	no	—
44	C₁₇×#125	—	4	1	no	—
45	C₁₇×C₂²⋊Q₈	—	2	2	no	—
46	C₁₇×C₂²⋊Q₈	—	4	2	no	—
47	C₄×Q₈×C₁₇	—	2	1	no	—
48	C₄×D₄×C₁₇	—	4	1	no	—
49	C₄×D₄×C₁₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
51	#10587624	—	4	1	no	—
52	#10587616	—	2	2	no	—
53	#10587612	—	2	2	no	—
54	C₂×D₁₃₆	C₂³	1	1	yes	characteristic
55	#10587578	—	4	1	no	—
56	#10587571	—	4	2	no	—
57	C₄×D₆₈	—	2	1	no	—
58	#10587555	—	2	1	no	—
59	C₁₇×#107	—	2	1	no	—
60	C₁₇×#107	—	2	2	no	—
61	C₁₇×#107	C₂³	1	2	yes	—
62	C₁₇×#107	C₂³	1	1	yes	characteristic
63	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₇×#103	—	2	1	no	—
65	C₁₇×#103	—	2	2	no	—
66	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
67	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	#10587499	—	2	2	no	—
69	#10587492	—	2	2	no	—
70	#10587492	—	2	1	no	—
71	#10587489	—	2	1	no	—
72	#10587489	C₂³	1	2	yes	—
73	#10587489	C₂³	1	1	yes	characteristic
74	#10587488	—	2	2	no	—
75	#10587483	—	2	4	no	—
76	C₂×#63214	C₂³	1	2	yes	—
77	C₂×#63214	C₂³	1	1	yes	characteristic
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	4	1	no	—
81	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₇	—	4	2	no	—
83	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
84	C₂×C₂²⋊D₁₇	—	4	2	no	—
85	C₂×D₆₈	—	2	1	no	—
86	C₂×C₄×D₁₇	—	4	1	no	—
87	C₂×Dic₃₄	—	4	1	no	—
88	Q₁₆×C₁₇	—	2	1	no	—
89	Q₁₆×C₁₇	—	2	1	no	—
90	C₂×C₈×C₁₇	—	2	2	no	—
91	C₂×C₈×C₁₇	—	2	2	no	—
92	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₁₇	—	2	2	no	—
96	C₄⋊C₄×C₁₇	—	2	1	no	—
97	C₄⋊C₄×C₁₇	—	4	2	no	—
98	C₄⋊C₄×C₁₇	—	4	1	no	—
99	C₄⋊C₄×C₁₇	—	4	2	no	—
100	C₄⋊C₄×C₁₇	—	4	1	no	—
101	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₇	—	4	2	no	—
103	C₂²⋊C₄×C₁₇	D₈	1	2	yes	—
104	C₄²×C₁₇	—	4	1	no	—
105	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₁₇	—	4	2	no	—
107	#63227	—	4	1	no	—
108	#63227	—	4	2	no	—
109	C₄⋊Dic₁₇	—	2	1	no	—
110	C₄⋊Dic₁₇	—	2	2	no	—
111	C₄⋊Dic₁₇	—	4	1	no	—
112	C₄×Dic₁₇	—	4	2	no	—
113	D₁₃₆	—	2	1	no	—
114	D₁₃₆	—	2	1	no	—
115	#63214	—	2	2	no	—
116	#63214	—	2	1	no	—
117	#63214	#41	1	2	yes	—
118	#33924	—	17	1	no	maximal, Sylow
119	C₂³×C₁₇	C₂×D₈	1	2	yes	—
120	C₂²×D₁₇	—	4	1	no	—
121	Q₈×C₁₇	—	4	1	no	—
122	Q₈×C₁₇	—	4	1	no	—
123	Q₈×C₁₇	—	16	1	no	—
124	D₄×C₁₇	—	2	2	no	—
125	C₂×C₄×C₁₇	—	2	2	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	2	no	—
130	C₂×C₄×C₁₇	—	16	2	no	—
131	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₁₇	—	16	2	no	—
136	C₂×Dic₁₇	—	4	1	no	—
137	C₂×Dic₁₇	—	4	2	no	—
138	D₆₈	—	4	1	no	—
139	D₆₈	—	4	1	no	—
140	C₄×D₁₇	—	8	2	no	—
141	Dic₃₄	—	16	1	no	—
142	C₈×C₁₇	—	2	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	#3341	—	17	1	no	—
147	#3338	D₁₇	1	1	yes	characteristic, complemented
148	#2252	—	17	1	no	—
149	#2237	—	17	2	no	—
150	#2232	—	17	2	no	—
151	#2227	—	17	1	no	—
152	#2226	—	17	1	no	—
153	#2220	—	17	2	no	—
154	C₄×SD₃₂	—	17	1	no	—
155	#2169	—	17	1	no	—
156	#2159	—	17	2	no	—
157	C₂²×C₁₇	—	2	2	no	—
158	C₂²×C₁₇	—	2	2	no	—
159	C₂²×C₁₇	—	2	2	no	—
160	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₁₇	—	8	1	no	—
162	C₂×D₁₇	—	8	1	no	—
163	C₂×D₁₇	—	8	1	no	—
164	C₄×C₁₇	—	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	2	no	—
169	C₄×C₁₇	#510	1	2	yes	—
170	C₄×C₁₇	#547	1	1	yes	characteristic
171	C₄×C₁₇	#578	1	1	yes	characteristic
172	Dic₁₇	—	8	1	no	—
173	Dic₁₇	—	8	2	no	—
174	#555	—	34	1	no	—
175	#554	—	2	1	no	—
176	C₂×SD₃₂	—	17	1	no	—
177	C₂×M₃₂	—	17	2	no	—
178	#500	—	17	1	no	—
179	#500	C₂×D₁₇	1	1	yes	characteristic
180	#484	—	2	2	no	—
181	#482	—	34	2	no	—
182	#477	—	2	1	no	—
183	#476	—	34	1	no	—
184	#467	C₂×D₁₇	1	2	yes	—
185	#466	—	17	2	no	—
186	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
187	C₄×D₈	—	17	1	no	—
188	C₈×D₄	C₂×D₁₇	1	1	yes	characteristic
189	#427	—	17	2	no	—
190	C₂×#107	C₂×D₁₇	1	2	yes	—
191	#367	—	17	1	no	—
192	#367	—	17	2	no	—
193	#366	—	17	2	no	—
194	#366	—	17	2	no	—
195	#366	—	17	2	no	—
196	C₄⋊C₁₆	—	17	1	no	—
197	#358	—	17	1	no	—
198	#358	—	17	2	no	—
199	#357	—	17	1	no	—
200	#357	—	17	2	no	—
201	C₂²⋊C₁₆	—	17	2	no	—
202	C₄×C₁₆	—	17	1	no	—
203	C₂×C₁₇	—	4	2	no	—
204	C₂×C₁₇	#3334	1	1	yes	characteristic
205	C₂×C₁₇	C₂×#510	1	1	yes	characteristic
206	C₂×C₁₇	#3415	1	1	yes	characteristic
207	D₁₇	—	16	1	no	—
208	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
209	C₂×D₈	—	17	1	no	—
210	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	34	1	no	—
213	#125	—	4	1	no	—
214	#124	—	68	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	#121	—	34	2	no	—
218	#121	—	68	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×D₄	—	4	1	no	—
221	C₄×D₄	—	34	1	no	—
222	C₄×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
223	#117	—	34	2	no	—
224	#117	—	68	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	SD₃₂	—	34	2	no	—
227	M₃₂	—	34	4	no	—
228	C₂×C₁₆	—	17	1	no	—
229	C₂×C₁₆	—	17	1	no	—
230	C₂×C₁₆	—	17	2	no	—
231	#107	—	2	2	no	—
232	#107	—	2	1	no	—
233	#107	—	17	2	no	—
234	#107	—	17	1	no	—
235	#107	—	34	1	no	—
236	#107	C₂²×D₁₇	1	1	yes	characteristic
237	#107	C₂²×D₁₇	1	2	yes	—
238	#106	—	34	2	no	—
239	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
240	#103	—	2	1	no	—
241	#103	—	2	2	no	—
242	#102	—	34	1	no	—
243	#102	—	34	2	no	—
244	C₂²⋊C₈	C₂²⋊D₁₇	1	2	yes	—
245	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₁₇	#33924	1	1	yes	characteristic, complemented, Sylow
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	68	1	no	—
250	C₂×D₄	—	34	1	no	—
251	C₂×D₄	—	68	2	no	—
252	C₂×D₄	#63228	1	1	yes	characteristic
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	68	1	no	—
255	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	D₈	—	34	1	no	—
259	D₈	—	34	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
264	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	2	no	—
273	C₄⋊C₄	—	68	1	no	—
274	C₄⋊C₄	#63228	1	1	yes	characteristic
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	68	2	no	—
277	C₂²⋊C₄	—	68	2	no	—
278	C₂²⋊C₄	—	68	1	no	—
279	C₂²⋊C₄	#63228	1	2	yes	—
280	C₄²	—	4	1	no	—
281	C₄²	—	68	2	no	—
282	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
283	C₁₆	—	17	2	no	—
284	C₁₆	—	34	2	no	—
285	C₁₆	—	34	1	no	—
286	C₂³	—	68	1	no	—
287	C₂³	C₂×#63228	1	2	yes	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	272	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	272	2	no	—
294	D₄	—	68	1	no	—
295	D₄	—	68	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	68	1	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	—	136	2	no	—
305	C₂×C₄	C₂×#63228	1	2	yes	—
306	C₂×C₄	C₂×#63228	1	1	yes	characteristic
307	C₂×C₄	C₂×#63228	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₁₇	1	1	yes	characteristic
312	C₈	#10587697	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	136	1	no	—
317	C₂²	—	136	1	no	—
318	C₂²	—	136	1	no	—
319	C₂²	C₂²×#63228	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	136	1	no	—
326	C₄	—	136	2	no	—
327	C₄	#11763250	1	2	yes	—
328	C₄	#11763458	1	1	yes	characteristic
329	C₄	#11763624	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	272	1	no	—
332	C₂	#14541127	1	1	yes	characteristic
333	C₂	C₂×#11763250	1	1	yes	characteristic
334	C₂	#14541923	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3338	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	#14541207	C₂
hypercenter	#3338	D₁₇
hypocenter	C₁₇	#33924
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3338

Complements

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

N	H ≅ G/N	type
2. #14531057	C₂	semidirect
3. #14530989	C₂	semidirect
4. #14530964	C₂	semidirect
5. #14530936	C₂	semidirect
6. #14530928	C₂	semidirect
7. #14530899	C₂	semidirect
8. C₄×#10587499	C₂	semidirect
9. #14530616	C₂	semidirect
10. #14530570	C₂	semidirect
11. C₁₇×#3338	C₂	semidirect
147. #3338	D₁₇	semidirect
246. C₁₇	#33924	semidirect

Maximal subgroups

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

subgroup	index
#14541207	2
#14531057	2
#14530989	2
#14530964	2
#14530936	2
#14530928	2
#14530899	2
C₄×#10587499	2
#14530616	2
#14530570	2
C₁₇×#3338	2
#33924	17