Subgroups — #14539507

Order 2176.

370 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763343	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763191	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763166	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763163	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763065	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763060	C₂	1	1	yes	maximal, characteristic
8	#11763057	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763053	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762919	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762907	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63223	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#126	—	2	1	no	—
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₁₇×#124	C₂²	1	1	yes	characteristic
19	C₁₇×#123	—	2	1	no	—
20	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×#121	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₇	—	2	1	no	—
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#117	—	2	1	no	—
26	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×#63231	C₂²	1	1	yes	characteristic
28	C₂×#63230	C₂²	1	1	yes	characteristic
29	C₂×#63229	C₂²	1	1	yes	characteristic
30	C₂×#63228	C₂²	1	1	yes	characteristic
31	#10587616	C₂²	1	1	yes	characteristic
32	#10587615	—	2	1	no	—
33	C₄×C₂²⋊D₁₇	—	2	1	no	—
34	#10587610	C₂²	1	1	yes	characteristic
35	C₂×#63223	C₂²	1	1	yes	characteristic
36	C₂×#63223	C₂²	1	1	yes	characteristic
37	#10587562	—	2	1	no	—
38	#10587561	C₂²	1	1	yes	characteristic
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587558	—	2	1	no	—
41	#10587556	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587481	C₂²	1	1	yes	characteristic
48	#10587481	C₂²	1	1	yes	characteristic
49	#10587480	C₂²	1	1	yes	characteristic
50	#10587480	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	#10587474	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	C₄×#3622	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
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	C₂²⋊C₄×C₁₇	—	2	1	no	—
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	D₄	1	1	yes	characteristic
83	C₄²×C₁₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	#63231	—	2	2	no	—
88	#63230	—	2	2	no	—
89	#63229	—	2	2	no	—
90	#63228	—	2	2	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#63223	—	2	2	no	—
103	#63223	—	2	2	no	—
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	4	1	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	4	2	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	D₆₈	—	2	1	no	—
135	D₆₈	—	2	1	no	—
136	C₄×D₁₇	—	4	2	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	#41	1	2	yes	—
142	#3622	#41	1	2	yes	—
143	#3183	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	—	4	1	no	—
150	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	#529	—	17	1	no	—
168	#529	D₁₇	1	1	yes	characteristic, complemented
169	#499	—	17	1	no	—
170	#495	—	17	1	no	—
171	#488	—	17	1	no	—
172	#488	—	17	1	no	—
173	#470	—	17	1	no	—
174	#469	—	17	1	no	—
175	#468	—	17	1	no	—
176	#468	—	17	1	no	—
177	#442	—	17	1	no	—
178	#441	—	17	1	no	—
179	#440	—	17	1	no	—
180	#440	—	17	1	no	—
181	C₄×M₁₆	—	17	1	no	—
182	C₂×C₁₇	—	4	1	no	—
183	C₂×C₁₇	—	8	1	no	—
184	C₂×C₁₇	#532	1	1	yes	characteristic
185	C₂×C₁₇	#575	1	1	yes	characteristic
186	C₂×C₁₇	#575	1	1	yes	characteristic
187	D₁₇	—	8	1	no	—
188	C₂×Q₁₆	—	17	1	no	—
189	C₂×SD₁₆	—	17	1	no	—
190	C₂×SD₁₆	—	17	1	no	—
191	C₂×D₈	—	17	1	no	—
192	C₂×M₁₆	—	17	1	no	—
193	C₂×M₁₆	—	17	1	no	—
194	#126	—	2	1	no	—
195	#126	—	34	1	no	—
196	#125	—	17	1	no	—
197	#125	C₂×D₁₇	1	1	yes	characteristic
198	#124	—	17	1	no	—
199	#124	C₂×D₁₇	1	1	yes	characteristic
200	#123	—	2	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	17	1	no	—
203	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
204	#121	—	17	1	no	—
205	#121	C₂×D₁₇	1	1	yes	characteristic
206	C₄×Q₈	—	17	1	no	—
207	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	—	17	1	no	—
210	C₄×D₄	—	34	1	no	—
211	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
212	#117	—	2	1	no	—
213	#117	—	34	1	no	—
214	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#97	—	17	1	no	—
228	#97	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	1	no	—
231	C₁₇	#3183	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	17	1	no	—
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	17	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
243	Q₁₆	—	34	2	no	—
244	SD₁₆	—	34	2	no	—
245	SD₁₆	—	34	2	no	—
246	D₈	—	34	2	no	—
247	M₁₆	—	34	2	no	—
248	M₁₆	—	34	2	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	17	1	no	—
251	C₂×C₈	—	17	1	no	—
252	C₂×C₈	—	17	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	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	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	34	1	no	—
277	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
278	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
279	C₄²	C₂²×D₁₇	1	1	yes	characteristic
280	C₄²	C₂²×D₁₇	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	34	1	no	—
283	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	34	1	no	—
287	Q₈	—	34	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	34	1	no	—
292	D₄	—	34	1	no	—
293	D₄	—	68	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	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	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
312	C₈	—	17	2	no	—
313	C₈	—	17	2	no	—
314	C₈	—	34	1	no	—
315	C₈	—	34	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	68	1	no	—
323	C₂²	—	68	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	68	1	no	—
335	C₄	—	68	1	no	—
336	C₄	—	68	1	no	—
337	C₄	#41×D₁₇	1	1	yes	characteristic
338	C₄	#41×D₁₇	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	136	1	no	—
342	C₂	#11763355	1	1	yes	characteristic
343	C₂	#11763607	1	1	yes	characteristic
344	C₂	#11763607	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#529	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763060	C₂
hypercenter	#529	D₁₇
hypocenter	C₁₇	#3183
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #11763343	C₂	semidirect
2. #11763208	C₂	semidirect
3. #11763191	C₂	semidirect
4. #11763166	C₂	semidirect
5. #11763163	C₂	semidirect
6. #11763065	C₂	semidirect
8. #11763057	C₂	semidirect
9. #11763053	C₂	semidirect
10. #11762919	C₂	semidirect
11. #11762914	C₂	semidirect
12. #11762909	C₂	semidirect
13. #11762907	C₂	semidirect
14. C₄×#63223	C₂	semidirect
15. C₁₇×#529	C₂	semidirect
168. #529	D₁₇	semidirect
231. C₁₇	#3183	semidirect

Maximal subgroups

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

subgroup	index
#11763343	2
#11763208	2
#11763191	2
#11763166	2
#11763163	2
#11763065	2
#11763060	2
#11763057	2
#11763053	2
#11762919	2
#11762914	2
#11762909	2
#11762907	2
C₄×#63223	2
C₁₇×#529	2
#3183	17