Subgroups — #14541268

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763520	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763512	C₂	1	1	yes	maximal, characteristic
3	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763124	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763086	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763056	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763039	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762882	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	#10587632	—	2	1	no	—
19	#10587632	—	2	1	no	—
20	C₂×#63219	C₂²	1	1	yes	characteristic
21	C₂×#63218	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	#10587584	—	2	1	no	—
24	#10587583	C₂²	1	1	yes	characteristic
25	#10587583	C₂²	1	1	yes	characteristic
26	#10587581	—	2	1	no	—
27	#10587581	C₂²	1	1	yes	characteristic
28	#10587581	C₂²	1	1	yes	characteristic
29	#10587580	—	2	1	no	—
30	#10587579	—	2	1	no	—
31	#10587579	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
36	#10587576	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×#97	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587481	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	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	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×D₆₈	—	2	1	no	—
61	C₂×D₆₈	—	2	1	no	—
62	C₂×D₆₈	D₄	1	1	yes	characteristic
63	C₂×D₆₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	D₄	1	1	yes	characteristic
88	#63227	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	#63219	—	2	2	no	—
103	#63218	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₁₇	—	8	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	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	D₆₈	—	2	2	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	—	2	1	no	—
124	D₆₈	—	4	2	no	—
125	D₆₈	—	8	1	no	—
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	C₄×D₁₇	—	8	1	no	—
133	C₄×D₁₇	—	8	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	#41	1	2	yes	—
138	#3622	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3346	—	17	1	no	maximal, Sylow
141	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₇	—	2	1	no	—
143	C₂×D₁₇	—	2	1	no	—
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	#137	1	2	yes	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	#554	—	17	1	no	—
165	#553	—	17	1	no	—
166	#501	D₁₇	1	1	yes	characteristic, complemented
167	#485	—	17	1	no	—
168	#484	—	17	1	no	—
169	#481	—	17	1	no	—
170	#480	—	17	1	no	—
171	#479	—	17	1	no	—
172	#474	—	17	1	no	—
173	#468	—	17	1	no	—
174	#465	—	17	1	no	—
175	#440	—	17	1	no	—
176	#435	—	17	1	no	—
177	#428	—	17	1	no	—
178	C₂×#107	—	17	1	no	—
179	C₂×C₁₇	#547	1	1	yes	characteristic
180	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
181	C₂×C₁₇	#576	1	1	yes	characteristic
182	D₁₇	—	2	2	no	—
183	D₁₇	—	4	1	no	—
184	D₁₇	—	8	1	no	—
185	C₂×SD₁₆	—	17	1	no	—
186	C₂×M₁₆	—	17	1	no	—
187	C₂²×C₈	—	17	1	no	—
188	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
189	#126	—	34	1	no	—
190	#125	—	34	1	no	—
191	#125	C₂×D₁₇	1	1	yes	characteristic
192	#123	—	34	1	no	—
193	C₂²⋊Q₈	—	17	1	no	—
194	C₂²⋊Q₈	—	17	1	no	—
195	C₂²⋊Q₈	—	34	1	no	—
196	C₂²⋊Q₈	—	34	1	no	—
197	#121	—	17	1	no	—
198	#121	—	17	1	no	—
199	#121	—	34	1	no	—
200	C₄×Q₈	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	34	1	no	—
204	C₄×D₄	—	34	1	no	—
205	#117	—	17	1	no	—
206	C₂×C₄⋊C₄	—	17	1	no	—
207	C₂×C₄⋊C₄	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	—	17	1	no	—
213	#107	C₂×D₁₇	1	1	yes	characteristic
214	#107	C₂×D₁₇	1	1	yes	characteristic
215	#106	—	17	1	no	—
216	#106	C₂×D₁₇	1	1	yes	characteristic
217	#106	C₂×D₁₇	1	1	yes	characteristic
218	C₄⋊C₈	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#102	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	C₂²⋊C₈	—	17	1	no	—
226	C₂²⋊C₈	—	17	1	no	—
227	#97	C₂×D₁₇	1	1	yes	characteristic
228	C₁₇	#3346	1	1	yes	characteristic, complemented, Sylow
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	17	1	no	—
231	C₂×D₄	—	17	1	no	—
232	C₂×D₄	—	17	1	no	—
233	C₂×D₄	—	34	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂²×C₄	—	17	1	no	—
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	SD₁₆	—	34	2	no	—
242	M₁₆	—	34	2	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	34	2	no	—
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	17	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	—	34	1	no	—
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₂²⋊C₄	—	17	1	no	—
265	C₂²⋊C₄	—	17	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	C₂²×D₁₇	1	1	yes	characteristic
275	C₂³	—	17	1	no	—
276	C₂³	—	17	1	no	—
277	C₂³	—	34	1	no	—
278	Q₈	—	8	1	no	—
279	Q₈	—	34	1	no	—
280	Q₈	—	34	1	no	—
281	D₄	—	34	2	no	—
282	D₄	—	34	1	no	—
283	D₄	—	34	1	no	—
284	D₄	—	68	2	no	—
285	D₄	—	136	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	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	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	136	1	no	—
300	C₂×C₄	—	136	1	no	—
301	C₂×C₄	—	136	1	no	—
302	C₂×C₄	—	136	1	no	—
303	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	34	1	no	—
308	C₈	—	34	1	no	—
309	C₈	#63256	1	2	yes	—
310	C₂²	—	17	2	no	—
311	C₂²	—	34	1	no	—
312	C₂²	—	34	1	no	—
313	C₂²	—	34	1	no	—
314	C₂²	—	34	1	no	—
315	C₂²	—	34	1	no	—
316	C₂²	—	68	1	no	—
317	C₂²	—	68	1	no	—
318	C₂²	—	68	1	no	—
319	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	34	1	no	—
327	C₄	—	34	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	—	68	1	no	—
331	C₄	C₂×#63256	1	1	yes	characteristic
332	C₄	#10587694	1	1	yes	characteristic
333	C₂	—	34	2	no	—
334	C₂	—	68	1	no	—
335	C₂	—	136	1	no	—
336	C₂	#11763462	1	1	yes	characteristic
337	C₂	#137×D₁₇	1	1	yes	characteristic
338	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×#501	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763512	C₂
hypercenter	#501	D₁₇
hypocenter	C₁₇	#3346
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #11763520	C₂	semidirect
3. #11763153	C₂	semidirect
4. #11763146	C₂	semidirect
5. #11763129	C₂	semidirect
6. #11763124	C₂	semidirect
7. #11763113	C₂	semidirect
8. #11763086	C₂	semidirect
9. #11763056	C₂	semidirect
10. #11763039	C₂	semidirect
11. #11762911	C₂	semidirect
12. #11762882	C₂	semidirect
13. #11762848	C₂	semidirect
14. #107×D₁₇	C₂	semidirect
15. C₁₇×#501	C₂	semidirect
166. #501	D₁₇	semidirect
228. C₁₇	#3346	semidirect

Maximal subgroups

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

subgroup	index
#11763520	2
#11763512	2
#11763153	2
#11763146	2
#11763129	2
#11763124	2
#11763113	2
#11763086	2
#11763056	2
#11763039	2
#11762911	2
#11762882	2
#11762848	2
#107×D₁₇	2
C₁₇×#501	2
#3346	17