Subgroups — #17191918

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	#14541215	C₂	1	1	yes	maximal, characteristic, complemented
2	#14531057	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530986	C₂	1	2	yes	maximal, complemented
4	#14530972	C₂	1	2	yes	maximal, complemented
5	#14530923	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530921	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530895	C₂	1	2	yes	maximal, complemented
8	C₄×#10587498	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×#3334	C₂	1	1	yes	maximal, characteristic
12	#11763529	—	2	1	no	—
13	C₂×#10587498	C₂²	1	1	yes	characteristic
14	C₂×#10587483	C₂²	1	2	yes	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763151	—	2	2	no	—
17	#11763095	—	2	1	no	—
18	#11763050	C₂²	1	2	yes	—
19	C₄×Dic₆₈	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	#11762612	C₂²	1	1	yes	characteristic
28	#11762612	C₂²	1	2	yes	—
29	#11762610	C₂²	1	1	yes	characteristic
30	#11762610	C₂²	1	2	yes	—
31	C₂²⋊#63214	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	C₁₇×#547	—	2	1	no	—
34	C₁₇×#500	C₂²	1	1	yes	characteristic
35	C₁₇×#480	—	2	2	no	—
36	C₁₇×#474	—	2	1	no	—
37	C₁₇×#466	C₂²	1	2	yes	—
38	C₄×D₈×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×#41×C₁₇	—	4	1	no	—
42	C₂×D₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
44	C₁₇×C₄⋊Q₈	—	2	1	no	—
45	C₁₇×#123	—	4	2	no	—
46	C₁₇×C₂²⋊Q₈	—	4	1	no	—
47	C₁₇×#121	—	2	2	no	—
48	C₄×D₄×C₁₇	—	2	1	no	—
49	C₄×D₄×C₁₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
51	#10587622	—	4	1	no	—
52	#10587612	—	2	2	no	—
53	#10587610	—	2	2	no	—
54	C₂×Dic₆₈	C₂³	1	1	yes	characteristic
55	#10587573	—	4	1	no	—
56	#10587565	—	4	2	no	—
57	#10587555	—	2	1	no	—
58	C₄×Dic₃₄	—	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×#102	—	2	1	no	—
65	C₁₇×#102	—	2	2	no	—
66	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
67	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	#10587498	—	2	2	no	—
69	#10587489	—	2	1	no	—
70	#10587489	C₂³	1	2	yes	—
71	#10587489	C₂³	1	1	yes	characteristic
72	#10587488	—	2	2	no	—
73	#10587487	—	2	2	no	—
74	#10587487	—	2	1	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	#41×C₁₇	—	8	4	no	—
80	C₂×Q₈×C₁₇	—	4	1	no	—
81	C₂×D₄×C₁₇	—	2	1	no	—
82	C₂×D₄×C₁₇	—	4	2	no	—
83	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	—	4	2	no	—
85	C₂²×C₄×C₁₇	—	4	1	no	—
86	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
87	C₂×Dic₃₄	—	2	1	no	—
88	C₂×Dic₃₄	—	4	1	no	—
89	D₈×C₁₇	—	2	1	no	—
90	D₈×C₁₇	—	2	1	no	—
91	C₂×C₈×C₁₇	—	2	2	no	—
92	C₂×C₈×C₁₇	—	2	2	no	—
93	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₁₇	—	2	2	no	—
97	C₄⋊C₄×C₁₇	—	2	1	no	—
98	C₄⋊C₄×C₁₇	—	4	2	no	—
99	C₄⋊C₄×C₁₇	—	4	1	no	—
100	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₇	—	4	2	no	—
102	C₂²⋊C₄×C₁₇	—	4	1	no	—
103	C₂²⋊C₄×C₁₇	D₈	1	2	yes	—
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₂²⋊Dic₁₇	—	4	2	no	—
106	C₂²⋊Dic₁₇	—	4	2	no	—
107	C₄⋊Dic₁₇	—	2	1	no	—
108	C₄⋊Dic₁₇	—	2	2	no	—
109	C₄⋊Dic₁₇	—	4	1	no	—
110	#63225	—	4	1	no	—
111	#63225	—	4	2	no	—
112	C₄×Dic₁₇	—	4	1	no	—
113	C₄×Dic₁₇	—	4	2	no	—
114	Dic₆₈	—	2	1	no	—
115	Dic₆₈	—	2	1	no	—
116	#63214	—	2	2	no	—
117	#63214	—	2	1	no	—
118	#63214	#41	1	2	yes	complemented
119	#33920	—	17	1	no	maximal, Sylow
120	C₂³×C₁₇	—	4	1	no	—
121	C₂³×C₁₇	C₂×D₈	1	2	yes	—
122	Q₈×C₁₇	—	8	2	no	—
123	D₄×C₁₇	—	2	2	no	—
124	D₄×C₁₇	—	4	1	no	—
125	D₄×C₁₇	—	4	1	no	—
126	D₄×C₁₇	—	8	4	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	2	2	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	—	8	4	no	—
132	C₂×C₄×C₁₇	—	8	2	no	—
133	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	—	4	1	no	—
138	C₂×Dic₁₇	—	4	2	no	—
139	C₂×Dic₁₇	—	4	1	no	—
140	Dic₃₄	—	4	1	no	—
141	Dic₃₄	—	4	1	no	—
142	Dic₃₄	—	8	2	no	—
143	C₈×C₁₇	—	2	2	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
147	#3342	—	17	1	no	—
148	#3334	D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
156	#2169	—	17	1	no	—
157	#2159	—	17	2	no	—
158	C₂²×C₁₇	—	2	2	no	—
159	C₂²×C₁₇	—	2	2	no	—
160	C₂²×C₁₇	—	2	2	no	—
161	C₂²×C₁₇	—	8	1	no	—
162	C₂²×C₁₇	—	8	1	no	—
163	C₂²×C₁₇	—	8	1	no	—
164	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₁₇	—	2	2	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	8	1	no	—
168	C₄×C₁₇	—	8	2	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	—	8	1	no	—
173	Dic₁₇	—	8	2	no	—
174	Dic₁₇	—	8	1	no	—
175	#556	—	34	1	no	—
176	#547	—	2	1	no	—
177	C₂×SD₃₂	—	17	1	no	—
178	C₂×M₃₂	—	17	2	no	—
179	#500	—	17	1	no	—
180	#500	C₂×D₁₇	1	1	yes	characteristic
181	#485	—	34	2	no	—
182	#480	—	2	2	no	—
183	#475	—	34	1	no	—
184	#474	—	2	1	no	—
185	#467	—	17	2	no	—
186	#466	C₂×D₁₇	1	2	yes	—
187	C₄×Q₁₆	—	17	1	no	—
188	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
189	C₈×D₄	C₂×D₁₇	1	1	yes	characteristic
190	#427	—	17	2	no	—
191	C₂×#107	C₂×D₁₇	1	2	yes	—
192	#367	—	17	1	no	—
193	#367	—	17	2	no	—
194	#366	—	17	2	no	—
195	#366	—	17	2	no	—
196	#366	—	17	2	no	—
197	C₄⋊C₁₆	—	17	1	no	—
198	#358	—	17	1	no	—
199	#358	—	17	2	no	—
200	#357	—	17	1	no	—
201	#357	—	17	2	no	—
202	C₂²⋊C₁₆	—	17	2	no	—
203	C₄×C₁₆	—	17	1	no	—
204	C₂×C₁₇	—	4	2	no	—
205	C₂×C₁₇	—	16	1	no	—
206	C₂×C₁₇	#3334	1	1	yes	characteristic
207	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
208	C₂×C₁₇	#3417	1	1	yes	characteristic
209	C₂×#41	—	4	1	no	—
210	C₂×Q₁₆	—	17	1	no	—
211	C₂×D₈	C₂²×D₁₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	34	1	no	—
215	#126	—	68	2	no	—
216	#124	—	68	1	no	—
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	4	1	no	—
219	C₂²⋊Q₈	—	34	2	no	—
220	#121	—	2	2	no	—
221	C₄×Q₈	—	34	1	no	—
222	C₄×Q₈	—	68	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
225	#117	—	34	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	SD₃₂	—	34	2	no	—
228	M₃₂	—	34	4	no	—
229	C₂×C₁₆	—	17	1	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	2	no	—
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	—	17	2	no	—
235	#107	—	17	1	no	—
236	#107	—	34	1	no	—
237	#107	C₂²×D₁₇	1	1	yes	characteristic
238	#107	C₂²×D₁₇	1	2	yes	—
239	#106	—	34	2	no	—
240	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
241	#103	—	34	1	no	—
242	#103	—	34	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	2	no	—
245	C₂²⋊C₈	C₂²⋊D₁₇	1	2	yes	—
246	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₁₇	#33920	1	1	yes	characteristic, complemented, Sylow
248	#41	—	8	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	34	1	no	—
251	C₂×Q₈	—	68	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	#63228	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
258	Q₁₆	—	34	1	no	—
259	Q₁₆	—	34	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
266	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	2	no	—
273	C₄⋊C₄	—	68	1	no	—
274	C₄⋊C₄	—	68	2	no	—
275	C₄⋊C₄	—	68	1	no	—
276	C₄⋊C₄	#63228	1	1	yes	characteristic
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	68	2	no	—
280	C₂²⋊C₄	—	68	2	no	—
281	C₂²⋊C₄	#63228	1	2	yes	—
282	C₄²	—	68	2	no	—
283	C₄²	—	68	1	no	—
284	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
285	C₁₆	—	17	2	no	—
286	C₁₆	—	34	2	no	—
287	C₁₆	—	34	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	C₂×#63228	1	2	yes	—
290	Q₈	—	8	2	no	—
291	Q₈	—	68	1	no	—
292	Q₈	—	68	1	no	—
293	Q₈	—	136	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	68	1	no	—
305	C₂×C₄	—	68	2	no	—
306	C₂×C₄	—	68	1	no	—
307	C₂×C₄	C₂×#63228	1	2	yes	—
308	C₂×C₄	C₂×#63228	1	1	yes	characteristic
309	C₂×C₄	C₂×#63228	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	#41×D₁₇	1	1	yes	characteristic
314	C₈	#10587697	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×#63228	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	136	1	no	—
327	C₄	—	136	2	no	—
328	C₄	—	136	1	no	—
329	C₄	#11763245	1	2	yes	—
330	C₄	#11763458	1	1	yes	characteristic
331	C₄	#11763624	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	16	1	no	—
334	C₂	#14541127	1	1	yes	characteristic
335	C₂	C₂×#11763245	1	1	yes	characteristic
336	C₂	#14541935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting, prime radical	C₁₇×#3334	C₂
socle	C₂²×C₁₇	C₂²×D₈
hypercenter	#3334	D₁₇
hypocenter	C₁₇	#33920
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #14541215	C₂	semidirect
2. #14531057	C₂	semidirect
3. #14530986	C₂	semidirect
4. #14530972	C₂	semidirect
5. #14530923	C₂	semidirect
6. #14530921	C₂	semidirect
7. #14530895	C₂	semidirect
8. C₄×#10587498	C₂	semidirect
9. #14530616	C₂	semidirect
10. #14530570	C₂	semidirect
118. #63214	#41	semidirect
247. C₁₇	#33920	semidirect

Maximal subgroups

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

subgroup	index
#14541215	2
#14531057	2
#14530986	2
#14530972	2
#14530923	2
#14530921	2
#14530895	2
C₄×#10587498	2
#14530616	2
#14530570	2
C₁₇×#3334	2
#33920	17