Subgroups — #14541573

Order 2176.

384 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	#11763541	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763474	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763170	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763154	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763027	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763021	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763003	C₂	1	1	yes	maximal, characteristic
11	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762918	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₇×#124	C₂²	1	1	yes	characteristic
19	#10587633	C₂²	1	1	yes	characteristic
20	#10587632	—	2	1	no	—
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587626	C₂²	1	1	yes	characteristic
24	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
25	C₂×#63228	C₂²	1	1	yes	characteristic
26	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
27	#10587582	—	2	1	no	—
28	#10587582	—	2	1	no	—
29	#10587579	C₂²	1	1	yes	characteristic
30	#10587579	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587574	C₂²	1	1	yes	characteristic
33	#10587572	—	2	1	no	—
34	#10587571	—	2	1	no	—
35	#10587569	—	2	1	no	—
36	#10587567	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#103	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	C₁₇×#97	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	Q₈×D₁₇	—	4	4	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	D₄	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	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	#63230	—	2	2	no	—
85	#63228	—	2	2	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	D₄	1	1	yes	characteristic
99	#63225	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	D₁₃₆	—	2	2	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	C₂²×D₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	4	2	no	—
112	D₄×C₁₇	—	2	1	no	—
113	D₄×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	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₇	—	4	2	no	—
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	8	1	no	—
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	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	#41	1	2	yes	—
133	C₄×D₁₇	—	4	2	no	—
134	C₄×D₁₇	—	4	2	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	8	1	no	—
137	Dic₃₄	—	2	2	no	—
138	Dic₃₄	—	4	2	no	—
139	Dic₃₄	—	4	2	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3373	—	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	—	2	2	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	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
168	#559	—	17	1	no	—
169	D₄×Q₈	—	17	1	no	—
170	#489	—	17	1	no	—
171	#489	D₁₇	1	1	yes	characteristic, complemented
172	#486	—	17	1	no	—
173	#476	—	17	1	no	—
174	#476	—	17	1	no	—
175	#475	—	17	1	no	—
176	#463	—	17	1	no	—
177	#463	—	17	1	no	—
178	#459	—	17	1	no	—
179	#446	—	17	1	no	—
180	#442	—	17	1	no	—
181	#442	—	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	—	4	2	no	—
188	D₁₇	—	8	1	no	—
189	C₂²×Q₈	—	34	1	no	—
190	C₂×SD₁₆	—	17	1	no	—
191	C₂×D₈	—	17	1	no	—
192	C₂×D₈	—	17	1	no	—
193	C₄⋊Q₈	—	17	1	no	—
194	C₄⋊Q₈	—	17	1	no	—
195	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
196	#127	—	17	1	no	—
197	#124	—	17	1	no	—
198	#124	—	34	1	no	—
199	#124	C₂×D₁₇	1	1	yes	characteristic
200	#123	—	34	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	C₂²⋊Q₈	—	34	1	no	—
204	C₂²⋊Q₈	—	34	1	no	—
205	C₄×Q₈	—	17	1	no	—
206	C₄×Q₈	—	17	1	no	—
207	C₄×D₄	—	17	1	no	—
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	17	1	no	—
210	C₄×D₄	—	17	1	no	—
211	C₄×D₄	—	17	1	no	—
212	C₄×D₄	—	34	1	no	—
213	#106	—	17	1	no	—
214	#106	—	17	1	no	—
215	#106	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	C₂×D₁₇	1	1	yes	characteristic
221	#103	C₂×D₁₇	1	1	yes	characteristic
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	C₂×D₁₇	1	1	yes	characteristic
231	C₄×C₈	—	17	1	no	—
232	C₁₇	#3373	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	17	1	no	—
235	C₂×Q₈	—	34	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×Q₈	—	68	4	no	—
238	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
244	C₂²×C₄	—	34	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	C₂²×C₄	—	34	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	SD₁₆	—	34	2	no	—
250	D₈	—	34	2	no	—
251	D₈	—	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	—	17	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	C₂²×D₁₇	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
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	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	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	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	34	2	no	—
287	Q₈	—	68	2	no	—
288	Q₈	—	68	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	17	2	no	—
292	D₄	—	34	1	no	—
293	D₄	—	34	1	no	—
294	D₄	—	34	1	no	—
295	D₄	—	68	2	no	—
296	D₄	—	136	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	68	2	no	—
307	C₂×C₄	—	68	2	no	—
308	C₂×C₄	—	68	2	no	—
309	C₂×C₄	—	136	1	no	—
310	C₂×C₄	—	136	1	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	—	34	2	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	1	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	68	1	no	—
338	C₄	—	68	1	no	—
339	C₄	#41×D₁₇	1	1	yes	characteristic
340	C₄	#10587697	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	68	2	no	—
343	C₂	—	136	1	no	—
344	C₂	#11763467	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
346	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×#489	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763003	C₂
hypercenter	#489	D₁₇
hypocenter	C₁₇	#3373
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #11763541	C₂	semidirect
2. #11763474	C₂	semidirect
3. #11763170	C₂	semidirect
4. #11763154	C₂	semidirect
5. #11763097	C₂	semidirect
6. #11763096	C₂	semidirect
7. #11763092	C₂	semidirect
8. #11763027	C₂	semidirect
9. #11763021	C₂	semidirect
11. #11762937	C₂	semidirect
12. #11762922	C₂	semidirect
13. #11762918	C₂	semidirect
14. #11762894	C₂	semidirect
15. C₁₇×#489	C₂	semidirect
171. #489	D₁₇	semidirect
232. C₁₇	#3373	semidirect

Maximal subgroups

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

subgroup	index
#11763541	2
#11763474	2
#11763170	2
#11763154	2
#11763097	2
#11763096	2
#11763092	2
#11763027	2
#11763021	2
#11763003	2
#11762937	2
#11762922	2
#11762918	2
#11762894	2
C₁₇×#489	2
#3373	17