Subgroups — #14541171

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763517	C₂	1	1	yes	maximal, characteristic
2	#11763514	C₂	1	1	yes	maximal, characteristic
3	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763145	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763143	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763103	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763049	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763048	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762901	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762872	C₂	1	1	yes	maximal, characteristic
13	C₂×#10587478	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#484	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
18	#10587632	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×#63231	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	—	2	1	no	—
22	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
24	#10587610	—	2	1	no	—
25	#10587610	C₂²	1	1	yes	characteristic
26	C₂²×#3622	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	#10587583	—	2	1	no	—
29	#10587583	C₂²	1	1	yes	characteristic
30	#10587582	—	2	1	no	—
31	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587575	—	2	1	no	—
34	#10587575	—	2	1	no	—
35	#10587574	C₂²	1	1	yes	characteristic
36	#10587574	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587567	—	2	1	no	—
39	#10587564	—	2	1	no	—
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587489	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
63	C₂²×Dic₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊Dic₁₇	—	2	1	no	—
79	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
80	#63231	—	2	2	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₂×#3622	—	2	2	no	—
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₈×D₁₇	—	2	2	no	—
105	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	8	1	no	—
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	Q₁₆	1	4	yes	complemented
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	8	1	no	—
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	8	1	no	—
130	C₄×D₁₇	—	8	1	no	—
131	Dic₃₄	—	2	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	8	1	no	—
134	Dic₃₄	—	8	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3622	#41	1	2	yes	—
139	#3338	—	17	1	no	maximal, Sylow
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	C₂×Q₁₆	1	2	yes	—
143	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	C₂×Q₁₆	1	2	yes	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	C₂×Q₁₆	1	2	yes	—
161	#554	—	17	1	no	—
162	#554	—	17	1	no	—
163	#500	—	17	1	no	—
164	#484	—	17	1	no	—
165	#484	—	17	1	no	—
166	#484	—	17	1	no	—
167	#484	D₁₇	1	1	yes	characteristic, complemented
168	#477	—	17	1	no	—
169	#477	—	17	1	no	—
170	#467	—	17	1	no	—
171	#467	—	17	1	no	—
172	C₄×Q₁₆	—	17	1	no	—
173	C₈×D₄	—	17	1	no	—
174	C₂×#107	—	17	1	no	—
175	C₂×#107	—	17	1	no	—
176	C₂×C₁₇	—	2	2	no	—
177	C₂×C₁₇	#547	1	1	yes	characteristic
178	C₂×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
179	C₂×C₁₇	#576	1	1	yes	characteristic
180	D₁₇	—	2	2	no	—
181	C₂×Q₁₆	—	17	1	no	—
182	C₂²×C₈	—	17	1	no	—
183	C₂²×C₈	—	17	1	no	—
184	C₄⋊Q₈	—	17	1	no	—
185	C₄⋊Q₈	—	17	1	no	—
186	#125	—	34	1	no	—
187	#125	—	34	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	—	34	1	no	—
194	C₂²⋊Q₈	—	34	1	no	—
195	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
196	C₄×Q₈	—	17	1	no	—
197	C₄×Q₈	—	17	1	no	—
198	C₄×D₄	—	17	1	no	—
199	C₄×D₄	—	34	1	no	—
200	C₄×D₄	—	34	1	no	—
201	C₂×C₄⋊C₄	—	17	1	no	—
202	C₂×C₄⋊C₄	—	17	1	no	—
203	C₂×C₄⋊C₄	—	17	1	no	—
204	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
205	#107	—	17	1	no	—
206	#107	—	17	1	no	—
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	C₂×D₁₇	1	1	yes	characteristic
213	#107	C₂×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	C₂×D₁₇	1	1	yes	characteristic
220	#103	C₂×D₁₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	17	1	no	—
222	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
223	C₄×C₈	—	17	1	no	—
224	C₁₇	#3338	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	17	1	no	—
226	C₂×Q₈	—	34	1	no	—
227	C₂×Q₈	—	34	1	no	—
228	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
229	C₂×D₄	—	17	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	17	1	no	—
232	C₂²×C₄	—	34	1	no	—
233	C₂²×C₄	—	34	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
236	Q₁₆	—	34	2	no	—
237	C₂×C₈	—	17	1	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	—	34	2	no	—
240	C₂×C₈	—	34	2	no	—
241	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	17	1	no	—
264	C₂²⋊C₄	—	17	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₄²	—	17	1	no	—
269	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₂³	—	17	1	no	—
272	C₂³	D₄×D₁₇	1	1	yes	characteristic
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	34	1	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	136	1	no	—
278	Q₈	—	136	1	no	—
279	D₄	—	17	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	17	1	no	—
288	C₂×C₄	—	17	1	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	136	1	no	—
297	C₂×C₄	—	136	1	no	—
298	C₂×C₄	—	136	1	no	—
299	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	17	2	no	—
304	C₈	—	34	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	17	2	no	—
308	C₂²	—	34	1	no	—
309	C₂²	—	34	1	no	—
310	C₂²	Q₁₆×D₁₇	1	2	yes	—
311	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	17	2	no	—
318	C₄	—	34	1	no	—
319	C₄	—	34	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	—	68	1	no	—
323	C₄	—	68	1	no	—
324	C₄	#41×D₁₇	1	1	yes	characteristic
325	C₄	#10587697	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	34	2	no	—
328	C₂	#11763457	1	1	yes	characteristic
329	C₂	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
330	C₂	#11763613	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×#484	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₇	D₄
hypercenter	#484	D₁₇
hypocenter	C₁₇	#3338
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #11763213	C₂	semidirect
4. #11763146	C₂	semidirect
5. #11763145	C₂	semidirect
6. #11763143	C₂	semidirect
7. #11763103	C₂	semidirect
8. #11763102	C₂	semidirect
9. #11763049	C₂	semidirect
10. #11763048	C₂	semidirect
11. #11762901	C₂	semidirect
13. C₂×#10587478	C₂	semidirect
14. #107×D₁₇	C₂	semidirect
15. C₁₇×#484	C₂	semidirect
118. C₂²⋊D₁₇	Q₁₆	semidirect
167. #484	D₁₇	semidirect
224. C₁₇	#3338	semidirect

Maximal subgroups

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

subgroup	index
#11763517	2
#11763514	2
#11763213	2
#11763146	2
#11763145	2
#11763143	2
#11763103	2
#11763102	2
#11763049	2
#11763048	2
#11762901	2
#11762872	2
C₂×#10587478	2
#107×D₁₇	2
C₁₇×#484	2
#3338	17