Subgroups — #14541356

Order 2176.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763500	C₂	1	1	yes	maximal, characteristic
2	#11763497	C₂	1	1	yes	maximal, characteristic
3	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763143	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763066	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763066	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762886	C₂	1	1	yes	maximal, characteristic
13	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	#10587632	C₂²	1	1	yes	characteristic
19	#10587632	C₂²	1	1	yes	characteristic
20	C₂×#63231	C₂²	1	1	yes	characteristic
21	C₂×#63218	C₂²	1	1	yes	characteristic
22	C₂×#63218	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	#10587580	—	2	1	no	—
27	#10587580	—	2	1	no	—
28	#10587578	—	2	1	no	—
29	#10587578	C₂²	1	1	yes	characteristic
30	#10587578	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587576	—	2	1	no	—
34	#10587575	—	2	1	no	—
35	#10587562	—	2	1	no	—
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	#10587558	—	2	1	no	—
38	#10587556	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	#10587506	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	#10587488	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587479	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	C₂³	1	1	yes	characteristic
60	C₂×D₆₈	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇	—	2	1	no	—
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	#63231	—	2	2	no	—
77	#63227	—	2	1	no	—
78	#63227	—	2	1	no	—
79	#63227	—	2	1	no	—
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	D₄	1	1	yes	characteristic
84	#63227	D₄	1	1	yes	characteristic
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
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	#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	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	#63218	—	2	2	no	—
104	#63218	—	2	2	no	—
105	C₂²×D₁₇	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	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	—	2	1	no	—
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	D₆₈	—	4	1	no	—
123	C₄×D₁₇	—	2	1	no	—
124	C₄×D₁₇	—	2	1	no	—
125	C₄×D₁₇	—	2	1	no	—
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	4	2	no	—
128	C₄×D₁₇	—	4	2	no	—
129	Dic₃₄	—	2	1	no	—
130	Dic₃₄	—	2	1	no	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	#3622	—	2	1	no	—
134	#3622	#41	1	2	yes	—
135	#3352	—	17	1	no	maximal, Sylow
136	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₇	—	2	1	no	—
138	C₂×D₁₇	—	2	1	no	—
139	C₂×D₁₇	—	2	1	no	—
140	C₂×D₁₇	—	2	1	no	—
141	C₂×D₁₇	—	2	1	no	—
142	C₂×D₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	—	2	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	#552	—	17	1	no	—
158	#552	—	17	1	no	—
159	#499	—	17	1	no	—
160	#485	—	17	1	no	—
161	#485	—	17	1	no	—
162	#484	—	17	1	no	—
163	#484	—	17	1	no	—
164	#479	—	17	1	no	—
165	#479	D₁₇	1	1	yes	characteristic, complemented
166	#470	—	17	1	no	—
167	#470	—	17	1	no	—
168	C₄×Q₁₆	—	17	1	no	—
169	#436	—	17	1	no	—
170	#428	—	17	1	no	—
171	#428	—	17	1	no	—
172	C₂×C₁₇	#547	1	1	yes	characteristic
173	C₂×C₁₇	#575	1	1	yes	characteristic
174	C₂×C₁₇	#578	1	1	yes	characteristic
175	D₁₇	—	4	1	no	—
176	D₁₇	—	4	1	no	—
177	C₂×Q₁₆	—	17	1	no	—
178	C₂×M₁₆	—	17	1	no	—
179	C₂×M₁₆	—	17	1	no	—
180	#126	—	34	1	no	—
181	#126	—	34	1	no	—
182	#125	—	17	1	no	—
183	#125	—	34	1	no	—
184	#125	—	34	1	no	—
185	#125	C₂×D₁₇	1	1	yes	characteristic
186	#123	—	34	1	no	—
187	#123	—	34	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	17	1	no	—
192	C₄×Q₈	—	17	1	no	—
193	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
194	C₄×D₄	—	17	1	no	—
195	#117	—	17	1	no	—
196	#117	—	17	1	no	—
197	#117	—	34	1	no	—
198	#117	—	34	1	no	—
199	C₂×C₄⋊C₄	—	17	1	no	—
200	C₂×C₄⋊C₄	—	17	1	no	—
201	#107	—	17	1	no	—
202	#107	—	17	1	no	—
203	#107	—	17	1	no	—
204	#107	—	17	1	no	—
205	#107	C₂×D₁₇	1	1	yes	characteristic
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	C₂×D₁₇	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	C₂×D₁₇	1	1	yes	characteristic
216	#103	C₂×D₁₇	1	1	yes	characteristic
217	C₂²⋊C₈	—	17	1	no	—
218	C₂²⋊C₈	—	17	1	no	—
219	C₄×C₈	—	17	1	no	—
220	C₁₇	#3352	1	1	yes	characteristic, complemented, Sylow
221	C₂×Q₈	—	17	1	no	—
222	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
223	C₂×D₄	—	17	1	no	—
224	C₂²×C₄	—	17	1	no	—
225	C₂²×C₄	—	17	1	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	—	34	1	no	—
228	Q₁₆	—	34	2	no	—
229	M₁₆	—	34	2	no	—
230	M₁₆	—	34	2	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	—	34	1	no	—
248	C₄⋊C₄	—	34	1	no	—
249	C₄⋊C₄	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
254	C₂²⋊C₄	—	17	1	no	—
255	C₂²⋊C₄	—	17	1	no	—
256	C₂²⋊C₄	—	34	1	no	—
257	C₂²⋊C₄	—	34	1	no	—
258	C₂²⋊C₄	—	34	1	no	—
259	C₂²⋊C₄	—	34	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	34	1	no	—
264	C₄²	—	34	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	C₂²×D₁₇	1	1	yes	characteristic
267	C₂³	—	17	1	no	—
268	C₂³	—	17	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	34	1	no	—
272	Q₈	—	34	1	no	—
273	D₄	—	68	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	17	1	no	—
279	C₂×C₄	—	17	1	no	—
280	C₂×C₄	—	34	1	no	—
281	C₂×C₄	—	34	1	no	—
282	C₂×C₄	—	34	1	no	—
283	C₂×C₄	—	34	1	no	—
284	C₂×C₄	—	34	1	no	—
285	C₂×C₄	—	34	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	68	2	no	—
289	C₂×C₄	—	68	2	no	—
290	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
291	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	17	2	no	—
296	C₈	—	34	1	no	—
297	C₂²	—	34	1	no	—
298	C₂²	—	34	1	no	—
299	C₂²	—	34	1	no	—
300	C₂²	—	34	1	no	—
301	C₂²	—	34	1	no	—
302	C₂²	—	34	1	no	—
303	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	34	1	no	—
311	C₄	—	34	1	no	—
312	C₄	—	68	1	no	—
313	C₄	—	68	1	no	—
314	C₄	—	68	1	no	—
315	C₄	—	68	1	no	—
316	C₄	#10587694	1	1	yes	characteristic
317	C₄	#41×D₁₇	1	1	yes	characteristic
318	C₂	—	68	1	no	—
319	C₂	—	68	1	no	—
320	C₂	#11763461	1	1	yes	characteristic
321	C₂	#11763606	1	1	yes	characteristic
322	C₂	#11763625	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×#479	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	D₄
hypercenter	#479	D₁₇
hypocenter	C₁₇	#3352
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #11763208	C₂	semidirect
4. #11763153	C₂	semidirect
5. #11763148	C₂	semidirect
6. #11763146	C₂	semidirect
7. #11763143	C₂	semidirect
8. #11763117	C₂	semidirect
9. #11763066	C₂	semidirect
10. #11763066	C₂	semidirect
11. C₄×#63231	C₂	semidirect
13. #11762850	C₂	semidirect
14. #11762848	C₂	semidirect
15. C₁₇×#479	C₂	semidirect
165. #479	D₁₇	semidirect
220. C₁₇	#3352	semidirect

Maximal subgroups

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

subgroup	index
#11763500	2
#11763497	2
#11763208	2
#11763153	2
#11763148	2
#11763146	2
#11763143	2
#11763117	2
#11763066	2
C₄×#63231	2
#11762886	2
#11762850	2
#11762848	2
C₁₇×#479	2
#3352	17