Subgroups — #14541224

Order 2176.

376 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	#11763463	C₂	1	1	yes	maximal, characteristic
2	#11763219	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763142	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763137	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763133	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763122	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763104	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763057	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763040	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762907	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762881	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762852	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10587478	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#552	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	—	2	1	no	—
17	C₁₇×#126	—	2	1	no	—
18	C₁₇×#125	—	2	1	no	—
19	C₁₇×#125	C₂²	1	1	yes	characteristic
20	C₁₇×#123	—	2	1	no	—
21	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#117	—	2	1	no	—
26	C₁₇×#117	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
28	C₂×#63230	C₂²	1	1	yes	characteristic
29	#10587616	C₂²	1	1	yes	characteristic
30	#10587616	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
33	C₂×#63223	C₂²	1	1	yes	characteristic
34	C₂²×#3622	C₂²	1	1	yes	characteristic
35	#10587583	—	2	1	no	—
36	#10587572	—	2	1	no	—
37	#10587572	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	#10587555	C₂²	1	1	yes	characteristic
40	#10587506	C₂²	1	1	yes	characteristic
41	#10587506	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	C₂²⋊#3622	C₂²	1	1	yes	characteristic
44	C₂²⋊#3622	C₂²	1	1	yes	characteristic
45	#10587480	C₂²	1	1	yes	characteristic
46	#10587480	C₂²	1	1	yes	characteristic
47	#10587480	C₂²	1	1	yes	characteristic
48	#10587479	C₂²	1	1	yes	characteristic
49	#10587479	C₂²	1	1	yes	characteristic
50	#10587479	C₂²	1	1	yes	characteristic
51	#10587478	C₂²	1	1	yes	characteristic
52	#10587478	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	C₄⋊#3622	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	C₂²×Dic₁₇	—	2	1	no	—
68	#63254	—	4	4	no	—
69	C₂×D₆₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×Dic₃₄	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	—	2	1	no	—
89	C₄²×C₁₇	C₂³	1	1	yes	characteristic
90	#63230	—	2	2	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	#63223	—	2	2	no	—
101	C₂×#3622	—	2	2	no	—
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×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	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	4	2	no	—
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₇	—	4	2	no	—
127	C₂²⋊D₁₇	—	4	2	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	4	2	no	—
132	C₂×Dic₁₇	—	4	2	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	2	1	no	—
135	C₄×D₁₇	—	4	2	no	—
136	Dic₃₄	—	4	2	no	—
137	#3622	—	2	1	no	—
138	#3622	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	#41	1	2	yes	complemented
141	#3343	—	17	1	no	maximal, Sylow
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
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	—	4	1	no	—
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	#552	D₁₇	1	1	yes	characteristic, complemented
166	#547	—	17	1	no	—
167	#501	—	17	1	no	—
168	#484	—	17	1	no	—
169	#483	—	17	1	no	—
170	#482	—	17	1	no	—
171	#480	—	17	1	no	—
172	#478	—	17	1	no	—
173	#477	—	17	1	no	—
174	#468	—	17	1	no	—
175	#465	—	17	1	no	—
176	#440	—	17	1	no	—
177	#435	—	17	1	no	—
178	#428	—	17	1	no	—
179	C₂×#107	—	17	1	no	—
180	C₂×C₁₇	—	2	2	no	—
181	C₂×C₁₇	—	4	1	no	—
182	C₂×C₁₇	#547	1	1	yes	characteristic
183	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₁₇	#578	1	1	yes	characteristic
185	D₁₇	—	8	1	no	—
186	C₂×#41	—	34	1	no	—
187	C₂×SD₁₆	—	17	1	no	—
188	C₂×M₁₆	—	17	1	no	—
189	C₂²×C₈	—	17	1	no	—
190	C₄⋊Q₈	—	17	1	no	—
191	#126	—	2	1	no	—
192	#125	—	2	1	no	—
193	#125	C₂×D₁₇	1	1	yes	characteristic
194	#123	—	2	1	no	—
195	#123	—	34	1	no	—
196	#123	—	34	1	no	—
197	C₂²⋊Q₈	—	34	1	no	—
198	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
199	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
200	#121	—	17	1	no	—
201	#121	—	17	1	no	—
202	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
205	#117	—	2	1	no	—
206	#117	C₂×D₁₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	—	17	1	no	—
213	#107	—	17	1	no	—
214	#107	—	17	1	no	—
215	#107	—	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	#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	C₂²⋊C₈	—	17	1	no	—
227	C₂²⋊C₈	—	17	1	no	—
228	#97	—	17	1	no	—
229	C₁₇	#3343	1	1	yes	characteristic, complemented, Sylow
230	#41	—	68	4	no	—
231	C₂×Q₈	—	34	1	no	—
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂×D₄	—	34	1	no	—
236	C₂×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
243	SD₁₆	—	34	2	no	—
244	M₁₆	—	34	2	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	34	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	17	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	C₂²⋊D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
273	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	C₂²×D₁₇	1	1	yes	characteristic
277	C₂³	—	34	1	no	—
278	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	68	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	34	1	no	—
285	D₄	—	34	1	no	—
286	D₄	—	68	2	no	—
287	D₄	—	68	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	68	2	no	—
301	C₂×C₄	—	68	2	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	17	2	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₈	—	34	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	68	1	no	—
318	C₂²	—	68	1	no	—
319	C₂²	—	68	1	no	—
320	C₂²	#10587635	1	2	yes	—
321	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	68	1	no	—
331	C₄	—	68	1	no	—
332	C₄	—	68	1	no	—
333	C₄	#41×D₁₇	1	1	yes	characteristic
334	C₄	#10587697	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	136	1	no	—
338	C₂	#11763458	1	1	yes	characteristic
339	C₂	C₂×#10587635	1	1	yes	characteristic
340	C₂	#11763624	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×#552	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763463	C₂
hypercenter	#552	D₁₇
hypocenter	C₁₇	#3343
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

Complements

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

N	H ≅ G/N	type
2. #11763219	C₂	semidirect
3. #11763142	C₂	semidirect
4. #11763137	C₂	semidirect
5. #11763133	C₂	semidirect
6. #11763122	C₂	semidirect
7. #11763106	C₂	semidirect
8. #11763104	C₂	semidirect
9. #11763057	C₂	semidirect
10. #11763040	C₂	semidirect
11. #11762907	C₂	semidirect
12. #11762881	C₂	semidirect
13. #11762852	C₂	semidirect
14. C₂×#10587478	C₂	semidirect
15. C₁₇×#552	C₂	semidirect
140. #3622	#41	semidirect
165. #552	D₁₇	semidirect
229. C₁₇	#3343	semidirect

Maximal subgroups

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

subgroup	index
#11763463	2
#11763219	2
#11763142	2
#11763137	2
#11763133	2
#11763122	2
#11763106	2
#11763104	2
#11763057	2
#11763040	2
#11762907	2
#11762881	2
#11762852	2
C₂×#10587478	2
C₁₇×#552	2
#3343	17