Subgroups — #17192063

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	#14541759	C₂	1	1	yes	maximal, characteristic
2	#14531027	C₂	1	2	yes	maximal, complemented
3	#14531009	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530938	C₂	1	2	yes	maximal, complemented
5	#14530918	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530824	C₂	1	2	yes	maximal, complemented
7	#14530824	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530693	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530646	C₂	1	2	yes	maximal, complemented
10	C₄×#10587499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763481	—	2	1	no	—
13	C₂×#10587499	C₂²	1	1	yes	characteristic
14	C₂×#10587499	C₂²	1	2	yes	—
15	#11763186	C₂²	1	1	yes	characteristic
16	#11763186	C₂²	1	2	yes	—
17	#11763005	—	2	1	no	—
18	#11763005	—	2	2	no	—
19	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
20	C₄×D₁₃₆	C₂²	1	2	yes	—
21	#11762636	C₂²	1	1	yes	characteristic
22	#11762634	C₂²	1	2	yes	—
23	C₄⋊#63214	C₂²	1	1	yes	characteristic
24	C₄⋊#63214	C₂²	1	2	yes	—
25	#11762613	C₂²	1	1	yes	characteristic
26	#11762609	C₂²	1	2	yes	—
27	#11762609	C₂²	1	2	yes	—
28	#11762609	C₂²	1	2	yes	—
29	#11762609	C₂²	1	1	yes	characteristic
30	#11762609	C₂²	1	2	yes	—
31	#11762585	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	C₁₇×#557	—	2	1	no	—
34	C₁₇×#500	C₂²	1	1	yes	characteristic
35	C₁₇×#499	C₂²	1	2	yes	—
36	C₁₇×#479	—	2	2	no	—
37	C₁₇×#475	—	2	1	no	—
38	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
41	C₂×#63256	—	4	1	no	—
42	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
43	C₁₇×C₄⋊Q₈	—	2	1	no	—
44	C₁₇×C₄⋊Q₈	—	4	1	no	—
45	C₁₇×#125	—	2	2	no	—
46	C₁₇×#125	—	4	2	no	—
47	C₄×Q₈×C₁₇	—	2	1	no	—
48	C₄×Q₈×C₁₇	—	2	2	no	—
49	C₄×Q₈×C₁₇	—	4	1	no	—
50	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
51	C₂×D₁₃₆	—	2	2	no	—
52	C₂×D₁₃₆	—	2	1	no	—
53	C₂×D₁₃₆	C₂³	1	1	yes	characteristic
54	C₂×D₁₃₆	C₂³	1	2	yes	—
55	#10587581	—	4	2	no	—
56	#10587581	—	4	1	no	—
57	#10587560	—	2	2	no	—
58	#10587560	—	2	1	no	—
59	C₄×D₆₈	—	2	2	no	—
60	C₄×D₆₈	—	2	1	no	—
61	C₁₇×#107	—	2	1	no	—
62	C₁₇×#107	C₂³	1	1	yes	characteristic
63	C₁₇×#107	C₂³	1	2	yes	—
64	C₁₇×#106	—	2	2	no	—
65	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₇×#103	—	2	2	no	—
68	C₁₇×#103	—	2	1	no	—
69	C₄×C₈×C₁₇	C₂³	1	2	yes	—
70	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	#10587499	—	2	2	no	—
72	#10587499	—	2	4	no	—
73	#10587492	—	2	2	no	—
74	#10587492	—	2	1	no	—
75	#10587489	C₂³	1	1	yes	characteristic
76	C₂×#63214	C₂³	1	1	yes	characteristic
77	C₂×#63214	C₂³	1	2	yes	—
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	4	1	no	—
81	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
82	#63256	—	8	4	no	—
83	C₂×D₆₈	—	2	2	no	—
84	C₂×D₆₈	—	2	1	no	—
85	C₂×D₆₈	—	4	2	no	—
86	C₂×D₆₈	—	4	1	no	—
87	C₂×D₆₈	—	4	2	no	—
88	C₂×D₆₈	—	4	1	no	—
89	C₂×C₄×D₁₇	—	4	2	no	—
90	C₂×C₄×D₁₇	—	4	1	no	—
91	Q₁₆×C₁₇	—	2	1	no	—
92	Q₁₆×C₁₇	#41	1	2	yes	complemented
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	1	no	—
97	C₄⋊C₄×C₁₇	—	2	2	no	—
98	C₄⋊C₄×C₁₇	—	4	2	no	—
99	C₄⋊C₄×C₁₇	—	4	2	no	—
100	C₄⋊C₄×C₁₇	—	4	1	no	—
101	C₄⋊C₄×C₁₇	—	4	1	no	—
102	C₄⋊C₄×C₁₇	—	4	2	no	—
103	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
104	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
105	C₄²×C₁₇	—	4	2	no	—
106	C₄²×C₁₇	—	4	1	no	—
107	C₄²×C₁₇	C₂×D₄	1	2	yes	—
108	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
109	#63227	—	4	2	no	—
110	#63227	—	4	1	no	—
111	C₄⋊Dic₁₇	—	2	1	no	—
112	D₁₃₆	—	2	1	no	—
113	D₁₃₆	—	2	2	no	—
114	D₁₃₆	—	2	1	no	—
115	D₁₃₆	—	2	2	no	—
116	D₁₃₆	—	8	2	no	—
117	D₁₃₆	—	8	1	no	—
118	#63214	—	2	1	no	—
119	#63214	—	2	2	no	—
120	#63214	—	2	1	no	—
121	#33935	—	17	1	no	maximal, Sylow
122	C₂²×D₁₇	—	4	2	no	—
123	C₂²×D₁₇	—	4	1	no	—
124	Q₈×C₁₇	—	2	2	no	—
125	Q₈×C₁₇	—	2	2	no	—
126	Q₈×C₁₇	—	4	1	no	—
127	Q₈×C₁₇	—	8	2	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	—	4	1	no	—
137	D₆₈	—	4	2	no	—
138	D₆₈	—	4	1	no	—
139	D₆₈	—	4	2	no	—
140	D₆₈	—	4	1	no	—
141	D₆₈	—	8	4	no	—
142	D₆₈	—	8	2	no	—
143	D₆₈	—	16	2	no	—
144	D₆₈	—	16	1	no	—
145	C₄×D₁₇	—	8	4	no	—
146	C₄×D₁₇	—	8	2	no	—
147	C₈×C₁₇	—	2	2	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
150	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
151	#3391	—	17	1	no	—
152	#3383	D₁₇	1	1	yes	characteristic, complemented
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	—	17	1	no	—
162	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₁₇	—	8	2	no	—
164	C₂×D₁₇	—	8	1	no	—
165	C₂×D₁₇	—	8	2	no	—
166	C₂×D₁₇	—	8	1	no	—
167	C₂×D₁₇	—	8	2	no	—
168	C₂×D₁₇	—	8	1	no	—
169	C₄×C₁₇	—	2	2	no	—
170	C₄×C₁₇	—	2	2	no	—
171	C₄×C₁₇	—	2	1	no	—
172	C₄×C₁₇	—	4	2	no	—
173	C₄×C₁₇	—	8	2	no	—
174	C₄×C₁₇	—	8	1	no	—
175	C₄×C₁₇	#509	1	2	yes	—
176	C₄×C₁₇	#548	1	1	yes	characteristic
177	C₄×C₁₇	#577	1	1	yes	characteristic
178	Dic₁₇	—	8	1	no	—
179	#557	—	2	1	no	—
180	#550	—	34	1	no	—
181	C₂×SD₃₂	—	17	1	no	—
182	C₂×SD₃₂	—	17	2	no	—
183	#500	C₂×D₁₇	1	1	yes	characteristic
184	#499	C₂×D₁₇	1	2	yes	—
185	#493	—	17	1	no	—
186	#493	—	17	2	no	—
187	#479	—	2	2	no	—
188	#475	—	2	1	no	—
189	#459	—	34	1	no	—
190	#459	—	34	2	no	—
191	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
192	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
193	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
194	C₄×D₈	—	17	1	no	—
195	C₄×D₈	—	17	2	no	—
196	#367	—	17	1	no	—
197	#366	—	17	2	no	—
198	C₄⋊C₁₆	—	17	1	no	—
199	C₄⋊C₁₆	—	17	2	no	—
200	#358	—	17	1	no	—
201	#357	—	17	1	no	—
202	#357	—	17	2	no	—
203	#357	—	17	2	no	—
204	#357	—	17	2	no	—
205	#357	—	17	2	no	—
206	#346	—	17	2	no	—
207	C₄×C₁₆	—	17	1	no	—
208	C₂×C₁₇	#3358	1	1	yes	characteristic
209	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
210	C₂×C₁₇	#3416	1	1	yes	characteristic
211	D₁₇	—	16	2	no	—
212	D₁₇	—	16	1	no	—
213	C₂×#41	—	68	1	no	—
214	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
215	C₂×D₈	—	17	1	no	—
216	C₂×D₈	—	17	2	no	—
217	C₂×D₈	—	34	2	no	—
218	C₂×D₈	—	34	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	4	1	no	—
221	#127	—	34	2	no	—
222	#127	—	34	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	2	no	—
225	#121	—	68	1	no	—
226	#121	—	68	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	4	1	no	—
230	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
231	C₄×D₄	—	34	2	no	—
232	C₄×D₄	—	34	1	no	—
233	SD₃₂	—	34	2	no	—
234	SD₃₂	—	34	4	no	—
235	C₂×C₁₆	—	17	1	no	—
236	C₂×C₁₆	—	17	1	no	—
237	C₂×C₁₆	—	17	2	no	—
238	#107	—	2	1	no	—
239	#107	—	17	1	no	—
240	#107	C₂²×D₁₇	1	2	yes	—
241	#107	C₂²×D₁₇	1	1	yes	characteristic
242	#106	—	2	2	no	—
243	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
244	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
245	#103	—	2	2	no	—
246	#103	—	2	1	no	—
247	#102	—	34	2	no	—
248	#102	—	34	1	no	—
249	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₇	1	2	yes	—
251	C₁₇	#33935	1	1	yes	characteristic, complemented, Sylow
252	#41	—	136	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	#63228	1	1	yes	characteristic
256	C₂×D₄	—	34	2	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂×D₄	—	68	2	no	—
259	C₂×D₄	—	68	1	no	—
260	C₂×D₄	—	68	2	no	—
261	C₂×D₄	—	68	1	no	—
262	C₂²×C₄	—	68	2	no	—
263	C₂²×C₄	—	68	1	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	#63256	1	2	yes	complemented
266	D₈	—	34	1	no	—
267	D₈	—	34	2	no	—
268	D₈	—	34	1	no	—
269	D₈	—	34	2	no	—
270	D₈	—	136	2	no	—
271	D₈	—	136	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	34	1	no	—
283	C₄⋊C₄	#63228	1	1	yes	characteristic
284	C₄⋊C₄	#63228	1	2	yes	—
285	C₂²⋊C₄	—	68	2	no	—
286	C₂²⋊C₄	—	68	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
291	C₁₆	—	34	1	no	—
292	C₁₆	—	34	2	no	—
293	C₁₆	—	34	1	no	—
294	C₂³	—	68	2	no	—
295	C₂³	—	68	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	272	2	no	—
301	D₄	—	272	1	no	—
302	D₄	—	68	1	no	—
303	D₄	—	68	2	no	—
304	D₄	—	68	1	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	2	no	—
311	C₂×C₄	—	68	1	no	—
312	C₂×C₄	—	136	4	no	—
313	C₂×C₄	—	136	2	no	—
314	C₂×C₄	C₂×#63228	1	2	yes	—
315	C₂×C₄	C₂×#63228	1	1	yes	characteristic
316	C₂×C₄	C₂×#63228	1	2	yes	—
317	C₂×C₄	C₂×#63228	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	C₂×#63256	1	1	yes	characteristic
322	C₈	#10587694	1	1	yes	characteristic
323	C₂²	—	136	2	no	—
324	C₂²	—	136	1	no	—
325	C₂²	—	136	2	no	—
326	C₂²	—	136	1	no	—
327	C₂²	—	136	2	no	—
328	C₂²	—	136	1	no	—
329	C₂²	C₂²×#63228	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	136	1	no	—
337	C₄	#11763244	1	2	yes	—
338	C₄	#11763470	1	1	yes	characteristic
339	C₄	#11763617	1	1	yes	characteristic
340	C₂	—	272	2	no	—
341	C₂	—	272	1	no	—
342	C₂	#14541419	1	1	yes	characteristic
343	C₂	C₂×#11763244	1	1	yes	characteristic
344	C₂	#14541928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3383	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	#14541759	C₂
hypercenter	#3383	D₁₇
hypocenter	C₁₇	#33935
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #14531027	C₂	semidirect
3. #14531009	C₂	semidirect
4. #14530938	C₂	semidirect
5. #14530918	C₂	semidirect
6. #14530824	C₂	semidirect
7. #14530824	C₂	semidirect
8. #14530693	C₂	semidirect
9. #14530646	C₂	semidirect
10. C₄×#10587499	C₂	semidirect
11. C₁₇×#3383	C₂	semidirect
92. Q₁₆×C₁₇	#41	semidirect
152. #3383	D₁₇	semidirect
251. C₁₇	#33935	semidirect
265. Q₁₆	#63256	semidirect

Maximal subgroups

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

subgroup	index
#14541759	2
#14531027	2
#14531009	2
#14530938	2
#14530918	2
#14530824	2
#14530693	2
#14530646	2
C₄×#10587499	2
C₁₇×#3383	2
#33935	17