Subgroups — #14539146

Order 2176.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763436	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763379	C₂	1	1	yes	maximal, characteristic
3	#11763178	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763176	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763163	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763161	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762807	C₂	1	1	yes	maximal, characteristic
12	#11762802	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762766	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63254	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#124	C₂²	1	1	yes	characteristic
20	#10587632	—	2	1	no	—
21	#10587630	—	2	1	no	—
22	#10587630	C₂²	1	1	yes	characteristic
23	#10587622	C₂²	1	1	yes	characteristic
24	C₂×#63218	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
26	#10587582	—	2	1	no	—
27	#10587578	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
29	#10587576	C₂²	1	1	yes	characteristic
30	#10587575	—	2	1	no	—
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587569	—	2	1	no	—
33	#10587569	—	2	1	no	—
34	#10587564	—	2	1	no	—
35	#10587564	—	2	1	no	—
36	#10587558	C₂²	1	1	yes	characteristic
37	#10587556	C₂²	1	1	yes	characteristic
38	#10587555	C₂²	1	1	yes	characteristic
39	C₁₇×#103	C₂²	1	1	yes	characteristic
40	C₁₇×#103	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587486	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587481	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	—	2	1	no	—
64	Q₈×D₁₇	—	2	2	no	—
65	Q₈×D₁₇	—	2	2	no	—
66	#63254	—	2	2	no	—
67	#63254	—	2	2	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	—	2	1	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	D₄	1	1	yes	characteristic
87	#63227	D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	D₄	1	1	yes	characteristic
101	#63225	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	8	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₇	—	4	2	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	—	4	2	no	—
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₄×D₁₇	—	4	2	no	—
131	C₄×D₁₇	—	8	1	no	—
132	C₄×D₁₇	#41	1	2	yes	—
133	C₄×D₁₇	#41	1	2	yes	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	4	2	no	—
139	Dic₃₄	—	8	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3137	—	17	1	no	maximal, Sylow
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	#124	1	2	yes	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	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	C₂×#41	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	Dic₁₇	—	4	1	no	—
165	Dic₁₇	#124	1	2	yes	—
166	#544	—	17	1	no	—
167	#536	—	17	1	no	—
168	#491	—	17	1	no	—
169	#491	—	17	1	no	—
170	#489	—	17	1	no	—
171	#489	D₁₇	1	1	yes	characteristic, complemented
172	#488	—	17	1	no	—
173	#488	—	17	1	no	—
174	#487	—	17	1	no	—
175	#487	—	17	1	no	—
176	#419	—	17	1	no	—
177	#417	—	17	1	no	—
178	#416	—	17	1	no	—
179	C₂×#103	—	17	1	no	—
180	#405	—	17	1	no	—
181	C₂×C₁₇	—	8	1	no	—
182	C₂×C₁₇	C₂×#124	1	1	yes	characteristic
183	C₂×C₁₇	#577	1	1	yes	characteristic
184	C₂×C₁₇	#578	1	1	yes	characteristic
185	D₁₇	—	2	2	no	—
186	C₂×#41	—	17	1	no	—
187	C₂²×Q₈	—	17	1	no	—
188	C₂×M₁₆	—	17	1	no	—
189	C₂²×C₈	—	17	1	no	—
190	C₄⋊Q₈	—	17	1	no	—
191	C₄⋊Q₈	—	17	1	no	—
192	C₄⋊Q₈	—	17	1	no	—
193	C₄⋊Q₈	—	34	1	no	—
194	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
195	#125	—	17	1	no	—
196	#125	—	17	1	no	—
197	#125	—	34	1	no	—
198	#124	—	17	1	no	—
199	#124	C₂×D₁₇	1	1	yes	characteristic
200	#123	—	34	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	C₂²⋊Q₈	—	34	1	no	—
204	C₂²⋊Q₈	—	34	1	no	—
205	C₂²⋊Q₈	—	34	1	no	—
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	#103	—	17	1	no	—
210	#103	—	17	1	no	—
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	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	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	#102	C₂×D₁₇	1	1	yes	characteristic
224	#102	C₂×D₁₇	1	1	yes	characteristic
225	#97	—	17	1	no	—
226	#97	C₂×D₁₇	1	1	yes	characteristic
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₁₇	#3137	1	1	yes	characteristic, complemented, Sylow
230	#41	—	34	2	no	—
231	#41	—	34	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	17	1	no	—
234	C₂×Q₈	—	17	1	no	—
235	C₂×Q₈	—	34	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×Q₈	—	34	2	no	—
238	C₂×Q₈	—	34	2	no	—
239	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×D₄	—	34	1	no	—
241	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
242	C₂²×C₄	—	17	1	no	—
243	C₂²×C₄	—	34	1	no	—
244	C₂²×C₄	—	34	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	M₁₆	—	34	2	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	34	2	no	—
250	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	17	1	no	—
254	C₄⋊C₄	—	17	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	—	17	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	34	1	no	—
278	C₄²	C₂²×D₁₇	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	17	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	8	1	no	—
284	Q₈	—	34	1	no	—
285	Q₈	—	34	1	no	—
286	Q₈	—	34	1	no	—
287	Q₈	—	34	1	no	—
288	Q₈	—	68	2	no	—
289	Q₈	—	136	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	68	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	17	2	no	—
298	C₂×C₄	—	17	2	no	—
299	C₂×C₄	—	17	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	68	2	no	—
306	C₂×C₄	—	68	2	no	—
307	C₂×C₄	—	136	1	no	—
308	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	34	1	no	—
314	C₈	—	34	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	17	2	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	1	no	—
321	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	17	2	no	—
328	C₄	—	34	1	no	—
329	C₄	—	34	1	no	—
330	C₄	—	68	1	no	—
331	C₄	—	68	1	no	—
332	C₄	—	68	1	no	—
333	C₄	—	68	1	no	—
334	C₄	#41×D₁₇	1	1	yes	characteristic
335	C₄	#41×D₁₇	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	34	2	no	—
338	C₂	#124×D₁₇	1	1	yes	characteristic
339	C₂	#11763619	1	1	yes	characteristic
340	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×#489	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63254	C₂²
hypercenter	#489	D₁₇
hypocenter	C₁₇	#3137
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #11763436	C₂	semidirect
3. #11763178	C₂	semidirect
4. #11763176	C₂	semidirect
5. #11763167	C₂	semidirect
6. #11763163	C₂	semidirect
7. #11763162	C₂	semidirect
8. #11763161	C₂	semidirect
9. #11763158	C₂	semidirect
10. #11762816	C₂	semidirect
13. #103×D₁₇	C₂	semidirect
14. #11762766	C₂	semidirect
15. C₁₇×#489	C₂	semidirect
171. #489	D₁₇	semidirect
229. C₁₇	#3137	semidirect

Maximal subgroups

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

subgroup	index
#11763436	2
#11763379	2
#11763178	2
#11763176	2
#11763167	2
#11763163	2
#11763162	2
#11763161	2
#11763158	2
#11762816	2
#11762807	2
#11762802	2
#103×D₁₇	2
#11762766	2
C₁₇×#489	2
#3137	17