Subgroups — #14541516

Order 2176.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763536	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763480	C₂	1	1	yes	maximal, characteristic
3	#11763175	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763172	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763154	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763108	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763010	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763008	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763008	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762908	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762896	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	#10587626	C₂²	1	1	yes	characteristic
21	C₂×#63229	C₂²	1	1	yes	characteristic
22	#10587616	—	2	1	no	—
23	C₂×#63219	C₂²	1	1	yes	characteristic
24	C₂×#63219	C₂²	1	1	yes	characteristic
25	#10587579	C₂²	1	1	yes	characteristic
26	#10587579	C₂²	1	1	yes	characteristic
27	#10587576	C₂²	1	1	yes	characteristic
28	#10587575	—	2	1	no	—
29	#10587575	—	2	1	no	—
30	#10587574	—	2	1	no	—
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587573	—	2	1	no	—
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587573	C₂²	1	1	yes	characteristic
35	#10587570	—	2	1	no	—
36	#10587570	—	2	1	no	—
37	#10587560	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	C₁₇×#102	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	#10587486	C₂²	1	1	yes	characteristic
49	#10587485	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587474	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	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	#63252	—	4	4	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×D₆₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	D₄	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	—	2	1	no	—
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	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	#63229	—	2	2	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	D₄	1	1	yes	characteristic
96	#63225	D₄	1	1	yes	characteristic
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
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	#63219	—	2	2	no	—
105	#63219	—	2	2	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	C₂²×D₁₇	—	2	1	no	—
109	D₄×C₁₇	—	2	1	no	—
110	D₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	4	2	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	4	2	no	—
119	C₂²⋊D₁₇	—	4	2	no	—
120	C₂²⋊D₁₇	—	8	1	no	—
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	D₆₈	—	2	1	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	—	4	2	no	—
133	D₆₈	—	8	1	no	—
134	C₄×D₁₇	—	4	2	no	—
135	C₄×D₁₇	—	4	2	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	2	2	no	—
138	Dic₃₄	—	4	2	no	—
139	Dic₃₄	#41	1	2	yes	complemented
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3368	—	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	—	4	1	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	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	—	2	2	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
168	#557	—	17	1	no	—
169	#550	—	17	1	no	—
170	#490	—	17	1	no	—
171	#490	—	17	1	no	—
172	#486	—	17	1	no	—
173	#478	—	17	1	no	—
174	#478	D₁₇	1	1	yes	characteristic, complemented
175	#474	—	17	1	no	—
176	#460	—	17	1	no	—
177	#460	—	17	1	no	—
178	#460	—	17	1	no	—
179	#446	—	17	1	no	—
180	#440	—	17	1	no	—
181	#440	—	17	1	no	—
182	C₄×SD₁₆	—	17	1	no	—
183	C₂×C₁₇	—	8	1	no	—
184	C₂×C₁₇	#548	1	1	yes	characteristic
185	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×C₁₇	#577	1	1	yes	characteristic
187	D₁₇	—	8	1	no	—
188	D₁₇	—	8	1	no	—
189	C₂×#41	—	34	1	no	—
190	C₂×SD₁₆	—	17	1	no	—
191	C₂×SD₁₆	—	17	1	no	—
192	C₂×SD₁₆	—	17	1	no	—
193	C₄⋊Q₈	—	17	1	no	—
194	C₄⋊Q₈	—	34	1	no	—
195	#127	—	17	1	no	—
196	#127	—	17	1	no	—
197	#127	—	17	1	no	—
198	#125	—	17	1	no	—
199	#125	—	34	1	no	—
200	#125	—	34	1	no	—
201	#125	C₂×D₁₇	1	1	yes	characteristic
202	#121	—	34	1	no	—
203	#121	—	34	1	no	—
204	#121	—	34	1	no	—
205	C₄×Q₈	—	17	1	no	—
206	C₄×Q₈	—	17	1	no	—
207	C₄×Q₈	—	17	1	no	—
208	C₄×Q₈	—	17	1	no	—
209	C₄×Q₈	—	34	1	no	—
210	C₄×D₄	—	17	1	no	—
211	C₄×D₄	—	17	1	no	—
212	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
213	#107	—	17	1	no	—
214	#107	C₂×D₁₇	1	1	yes	characteristic
215	#106	C₂×D₁₇	1	1	yes	characteristic
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	—	17	1	no	—
218	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	C₂×D₁₇	1	1	yes	characteristic
228	#102	C₂×D₁₇	1	1	yes	characteristic
229	#97	—	17	1	no	—
230	#97	—	17	1	no	—
231	C₄×C₈	—	17	1	no	—
232	C₁₇	#3368	1	1	yes	characteristic, complemented, Sylow
233	#41	—	68	4	no	—
234	C₂×Q₈	—	17	1	no	—
235	C₂×Q₈	—	17	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×D₄	—	17	1	no	—
238	C₂×D₄	—	17	1	no	—
239	C₂×D₄	—	34	1	no	—
240	C₂×D₄	—	34	1	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂×D₄	—	34	1	no	—
244	C₂×D₄	—	34	1	no	—
245	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	SD₁₆	—	34	2	no	—
250	SD₁₆	—	34	2	no	—
251	SD₁₆	—	34	2	no	—
252	C₂×C₈	—	17	1	no	—
253	C₂×C₈	—	17	1	no	—
254	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	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	—	34	1	no	—
268	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₄²	—	17	1	no	—
275	C₄²	—	17	1	no	—
276	C₄²	—	34	1	no	—
277	C₄²	—	34	1	no	—
278	C₄²	—	34	1	no	—
279	C₄²	C₂²×D₁₇	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	34	1	no	—
282	C₂³	—	34	1	no	—
283	Q₈	—	17	2	no	—
284	Q₈	—	34	2	no	—
285	Q₈	—	34	1	no	—
286	Q₈	—	68	2	no	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	34	1	no	—
290	D₄	—	34	1	no	—
291	D₄	—	34	1	no	—
292	D₄	—	34	1	no	—
293	D₄	—	68	2	no	—
294	D₄	—	68	2	no	—
295	D₄	—	68	2	no	—
296	D₄	—	136	1	no	—
297	D₄	—	136	1	no	—
298	D₄	—	136	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	17	1	no	—
306	C₂×C₄	—	17	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	68	2	no	—
310	C₂×C₄	—	68	2	no	—
311	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
312	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	34	1	no	—
317	C₈	—	34	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	68	1	no	—
322	C₂²	—	68	1	no	—
323	C₂²	—	68	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	—	68	1	no	—
326	C₂²	—	68	1	no	—
327	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	17	2	no	—
334	C₄	—	34	2	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	34	1	no	—
338	C₄	—	68	1	no	—
339	C₄	#10587694	1	1	yes	characteristic
340	C₄	#41×D₁₇	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	136	1	no	—
343	C₂	—	136	1	no	—
344	C₂	#11763469	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
346	C₂	#11763616	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	#11763480	C₂
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3368
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. #11763536	C₂	semidirect
3. #11763175	C₂	semidirect
4. #11763172	C₂	semidirect
5. #11763154	C₂	semidirect
6. #11763108	C₂	semidirect
7. #11763088	C₂	semidirect
8. #11763010	C₂	semidirect
9. #11763008	C₂	semidirect
10. #11763008	C₂	semidirect
11. #11762938	C₂	semidirect
12. #11762909	C₂	semidirect
13. #11762908	C₂	semidirect
14. #11762896	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
139. Dic₃₄	#41	semidirect
174. #478	D₁₇	semidirect
232. C₁₇	#3368	semidirect

Maximal subgroups

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

subgroup	index
#11763536	2
#11763480	2
#11763175	2
#11763172	2
#11763154	2
#11763108	2
#11763088	2
#11763010	2
#11763008	2
#11762938	2
#11762909	2
#11762908	2
#11762896	2
C₁₇×#478	2
#3368	17