Subgroups — #14538608

Order 2176.

398 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	#11763436	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763281	C₂	1	1	yes	maximal, characteristic
3	#11763112	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763107	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762846	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762834	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762812	C₂	1	1	yes	maximal, characteristic
14	#102×D₁₇	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587632	—	2	1	no	—
20	#10587630	—	2	1	no	—
21	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	C₂×#63227	—	2	1	no	—
24	C₂×#63218	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
26	#10587582	—	2	1	no	—
27	#10587579	C₂²	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	#10587578	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
31	#10587576	C₂²	1	1	yes	characteristic, complemented
32	#10587575	—	2	1	no	—
33	#10587574	C₂²	1	1	yes	characteristic, complemented
34	#10587567	—	2	1	no	—
35	C₂²⋊C₄×D₁₇	—	2	1	no	—
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	#10587558	C₂²	1	1	yes	characteristic
38	#10587556	C₂²	1	1	yes	characteristic, complemented
39	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
40	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic, complemented
49	#10587488	C₂²	1	1	yes	characteristic, complemented
50	#10587485	C₂²	1	1	yes	characteristic, complemented
51	#10587485	C₂²	1	1	yes	characteristic, complemented
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic, complemented
55	#10587479	C₂²	1	1	yes	characteristic, complemented
56	#10587479	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊#3622	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	D₄×D₁₇	—	2	2	no	—
66	D₄×D₁₇	Q₈	1	4	yes	complemented
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	—	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	D₄	1	1	yes	characteristic
87	#63227	D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	—	2	1	no	—
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	#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	D₄	1	1	yes	characteristic
98	#63225	D₄	1	1	yes	characteristic
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	#63218	—	2	2	no	—
105	C₈×D₁₇	—	2	2	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	C₂²×D₁₇	—	2	4	no	—
109	C₂²×D₁₇	—	8	1	no	—
110	C₂²×D₁₇	—	8	1	no	—
111	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₇	—	8	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	C₂×Q₈	1	2	yes	—
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	—	8	1	no	—
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	2	4	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	8	1	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	D₆₈	—	2	1	no	—
131	D₆₈	C₂×Q₈	1	2	yes	—
132	C₄×D₁₇	—	8	1	no	—
133	C₄×D₁₇	—	8	1	no	—
134	C₄×D₁₇	—	8	1	no	—
135	C₄×D₁₇	C₂×Q₈	1	2	yes	—
136	C₄×D₁₇	#41	1	2	yes	—
137	Dic₃₄	—	8	1	no	—
138	C₈×C₁₇	—	2	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3081	—	17	1	no	maximal, Sylow
143	C₂²×C₁₇	—	2	2	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	2	1	no	—
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	2	2	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	8	4	no	—
153	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	C₂²⋊Q₈	1	2	yes	—
167	#544	—	17	1	no	—
168	#518	—	17	1	no	—
169	#478	—	17	1	no	—
170	#478	—	17	1	no	—
171	#478	—	17	1	no	—
172	#478	—	17	1	no	—
173	#476	—	17	1	no	—
174	#476	—	17	1	no	—
175	#474	—	17	1	no	—
176	#474	D₁₇	1	1	yes	characteristic, complemented
177	#428	—	17	1	no	—
178	#427	—	17	1	no	—
179	#424	—	17	1	no	—
180	#418	—	17	1	no	—
181	C₂×#102	—	17	1	no	—
182	C₂×C₁₇	—	4	2	no	—
183	C₂×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₁₇	#576	1	1	yes	characteristic
185	C₂×C₁₇	#577	1	1	yes	characteristic
186	D₁₇	—	2	2	no	—
187	D₁₇	—	4	2	no	—
188	C₂²×D₄	—	17	1	no	—
189	C₂×M₁₆	—	17	1	no	—
190	C₂²×C₈	—	17	1	no	—
191	C₄⋊Q₈	—	17	1	no	—
192	C₄⋊Q₈	—	34	1	no	—
193	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic, complemented
194	#125	—	17	1	no	—
195	#125	—	17	1	no	—
196	#125	—	34	1	no	—
197	C₂²⋊Q₈	—	34	1	no	—
198	C₂²⋊Q₈	—	34	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	34	1	no	—
203	C₄×D₄	—	34	1	no	—
204	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
205	#117	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	C₂×C₂²⋊C₄	—	34	1	no	—
210	C₂×C₂²⋊C₄	—	34	1	no	—
211	#107	—	17	1	no	—
212	#107	—	17	1	no	—
213	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
214	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
215	#106	—	17	1	no	—
216	#106	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	C₄⋊C₈	—	17	1	no	—
221	C₄⋊C₈	—	17	1	no	—
222	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	#102	C₂×D₁₇	1	1	yes	characteristic
230	#102	C₂×D₁₇	1	1	yes	characteristic
231	C₁₇	#3081	1	1	yes	characteristic, complemented, Sylow
232	C₂⁴	—	34	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	34	1	no	—
235	C₂×D₄	—	17	4	no	—
236	C₂×D₄	—	17	1	no	—
237	C₂×D₄	—	34	2	no	—
238	C₂×D₄	—	34	1	no	—
239	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	17	1	no	—
242	C₂²×C₄	—	34	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	—	17	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	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	17	1	no	—
269	C₂²⋊C₄	—	17	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	—	17	1	no	—
276	C₄²	—	34	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	C₂³	—	34	4	no	—
282	C₂³	—	34	1	no	—
283	C₂³	—	136	1	no	—
284	C₂³	—	136	1	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	136	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	17	2	no	—
289	D₄	—	34	4	no	—
290	D₄	—	34	1	no	—
291	D₄	Q₈×D₁₇	1	2	yes	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	8	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	—	136	1	no	—
305	C₂×C₄	—	136	1	no	—
306	C₂×C₄	—	136	1	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	—	2	2	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	—	34	2	no	—
322	C₂²	—	68	1	no	—
323	C₂²	—	68	1	no	—
324	C₂²	—	136	4	no	—
325	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	17	2	no	—
332	C₄	—	34	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	68	1	no	—
335	C₄	—	68	1	no	—
336	C₄	—	68	1	no	—
337	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
338	C₄	#41×D₁₇	1	1	yes	characteristic
339	C₂	—	4	2	no	—
340	C₂	—	34	2	no	—
341	C₂	—	68	2	no	—
342	C₂	C₂²⋊Q₈×D₁₇	1	1	yes	characteristic
343	C₂	#11763610	1	1	yes	characteristic
344	C₂	#11763618	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×#474	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×D₁₇	C₂²
hypercenter	#474	D₁₇
hypocenter	C₁₇	#3081
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #11763436	C₂	semidirect
3. #11763112	C₂	semidirect
4. #11763110	C₂	semidirect
5. #11763109	C₂	semidirect
6. #11763107	C₂	semidirect
7. #11763097	C₂	semidirect
8. #11763096	C₂	semidirect
9. #11763088	C₂	semidirect
10. #11762850	C₂	semidirect
11. #11762846	C₂	semidirect
12. #11762834	C₂	semidirect
15. C₁₇×#474	C₂	semidirect
17. C₁₇×C₄⋊Q₈	C₂²	semidirect
31. #10587576	C₂²	semidirect
33. #10587574	C₂²	semidirect
38. #10587556	C₂²	semidirect
39. C₁₇×#107	C₂²	semidirect
40. C₁₇×#107	C₂²	semidirect
41. C₁₇×C₄⋊C₈	C₂²	semidirect
48. #10587489	C₂²	semidirect
49. #10587488	C₂²	semidirect
50. #10587485	C₂²	semidirect
51. #10587485	C₂²	semidirect
54. #10587479	C₂²	semidirect
55. #10587479	C₂²	semidirect
56. #10587479	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. C₄⋊#3622	C₂²	semidirect
66. D₄×D₁₇	Q₈	semidirect
176. #474	D₁₇	semidirect
193. C₄⋊Q₈	C₂×D₁₇	semidirect
213. #107	C₂×D₁₇	semidirect
214. #107	C₂×D₁₇	semidirect
222. C₄⋊C₈	C₂×D₁₇	semidirect
231. C₁₇	#3081	semidirect

Maximal subgroups

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

subgroup	index
#11763436	2
#11763281	2
#11763112	2
#11763110	2
#11763109	2
#11763107	2
#11763097	2
#11763096	2
#11763088	2
#11762850	2
#11762846	2
#11762834	2
#11762812	2
#102×D₁₇	2
C₁₇×#474	2
#3081	17