Subgroups — #14541849

Order 2176.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763541	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763533	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763195	C₂	1	1	yes	maximal, characteristic
4	#11763194	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763033	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763033	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763026	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763021	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763011	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763007	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762916	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×#124	C₂²	1	1	yes	characteristic
22	#10587633	C₂²	1	1	yes	characteristic
23	#10587633	C₂²	1	1	yes	characteristic
24	#10587633	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
28	C₂×#63230	C₂²	1	1	yes	characteristic
29	C₂×#63230	C₂²	1	1	yes	characteristic
30	C₂×#63230	C₂²	1	1	yes	characteristic
31	#10587626	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
33	C₂×#63228	C₂²	1	1	yes	characteristic
34	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
35	#10587584	—	2	1	no	—
36	#10587584	—	2	1	no	—
37	#10587583	—	2	1	no	—
38	#10587578	—	2	1	no	—
39	#10587572	—	2	1	no	—
40	#10587571	—	2	1	no	—
41	#10587569	—	2	1	no	—
42	#10587567	—	2	1	no	—
43	C₄×D₆₈	C₂²	1	1	yes	characteristic
44	C₄×D₆₈	C₂²	1	1	yes	characteristic
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	C₁₇×#97	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	#10587503	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587492	C₂²	1	1	yes	characteristic
53	#10587488	C₂²	1	1	yes	characteristic
54	#10587486	C₂²	1	1	yes	characteristic
55	#10587485	C₂²	1	1	yes	characteristic
56	#10587485	C₂²	1	1	yes	characteristic
57	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×Dic₃₄	D₄	1	1	yes	characteristic
71	Q₁₆×C₁₇	—	2	2	no	—
72	Q₁₆×C₁₇	—	4	1	no	—
73	SD₁₆×C₁₇	—	2	2	no	—
74	SD₁₆×C₁₇	—	4	1	no	—
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63230	—	4	1	no	—
84	#63230	—	4	1	no	—
85	#63230	—	4	1	no	—
86	#63228	—	4	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	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	—	2	1	no	—
98	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
101	#63225	D₄	1	1	yes	characteristic
102	#63225	D₄	1	1	yes	characteristic
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
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	D₁₃₆	—	2	2	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	Q₈×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	no	—
121	D₄×C₁₇	—	2	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	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₇	—	4	2	no	—
129	C₂²⋊D₁₇	—	8	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	8	1	no	—
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	D₆₈	—	2	1	no	—
138	D₆₈	—	2	1	no	—
139	D₆₈	—	2	1	no	—
140	D₆₈	—	2	1	no	—
141	C₄×D₁₇	—	4	2	no	—
142	C₄×D₁₇	—	8	1	no	—
143	Dic₃₄	—	4	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	#41	1	2	yes	—
146	#3622	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3401	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	4	1	no	—
150	C₂²×C₁₇	—	4	1	no	—
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	#559	—	17	1	no	—
173	#556	—	17	1	no	—
174	#497	D₁₇	1	1	yes	characteristic, complemented
175	#496	—	17	1	no	—
176	#495	—	17	1	no	—
177	#464	—	17	1	no	—
178	#464	—	17	1	no	—
179	#463	—	17	1	no	—
180	#463	—	17	1	no	—
181	#461	—	17	1	no	—
182	#460	—	17	1	no	—
183	#446	—	17	1	no	—
184	#442	—	17	1	no	—
185	#441	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	C₂×C₁₇	—	8	1	no	—
188	C₂×C₁₇	#550	1	1	yes	characteristic
189	C₂×C₁₇	#575	1	1	yes	characteristic
190	C₂×C₁₇	#577	1	1	yes	characteristic
191	D₁₇	—	8	1	no	—
192	D₁₇	—	8	1	no	—
193	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
194	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
195	C₂×SD₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	17	1	no	—
197	C₂×SD₁₆	—	17	1	no	—
198	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
199	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
200	C₂×D₈	—	17	1	no	—
201	C₂×D₈	—	17	1	no	—
202	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
203	#127	—	17	1	no	—
204	#126	—	34	1	no	—
205	#126	—	34	1	no	—
206	#124	—	17	1	no	—
207	#124	—	17	1	no	—
208	#124	—	17	1	no	—
209	#124	—	34	1	no	—
210	#124	C₂×D₁₇	1	1	yes	characteristic
211	#123	—	34	1	no	—
212	#123	—	34	1	no	—
213	C₂²⋊Q₈	—	34	1	no	—
214	C₄×Q₈	—	17	1	no	—
215	C₄×Q₈	—	17	1	no	—
216	C₄×Q₈	—	17	1	no	—
217	C₄×Q₈	—	17	1	no	—
218	C₄×D₄	—	17	1	no	—
219	C₄×D₄	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	34	1	no	—
222	#117	—	34	1	no	—
223	#106	—	17	1	no	—
224	C₄⋊C₈	—	17	1	no	—
225	C₄⋊C₈	—	17	1	no	—
226	C₄⋊C₈	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#97	—	17	1	no	—
234	#97	C₂×D₁₇	1	1	yes	characteristic
235	C₄×C₈	—	17	1	no	—
236	C₁₇	#3401	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₈	—	17	1	no	—
238	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×D₄	—	17	1	no	—
242	C₂×D₄	—	17	1	no	—
243	C₂×D₄	—	34	1	no	—
244	C₂×D₄	—	34	1	no	—
245	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	—	34	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	4	1	no	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	4	1	no	—
253	SD₁₆	—	68	1	no	—
254	SD₁₆	—	68	1	no	—
255	SD₁₆	—	68	1	no	—
256	D₈	—	34	2	no	—
257	D₈	—	68	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	17	1	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₄²	—	17	1	no	—
284	C₄²	—	17	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	—	34	1	no	—
287	C₄²	—	34	1	no	—
288	C₄²	C₂²×D₁₇	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	34	1	no	—
291	C₂³	—	34	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	68	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	34	1	no	—
302	D₄	—	34	1	no	—
303	D₄	—	34	1	no	—
304	D₄	—	34	1	no	—
305	D₄	—	68	2	no	—
306	D₄	—	136	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	17	1	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	68	2	no	—
317	C₂×C₄	—	136	1	no	—
318	C₂×C₄	—	136	1	no	—
319	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
320	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	34	1	no	—
324	C₈	—	34	1	no	—
325	C₈	#63254	1	2	yes	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	68	1	no	—
330	C₂²	—	68	1	no	—
331	C₂²	—	68	1	no	—
332	C₂²	—	68	1	no	—
333	C₂²	—	68	1	no	—
334	C₂²	—	68	1	no	—
335	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	34	1	no	—
342	C₄	—	34	1	no	—
343	C₄	—	34	1	no	—
344	C₄	—	34	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	C₂×#63254	1	1	yes	characteristic
348	C₄	#10587691	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	136	1	no	—
351	C₂	—	136	1	no	—
352	C₂	#11763479	1	1	yes	characteristic
353	C₂	#11763608	1	1	yes	characteristic
354	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×#497	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763195	C₂
hypercenter	#497	D₁₇
hypocenter	C₁₇	#3401
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. #11763541	C₂	semidirect
2. #11763533	C₂	semidirect
4. #11763194	C₂	semidirect
5. #11763033	C₂	semidirect
6. #11763033	C₂	semidirect
7. #11763026	C₂	semidirect
8. #11763021	C₂	semidirect
9. #11763011	C₂	semidirect
10. #11763007	C₂	semidirect
11. #11762937	C₂	semidirect
12. #11762922	C₂	semidirect
13. #11762916	C₂	semidirect
14. SD₁₆×Dic₁₇	C₂	semidirect
15. C₁₇×#497	C₂	semidirect
174. #497	D₁₇	semidirect
236. C₁₇	#3401	semidirect

Maximal subgroups

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

subgroup	index
#11763541	2
#11763533	2
#11763195	2
#11763194	2
#11763033	2
#11763026	2
#11763021	2
#11763011	2
#11763007	2
#11762937	2
#11762922	2
#11762916	2
SD₁₆×Dic₁₇	2
C₁₇×#497	2
#3401	17