Subgroups — #14541165

Order 2176.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763505	C₂	1	1	yes	maximal, characteristic
2	#11763499	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763145	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763134	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763123	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762847	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10587488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587625	—	2	1	no	—
20	#10587621	—	2	1	no	—
21	#10587616	C₂²	1	1	yes	characteristic
22	#10587616	C₂²	1	1	yes	characteristic
23	#10587612	C₂²	1	1	yes	characteristic
24	#10587612	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
27	#10587610	C₂²	1	1	yes	characteristic
28	#10587610	C₂²	1	1	yes	characteristic
29	C₂×#63219	C₂²	1	1	yes	characteristic
30	#10587584	—	2	1	no	—
31	#10587575	—	2	1	no	—
32	#10587569	—	2	1	no	—
33	#10587567	—	2	1	no	—
34	#10587565	—	2	1	no	—
35	#10587563	—	2	1	no	—
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	#10587556	C₂²	1	1	yes	characteristic
38	#10587556	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
44	#10587492	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587489	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587488	C₂²	1	1	yes	characteristic
54	#10587488	C₂²	1	1	yes	characteristic
55	#10587488	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	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	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	#63219	—	2	2	no	—
105	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	—	2	1	no	—
108	D₄×C₁₇	—	2	2	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	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	4	2	no	—
119	C₂²⋊D₁₇	—	8	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	4	2	no	—
128	C₂×Dic₁₇	—	8	1	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	2	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	#41	1	2	yes	—
138	#3337	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	#135	1	2	yes	—
145	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₇	—	4	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	#553	—	17	1	no	—
163	#552	—	17	1	no	—
164	#499	—	17	1	no	—
165	#485	—	17	1	no	—
166	#484	—	17	1	no	—
167	#482	—	17	1	no	—
168	#480	—	17	1	no	—
169	#479	—	17	1	no	—
170	#478	—	17	1	no	—
171	#465	—	17	1	no	—
172	#465	—	17	1	no	—
173	C₄×SD₁₆	—	17	1	no	—
174	C₈×D₄	D₁₇	1	1	yes	characteristic, complemented
175	#427	—	17	1	no	—
176	C₂×#106	—	17	1	no	—
177	C₂×C₁₇	—	2	2	no	—
178	C₂×C₁₇	—	4	1	no	—
179	C₂×C₁₇	#547	1	1	yes	characteristic
180	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
181	C₂×C₁₇	#577	1	1	yes	characteristic
182	D₁₇	—	8	1	no	—
183	C₂×SD₁₆	—	17	1	no	—
184	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
185	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
186	#126	—	34	1	no	—
187	#126	—	34	1	no	—
188	#125	—	17	1	no	—
189	#125	—	17	1	no	—
190	#125	—	34	1	no	—
191	#123	—	34	1	no	—
192	#123	—	34	1	no	—
193	C₂²⋊Q₈	—	17	1	no	—
194	C₂²⋊Q₈	—	17	1	no	—
195	#121	—	17	1	no	—
196	#121	—	17	1	no	—
197	#121	—	34	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	34	1	no	—
201	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
202	#117	—	17	1	no	—
203	#117	—	17	1	no	—
204	#117	—	34	1	no	—
205	C₂×C₄⋊C₄	—	17	1	no	—
206	C₂×C₄⋊C₄	—	17	1	no	—
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	#106	—	17	1	no	—
214	#106	—	17	1	no	—
215	#106	—	17	1	no	—
216	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
224	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
225	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
226	C₁₇	#3337	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	17	1	no	—
228	C₂×D₄	—	17	1	no	—
229	C₂×D₄	—	34	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	D₆₈	1	1	yes	characteristic
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	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
237	SD₁₆	—	34	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄⋊C₄	—	17	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	—	17	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	D₆₈	1	1	yes	characteristic
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
268	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
269	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	—	34	1	no	—
272	C₄²	C₂²×D₁₇	1	1	yes	characteristic
273	C₂³	—	34	1	no	—
274	C₂³	C₂×D₆₈	1	1	yes	characteristic
275	C₂³	C₂×D₆₈	1	1	yes	characteristic
276	Q₈	—	34	1	no	—
277	Q₈	—	34	1	no	—
278	D₄	—	2	2	no	—
279	D₄	—	34	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	68	2	no	—
282	D₄	—	136	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	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	—	68	2	no	—
295	C₂×C₄	—	136	1	no	—
296	C₂×C₄	—	136	1	no	—
297	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
298	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
299	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
300	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#63254	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	68	1	no	—
312	C₂²	—	68	1	no	—
313	C₂²	—	68	1	no	—
314	C₂²	#10587590	1	2	yes	—
315	C₂²	C₂²×D₆₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	—	68	1	no	—
323	C₄	—	68	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	—	68	1	no	—
327	C₄	C₂×#63254	1	1	yes	characteristic
328	C₄	#10587697	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	4	1	no	—
331	C₂	—	136	1	no	—
332	C₂	#11763463	1	1	yes	characteristic
333	C₂	C₂×#10587590	1	1	yes	characteristic
334	C₂	#11763621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₈×D₄×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763505	C₂
hypercenter	C₈×D₄	D₁₇
hypocenter	C₁₇	#3337
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #11763499	C₂	semidirect
3. #11763211	C₂	semidirect
4. #11763151	C₂	semidirect
5. #11763145	C₂	semidirect
6. #11763134	C₂	semidirect
7. #11763123	C₂	semidirect
8. #11763117	C₂	semidirect
9. #11763110	C₂	semidirect
10. #11763038	C₂	semidirect
11. #11763038	C₂	semidirect
12. C₄×#63219	C₂	semidirect
13. #11762847	C₂	semidirect
14. C₂×#10587488	C₂	semidirect
15. C₈×D₄×C₁₇	C₂	semidirect
174. C₈×D₄	D₁₇	semidirect
226. C₁₇	#3337	semidirect

Maximal subgroups

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

subgroup	index
#11763505	2
#11763499	2
#11763211	2
#11763151	2
#11763145	2
#11763134	2
#11763123	2
#11763117	2
#11763110	2
#11763038	2
C₄×#63219	2
#11762847	2
C₂×#10587488	2
C₈×D₄×C₁₇	2
#3337	17