Subgroups — #14539896

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#11763384	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763201	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763193	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763171	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763034	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763027	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763013	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762974	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762966	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762957	C₂	1	1	yes	maximal, characteristic
13	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762868	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63256	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	#10587633	—	2	1	no	—
21	#10587633	C₂²	1	1	yes	characteristic
22	#10587632	—	2	1	no	—
23	#10587630	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	C₂×#63231	C₂²	1	1	yes	characteristic
26	C₂×#63230	C₂²	1	1	yes	characteristic
27	C₂×#63230	C₂²	1	1	yes	characteristic
28	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
29	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
30	C₂×#63219	C₂²	1	1	yes	characteristic
31	C₂×#63219	C₂²	1	1	yes	characteristic
32	#10587584	—	2	1	no	—
33	#10587583	C₂²	1	1	yes	characteristic
34	#10587582	—	2	1	no	—
35	#10587582	—	2	1	no	—
36	#10587581	C₂²	1	1	yes	characteristic
37	#10587580	—	2	1	no	—
38	#10587580	—	2	1	no	—
39	#10587580	—	2	1	no	—
40	#10587579	C₂²	1	1	yes	characteristic
41	#10587573	C₂²	1	1	yes	characteristic
42	#10587561	C₂²	1	1	yes	characteristic
43	#10587561	C₂²	1	1	yes	characteristic
44	#10587558	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×#103	C₂²	1	1	yes	characteristic
47	C₁₇×#103	C₂²	1	1	yes	characteristic
48	C₁₇×#103	C₂²	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
50	#10587491	C₂²	1	1	yes	characteristic
51	#10587491	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587480	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	#63256	—	4	1	no	—
63	#63256	—	4	1	no	—
64	Q₈×D₁₇	—	4	1	no	—
65	Q₈×D₁₇	—	4	1	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×D₆₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	#63231	—	4	1	no	—
86	#63231	—	4	1	no	—
87	#63230	—	4	1	no	—
88	#63230	—	4	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	—	2	1	no	—
96	#63227	—	2	1	no	—
97	#63227	—	2	1	no	—
98	#63227	D₄	1	1	yes	characteristic
99	#63227	D₄	1	1	yes	characteristic
100	C₄⋊Dic₁₇	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	D₄	1	1	yes	characteristic
105	#63225	D₄	1	1	yes	characteristic
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	Dic₆₈	—	4	1	no	—
111	D₁₃₆	—	4	1	no	—
112	#63219	—	4	1	no	—
113	#63219	—	4	1	no	—
114	C₂²×D₁₇	—	2	1	no	—
115	C₂²×D₁₇	—	2	1	no	—
116	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	Q₈×C₁₇	—	2	1	no	—
120	Q₈×C₁₇	—	2	1	no	—
121	Q₈×C₁₇	—	8	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	D₆₈	—	2	1	no	—
135	D₆₈	—	2	1	no	—
136	D₆₈	—	2	1	no	—
137	D₆₈	—	2	1	no	—
138	D₆₈	—	4	2	no	—
139	D₆₈	—	8	1	no	—
140	C₄×D₁₇	—	2	1	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	4	2	no	—
143	C₄×D₁₇	—	4	2	no	—
144	C₄×D₁₇	—	8	1	no	—
145	Dic₃₄	—	2	1	no	—
146	Dic₃₄	—	2	1	no	—
147	Dic₃₄	—	2	1	no	—
148	Dic₃₄	—	2	1	no	—
149	Dic₃₄	—	4	2	no	—
150	C₈×C₁₇	—	2	1	no	—
151	C₈×C₁₇	—	2	1	no	—
152	#3622	—	2	1	no	—
153	#3622	—	2	1	no	—
154	#3225	—	17	1	no	maximal, Sylow
155	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	4	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	—	4	1	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	#542	—	17	1	no	—
179	#536	—	17	1	no	—
180	#497	—	17	1	no	—
181	#495	—	17	1	no	—
182	#489	—	17	1	no	—
183	#487	D₁₇	1	1	yes	characteristic, complemented
184	#464	—	17	1	no	—
185	#463	—	17	1	no	—
186	#462	—	17	1	no	—
187	#461	—	17	1	no	—
188	#452	—	17	1	no	—
189	#451	—	17	1	no	—
190	#449	—	17	1	no	—
191	#448	—	17	1	no	—
192	#433	—	17	1	no	—
193	C₂×C₁₇	#535	1	1	yes	characteristic
194	C₂×C₁₇	#577	1	1	yes	characteristic
195	C₂×C₁₇	#578	1	1	yes	characteristic
196	D₁₇	—	4	1	no	—
197	D₁₇	—	8	1	no	—
198	D₁₇	—	8	1	no	—
199	C₂×#41	—	17	1	no	—
200	C₂²×Q₈	—	17	1	no	—
201	C₂×Q₁₆	—	17	1	no	—
202	C₂×Q₁₆	—	17	1	no	—
203	C₂×Q₁₆	—	17	1	no	—
204	C₂×SD₁₆	—	17	1	no	—
205	C₂×SD₁₆	—	17	1	no	—
206	C₂×SD₁₆	—	17	1	no	—
207	C₂×SD₁₆	—	17	1	no	—
208	C₂×D₈	—	17	1	no	—
209	C₄⋊Q₈	—	17	1	no	—
210	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
211	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
212	#126	—	34	1	no	—
213	#124	—	17	1	no	—
214	#124	—	17	1	no	—
215	#124	—	17	1	no	—
216	#124	—	34	1	no	—
217	#123	—	34	1	no	—
218	#123	—	34	1	no	—
219	#123	—	34	1	no	—
220	C₂²⋊Q₈	—	17	1	no	—
221	C₂²⋊Q₈	—	34	1	no	—
222	C₂²⋊Q₈	—	34	1	no	—
223	C₂²⋊Q₈	—	34	1	no	—
224	#121	—	17	1	no	—
225	C₄×Q₈	—	17	1	no	—
226	C₄×D₄	—	17	1	no	—
227	#117	—	17	1	no	—
228	C₄⋊C₈	—	17	1	no	—
229	C₄⋊C₈	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	C₂×D₁₇	1	1	yes	characteristic
233	#103	C₂×D₁₇	1	1	yes	characteristic
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#103	C₂×D₁₇	1	1	yes	characteristic
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	C₂²⋊C₈	—	17	1	no	—
239	C₂²⋊C₈	—	17	1	no	—
240	#97	—	17	1	no	—
241	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
242	C₁₇	#3225	1	1	yes	characteristic, complemented, Sylow
243	#41	—	68	1	no	—
244	#41	—	68	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	17	1	no	—
247	C₂×Q₈	—	17	1	no	—
248	C₂×Q₈	—	34	1	no	—
249	C₂×Q₈	—	68	1	no	—
250	C₂×Q₈	—	68	1	no	—
251	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₂×D₄	—	17	1	no	—
254	C₂×D₄	—	17	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂²×C₄	—	17	1	no	—
258	C₂²×C₄	—	34	1	no	—
259	C₂²×C₄	—	34	1	no	—
260	C₂²×C₄	—	34	1	no	—
261	Q₁₆	—	68	1	no	—
262	Q₁₆	—	68	1	no	—
263	Q₁₆	—	68	1	no	—
264	SD₁₆	—	68	1	no	—
265	SD₁₆	—	68	1	no	—
266	SD₁₆	—	68	1	no	—
267	SD₁₆	—	68	1	no	—
268	D₈	—	68	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	17	1	no	—
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
284	C₂²⋊C₄	—	17	1	no	—
285	C₂²⋊C₄	—	17	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	—	34	1	no	—
291	C₂²⋊C₄	—	34	1	no	—
292	C₂²⋊C₄	—	34	1	no	—
293	C₂²⋊C₄	—	34	1	no	—
294	C₂²⋊C₄	—	34	1	no	—
295	C₄²	—	17	1	no	—
296	C₄²	—	34	1	no	—
297	C₄²	C₂²×D₁₇	1	1	yes	characteristic
298	C₂³	—	17	1	no	—
299	C₂³	—	34	1	no	—
300	C₂³	—	34	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	8	1	no	—
306	Q₈	—	34	1	no	—
307	Q₈	—	34	1	no	—
308	Q₈	—	34	1	no	—
309	Q₈	—	34	1	no	—
310	Q₈	—	68	2	no	—
311	D₄	—	34	1	no	—
312	D₄	—	34	1	no	—
313	D₄	—	34	1	no	—
314	D₄	—	34	1	no	—
315	D₄	—	68	2	no	—
316	D₄	—	136	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	17	1	no	—
322	C₂×C₄	—	17	1	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	34	1	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	—	68	2	no	—
330	C₂×C₄	—	136	1	no	—
331	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
332	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	34	1	no	—
337	C₈	—	34	1	no	—
338	C₂²	—	34	1	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	68	1	no	—
342	C₂²	—	68	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	68	1	no	—
345	C₂²	—	68	1	no	—
346	C₂²	—	68	1	no	—
347	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	34	1	no	—
355	C₄	—	34	1	no	—
356	C₄	—	34	1	no	—
357	C₄	—	68	1	no	—
358	C₄	—	68	1	no	—
359	C₄	#10587696	1	1	yes	characteristic
360	C₄	#10587696	1	1	yes	characteristic
361	C₂	—	68	1	no	—
362	C₂	—	136	1	no	—
363	C₂	—	136	1	no	—
364	C₂	#11763377	1	1	yes	characteristic
365	C₂	#11763615	1	1	yes	characteristic
366	C₂	#11763622	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×#487	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11762957	C₂
hypercenter	#487	D₁₇
hypocenter	C₁₇	#3225
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

Complements

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

N	H ≅ G/N	type
1. #11763433	C₂	semidirect
2. #11763384	C₂	semidirect
3. #11763201	C₂	semidirect
4. #11763193	C₂	semidirect
5. #11763171	C₂	semidirect
6. #11763034	C₂	semidirect
7. #11763027	C₂	semidirect
8. #11763018	C₂	semidirect
9. #11763013	C₂	semidirect
10. #11762974	C₂	semidirect
11. #11762966	C₂	semidirect
13. #11762949	C₂	semidirect
14. #11762868	C₂	semidirect
15. C₁₇×#487	C₂	semidirect
183. #487	D₁₇	semidirect
242. C₁₇	#3225	semidirect

Maximal subgroups

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

subgroup	index
#11763433	2
#11763384	2
#11763201	2
#11763193	2
#11763171	2
#11763034	2
#11763027	2
#11763018	2
#11763013	2
#11762974	2
#11762966	2
#11762957	2
#11762949	2
#11762868	2
C₁₇×#487	2
#3225	17