Subgroups — #14539068

Order 2176.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763433	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763291	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763031	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762974	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762957	C₂	1	1	yes	maximal, characteristic
10	#11762920	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762913	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762896	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63256	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
19	#10587633	—	2	1	no	—
20	#10587632	—	2	1	no	—
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63231	C₂²	1	1	yes	characteristic
23	C₂×#63230	C₂²	1	1	yes	characteristic
24	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
25	C₂×#63219	C₂²	1	1	yes	characteristic
26	#10587584	—	2	1	no	—
27	#10587583	C₂²	1	1	yes	characteristic
28	#10587581	C₂²	1	1	yes	characteristic
29	#10587580	—	2	1	no	—
30	#10587579	—	2	1	no	—
31	#10587579	C₂²	1	1	yes	characteristic
32	#10587579	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	—	2	1	no	—
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	#10587573	C₂²	1	1	yes	characteristic
36	#10587561	C₂²	1	1	yes	characteristic
37	C₄×D₆₈	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	#10587558	—	2	1	no	—
40	#10587558	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	#63256	—	2	2	no	—
62	#63256	—	2	2	no	—
63	C₂×D₆₈	—	2	1	no	—
64	C₂×D₆₈	—	2	1	no	—
65	C₂×D₆₈	C₂³	1	1	yes	characteristic
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	#63231	—	2	2	no	—
84	#63230	—	2	2	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	D₄	1	1	yes	characteristic
93	#63227	D₄	1	1	yes	characteristic
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	D₄	1	1	yes	characteristic
100	#63225	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	D₁₃₆	—	2	2	no	—
107	#63219	—	2	2	no	—
108	C₂²×D₁₇	—	2	1	no	—
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	8	1	no	—
113	Q₈×C₁₇	#41	1	2	yes	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	D₆₈	—	2	4	no	—
127	D₆₈	—	2	1	no	—
128	D₆₈	—	2	1	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	8	1	no	—
131	D₆₈	#41	1	2	yes	—
132	C₄×D₁₇	—	2	4	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	C₄×D₁₇	—	8	1	no	—
137	C₄×D₁₇	—	8	1	no	—
138	C₄×D₁₇	—	8	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3125	—	17	1	no	maximal, Sylow
146	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	2	1	no	—
148	C₂×D₁₇	—	2	2	no	—
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₄×C₁₇	—	2	2	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	#542	—	17	1	no	—
170	#520	—	17	1	no	—
171	#483	—	17	1	no	—
172	#481	—	17	1	no	—
173	#476	—	17	1	no	—
174	#475	D₁₇	1	1	yes	characteristic, complemented
175	#464	—	17	1	no	—
176	#463	—	17	1	no	—
177	#452	—	17	1	no	—
178	#449	—	17	1	no	—
179	#442	—	17	1	no	—
180	#441	—	17	1	no	—
181	C₄×SD₁₆	—	17	1	no	—
182	C₄×SD₁₆	—	17	1	no	—
183	#433	—	17	1	no	—
184	C₂×C₁₇	#525	1	1	yes	characteristic
185	C₂×C₁₇	#577	1	1	yes	characteristic
186	C₂×C₁₇	#577	1	1	yes	characteristic
187	D₁₇	—	4	1	no	—
188	D₁₇	—	4	2	no	—
189	D₁₇	—	8	1	no	—
190	C₂×#41	—	17	1	no	—
191	C₂×Q₁₆	—	17	1	no	—
192	C₂×SD₁₆	—	17	1	no	—
193	C₂×SD₁₆	—	17	1	no	—
194	C₂×D₈	—	17	1	no	—
195	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
196	#126	—	34	1	no	—
197	#124	—	17	1	no	—
198	#124	—	34	1	no	—
199	#123	—	34	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	C₂²⋊Q₈	—	34	1	no	—
202	#121	—	17	1	no	—
203	C₄×Q₈	—	17	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	—	17	1	no	—
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	34	1	no	—
210	C₄×D₄	—	34	1	no	—
211	#117	—	17	1	no	—
212	#117	—	34	1	no	—
213	C₂×C₄⋊C₄	—	17	1	no	—
214	C₂×C₄⋊C₄	—	34	1	no	—
215	#106	—	17	1	no	—
216	#106	—	17	1	no	—
217	#106	C₂×D₁₇	1	1	yes	characteristic
218	#106	C₂×D₁₇	1	1	yes	characteristic
219	C₄⋊C₈	—	17	1	no	—
220	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	C₂×D₁₇	1	1	yes	characteristic
224	#103	C₂×D₁₇	1	1	yes	characteristic
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	C₂²⋊C₈	—	17	1	no	—
230	C₂²⋊C₈	—	17	1	no	—
231	#97	—	17	1	no	—
232	C₄×C₈	—	17	1	no	—
233	C₁₇	#3125	1	1	yes	characteristic, complemented, Sylow
234	#41	—	34	2	no	—
235	#41	—	34	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	17	1	no	—
238	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂²×C₄	—	17	1	no	—
244	C₂²×C₄	—	34	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	C₂²×C₄	—	34	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	Q₁₆	—	34	2	no	—
250	SD₁₆	—	34	2	no	—
251	SD₁₆	—	34	2	no	—
252	D₈	—	34	2	no	—
253	C₂×C₈	—	17	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
269	C₂²⋊C₄	—	17	1	no	—
270	C₂²⋊C₄	—	17	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	34	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	C₂²×D₁₇	1	1	yes	characteristic
283	C₂³	—	17	1	no	—
284	C₂³	—	34	1	no	—
285	C₂³	—	34	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	8	1	no	—
288	Q₈	—	34	1	no	—
289	Q₈	—	34	1	no	—
290	Q₈	#63256	1	2	yes	—
291	D₄	—	17	2	no	—
292	D₄	—	34	4	no	—
293	D₄	—	34	1	no	—
294	D₄	—	34	1	no	—
295	D₄	—	34	1	no	—
296	D₄	—	136	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	34	4	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	136	1	no	—
310	C₂×C₄	—	136	1	no	—
311	C₂×C₄	—	136	1	no	—
312	C₂×C₄	—	136	1	no	—
313	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
314	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	34	1	no	—
319	C₈	—	34	1	no	—
320	C₂²	—	34	2	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	—	68	1	no	—
326	C₂²	—	68	1	no	—
327	C₂²	—	68	1	no	—
328	C₂²	—	68	1	no	—
329	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	34	1	no	—
338	C₄	—	34	1	no	—
339	C₄	—	68	1	no	—
340	C₄	—	68	1	no	—
341	C₄	C₂×#63256	1	1	yes	characteristic
342	C₄	#41×D₁₇	1	1	yes	characteristic
343	C₂	—	68	1	no	—
344	C₂	—	68	2	no	—
345	C₂	—	136	1	no	—
346	C₂	#11763322	1	1	yes	characteristic
347	C₂	#11763615	1	1	yes	characteristic
348	C₂	#11763618	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#475	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11762957	C₂
hypercenter	#475	D₁₇
hypocenter	C₁₇	#3125
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
1. #11763433	C₂	semidirect
2. #11763291	C₂	semidirect
3. #11763141	C₂	semidirect
4. #11763129	C₂	semidirect
5. #11763096	C₂	semidirect
6. #11763031	C₂	semidirect
7. #11763025	C₂	semidirect
8. #11762974	C₂	semidirect
10. #11762920	C₂	semidirect
11. #11762913	C₂	semidirect
12. #11762896	C₂	semidirect
13. #11762894	C₂	semidirect
14. #11762869	C₂	semidirect
15. C₁₇×#475	C₂	semidirect
174. #475	D₁₇	semidirect
233. C₁₇	#3125	semidirect

Maximal subgroups

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

subgroup	index
#11763433	2
#11763291	2
#11763141	2
#11763129	2
#11763096	2
#11763031	2
#11763025	2
#11762974	2
#11762957	2
#11762920	2
#11762913	2
#11762896	2
#11762894	2
#11762869	2
C₁₇×#475	2
#3125	17