Subgroups — #14539919

Order 2176.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763365	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763356	C₂	1	1	yes	maximal, characteristic
3	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763154	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763154	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763136	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763135	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762861	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	—	2	1	no	—
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587626	C₂²	1	1	yes	characteristic
20	#10587624	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	—	2	1	no	—
22	#10587582	—	2	1	no	—
23	#10587582	—	2	1	no	—
24	#10587581	C₂²	1	1	yes	characteristic
25	#10587578	C₂²	1	1	yes	characteristic
26	#10587578	C₂²	1	1	yes	characteristic
27	#10587574	C₂²	1	1	yes	characteristic
28	#10587574	C₂²	1	1	yes	characteristic
29	#10587574	C₂²	1	1	yes	characteristic
30	#10587573	—	2	1	no	—
31	#10587573	—	2	1	no	—
32	#10587571	—	2	1	no	—
33	#10587570	—	2	1	no	—
34	C₄×D₆₈	C₂²	1	1	yes	characteristic
35	C₄²×D₁₇	C₂²	1	1	yes	characteristic
36	#10587556	C₂²	1	1	yes	characteristic
37	C₁₇×#107	C₂²	1	1	yes	characteristic
38	C₁₇×#106	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₇×#102	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	#10587503	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587492	C₂²	1	1	yes	characteristic
45	#10587492	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	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	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×D₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	#63252	—	4	4	no	—
65	C₂×D₆₈	—	2	1	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	—	2	1	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	1	no	—
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	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
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	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	D₄	1	1	yes	characteristic
99	C₄×Dic₁₇	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂³×C₁₇	—	2	1	no	—
104	C₂²×D₁₇	—	2	1	no	—
105	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
106	D₄×C₁₇	—	2	1	no	—
107	D₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	4	2	no	—
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₁₇	—	4	2	no	—
116	C₂²⋊D₁₇	—	4	2	no	—
117	C₂²⋊D₁₇	—	4	2	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	D₆₈	—	2	1	no	—
125	D₆₈	—	2	1	no	—
126	D₆₈	—	4	2	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	2	2	no	—
130	C₄×D₁₇	—	2	2	no	—
131	C₄×D₁₇	—	4	2	no	—
132	Dic₃₄	—	4	2	no	—
133	Dic₃₄	—	4	2	no	—
134	Dic₃₄	—	4	2	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3622	—	2	1	no	—
139	#3228	—	17	1	no	maximal, Sylow
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₁₇	—	2	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	#135	1	2	yes	—
162	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
163	#533	—	17	1	no	—
164	#532	—	17	1	no	—
165	#500	—	17	1	no	—
166	#498	—	17	1	no	—
167	#486	—	17	1	no	—
168	#486	—	17	1	no	—
169	#482	—	17	1	no	—
170	#482	—	17	1	no	—
171	#482	—	17	1	no	—
172	#482	—	17	1	no	—
173	#478	—	17	1	no	—
174	#478	—	17	1	no	—
175	#478	—	17	1	no	—
176	#478	D₁₇	1	1	yes	characteristic, complemented
177	#431	—	17	1	no	—
178	C₂×C₁₇	—	8	1	no	—
179	C₂×C₁₇	#536	1	1	yes	characteristic
180	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
181	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	D₁₇	—	4	1	no	—
183	D₁₇	—	8	1	no	—
184	C₂×#41	—	34	1	no	—
185	C₄⋊Q₈	—	17	1	no	—
186	C₄⋊Q₈	—	17	1	no	—
187	C₄⋊Q₈	—	17	1	no	—
188	#127	—	17	1	no	—
189	#125	—	17	1	no	—
190	#125	C₂×D₁₇	1	1	yes	characteristic
191	#124	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	—	34	1	no	—
194	#121	—	17	1	no	—
195	#121	—	17	1	no	—
196	#121	—	34	1	no	—
197	C₄×Q₈	—	34	1	no	—
198	C₄×Q₈	—	34	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	34	1	no	—
201	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
202	#117	—	17	1	no	—
203	#117	—	17	1	no	—
204	C₂×C₄²	—	17	1	no	—
205	#107	—	17	1	no	—
206	#107	—	17	1	no	—
207	#107	—	17	1	no	—
208	#107	C₂×D₁₇	1	1	yes	characteristic
209	#106	—	17	1	no	—
210	#106	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	C₂×D₁₇	1	1	yes	characteristic
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
215	#102	—	17	1	no	—
216	#102	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	—	17	1	no	—
219	#102	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	C₂×D₁₇	1	1	yes	characteristic
222	#102	C₂×D₁₇	1	1	yes	characteristic
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	C₄×C₈	—	17	1	no	—
226	C₄×C₈	—	17	1	no	—
227	C₁₇	#3228	1	1	yes	characteristic, complemented, Sylow
228	#41	—	68	4	no	—
229	C₂×Q₈	—	34	1	no	—
230	C₂×Q₈	—	34	1	no	—
231	C₂×Q₈	—	34	1	no	—
232	C₂×D₄	—	17	1	no	—
233	C₂×D₄	—	34	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂×D₄	—	34	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	—	17	1	no	—
240	C₂²×C₄	—	17	1	no	—
241	C₂²×C₄	—	17	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₄²	—	17	1	no	—
267	C₄²	—	17	1	no	—
268	C₄²	—	17	1	no	—
269	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	C₂²×D₁₇	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	17	1	no	—
274	C₂³	—	34	1	no	—
275	Q₈	—	68	2	no	—
276	Q₈	—	68	2	no	—
277	Q₈	—	68	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	34	1	no	—
282	D₄	—	68	2	no	—
283	D₄	—	68	2	no	—
284	D₄	—	68	2	no	—
285	D₄	—	68	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	34	2	no	—
294	C₂×C₄	—	34	2	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	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	D₄×D₁₇	1	1	yes	characteristic
302	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	34	1	no	—
307	C₈	—	34	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	34	1	no	—
312	C₂²	—	34	1	no	—
313	C₂²	—	34	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	—	68	1	no	—
316	C₂²	—	68	1	no	—
317	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	17	2	no	—
324	C₄	—	17	2	no	—
325	C₄	—	34	1	no	—
326	C₄	—	68	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	#10587694	1	1	yes	characteristic
330	C₄	#10587697	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	68	1	no	—
333	C₂	—	136	1	no	—
334	C₂	#11763382	1	1	yes	characteristic
335	C₂	#135×D₁₇	1	1	yes	characteristic
336	C₂	AGL(1,ℤ/8)×D₁₇	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×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763356	C₂
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3228
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763365	C₂	semidirect
3. #11763213	C₂	semidirect
4. #11763206	C₂	semidirect
5. #11763154	C₂	semidirect
6. #11763154	C₂	semidirect
7. #11763136	C₂	semidirect
8. #11763135	C₂	semidirect
9. #11763132	C₂	semidirect
10. #11763131	C₂	semidirect
11. #11763110	C₂	semidirect
12. #11763109	C₂	semidirect
13. #11763106	C₂	semidirect
14. #11762861	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
176. #478	D₁₇	semidirect
227. C₁₇	#3228	semidirect

Maximal subgroups

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

subgroup	index
#11763365	2
#11763356	2
#11763213	2
#11763206	2
#11763154	2
#11763136	2
#11763135	2
#11763132	2
#11763131	2
#11763110	2
#11763109	2
#11763106	2
#11762861	2
C₁₇×#478	2
#3228	17