Subgroups — #14540145

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763356	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763344	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763093	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763063	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763061	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763056	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763054	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763024	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763008	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763003	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762831	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	C₂×#63229	C₂²	1	1	yes	characteristic
22	C₂×#63228	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₇	—	2	1	no	—
24	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
25	C₂×#63219	C₂²	1	1	yes	characteristic
26	C₂×#63218	C₂²	1	1	yes	characteristic
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	#10587584	—	2	1	no	—
29	#10587583	C₂²	1	1	yes	characteristic
30	#10587581	C₂²	1	1	yes	characteristic
31	#10587581	C₂²	1	1	yes	characteristic
32	#10587580	—	2	1	no	—
33	#10587579	—	2	1	no	—
34	#10587579	C₂²	1	1	yes	characteristic
35	#10587578	—	2	1	no	—
36	#10587574	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587571	—	2	1	no	—
39	#10587570	—	2	1	no	—
40	#10587561	C₂²	1	1	yes	characteristic
41	C₄×D₆₈	C₂²	1	1	yes	characteristic
42	C₄²×D₁₇	C₂²	1	1	yes	characteristic
43	C₁₇×#107	C₂²	1	1	yes	characteristic
44	C₁₇×#106	C₂²	1	1	yes	characteristic
45	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₇×#102	C₂²	1	1	yes	characteristic
47	C₁₇×#102	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	#10587485	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	#10587479	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	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	#63229	—	4	1	no	—
87	#63228	—	4	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	D₄	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	—	4	1	no	—
106	#63219	—	4	1	no	—
107	#63218	—	4	1	no	—
108	#63218	—	4	1	no	—
109	C₂³×C₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
113	D₄×C₁₇	—	2	1	no	—
114	D₄×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	—	4	2	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	—	4	2	no	—
123	C₂²⋊D₁₇	—	4	2	no	—
124	C₂²⋊D₁₇	—	4	2	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	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	—	2	1	no	—
132	D₆₈	—	2	1	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	4	2	no	—
135	D₆₈	—	4	2	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	2	2	no	—
139	C₄×D₁₇	—	4	1	no	—
140	C₄×D₁₇	—	4	2	no	—
141	C₄×D₁₇	—	4	2	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	4	2	no	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3251	—	17	1	no	maximal, Sylow
150	C₂²×C₁₇	—	4	1	no	—
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	—	4	1	no	—
153	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₂×D₁₇	—	4	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	—	4	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	2	1	no	—
172	Dic₁₇	—	4	1	no	—
173	Dic₁₇	—	4	1	no	—
174	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
175	#532	—	17	1	no	—
176	#529	—	17	1	no	—
177	#478	—	17	1	no	—
178	#478	D₁₇	1	1	yes	characteristic, complemented
179	#475	—	17	1	no	—
180	#474	—	17	1	no	—
181	#469	—	17	1	no	—
182	#469	—	17	1	no	—
183	#468	—	17	1	no	—
184	#468	—	17	1	no	—
185	#463	—	17	1	no	—
186	#463	—	17	1	no	—
187	#460	—	17	1	no	—
188	#459	—	17	1	no	—
189	#423	—	17	1	no	—
190	C₂×C₁₇	—	8	1	no	—
191	C₂×C₁₇	#537	1	1	yes	characteristic
192	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂×C₁₇	#575	1	1	yes	characteristic
194	D₁₇	—	4	1	no	—
195	D₁₇	—	8	1	no	—
196	D₁₇	—	8	1	no	—
197	C₂×#41	—	34	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×D₈	—	17	1	no	—
201	C₂×D₈	—	17	1	no	—
202	C₂×M₁₆	—	17	1	no	—
203	C₂×M₁₆	—	17	1	no	—
204	C₄⋊Q₈	—	17	1	no	—
205	#127	—	17	1	no	—
206	#126	—	34	1	no	—
207	#125	C₂×D₁₇	1	1	yes	characteristic
208	#124	—	17	1	no	—
209	#124	—	34	1	no	—
210	#123	—	34	1	no	—
211	C₂²⋊Q₈	—	17	1	no	—
212	#121	—	17	1	no	—
213	#121	—	17	1	no	—
214	#121	—	17	1	no	—
215	#121	—	34	1	no	—
216	C₄×Q₈	—	17	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	17	1	no	—
219	C₄×D₄	—	34	1	no	—
220	C₄×D₄	—	34	1	no	—
221	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
222	#117	—	34	1	no	—
223	C₂×C₄²	—	17	1	no	—
224	#107	—	17	1	no	—
225	#107	C₂×D₁₇	1	1	yes	characteristic
226	#106	—	17	1	no	—
227	#106	C₂×D₁₇	1	1	yes	characteristic
228	C₄⋊C₈	—	17	1	no	—
229	C₄⋊C₈	—	17	1	no	—
230	C₄⋊C₈	—	17	1	no	—
231	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	C₂×D₁₇	1	1	yes	characteristic
239	#102	C₂×D₁₇	1	1	yes	characteristic
240	C₁₇	#3251	1	1	yes	characteristic, complemented, Sylow
241	#41	—	68	4	no	—
242	C₂×Q₈	—	17	1	no	—
243	C₂×Q₈	—	34	1	no	—
244	C₂×D₄	—	17	1	no	—
245	C₂×D₄	—	17	1	no	—
246	C₂×D₄	—	34	1	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	—	34	1	no	—
250	C₂×D₄	—	34	1	no	—
251	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	17	1	no	—
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	17	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	C₂²×C₄	—	34	1	no	—
258	SD₁₆	—	68	1	no	—
259	SD₁₆	—	68	1	no	—
260	D₈	—	68	1	no	—
261	D₈	—	68	1	no	—
262	M₁₆	—	68	1	no	—
263	M₁₆	—	68	1	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	—	17	1	no	—
266	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	17	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₄²	—	17	1	no	—
287	C₄²	—	17	1	no	—
288	C₄²	—	17	1	no	—
289	C₄²	—	34	1	no	—
290	C₄²	C₂²×D₁₇	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	17	1	no	—
293	C₂³	—	34	1	no	—
294	C₂³	—	34	1	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	68	2	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	34	1	no	—
301	D₄	—	34	1	no	—
302	D₄	—	34	1	no	—
303	D₄	—	34	1	no	—
304	D₄	—	68	2	no	—
305	D₄	—	68	2	no	—
306	D₄	—	68	2	no	—
307	D₄	—	68	2	no	—
308	D₄	—	68	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	34	2	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	1	no	—
321	C₂×C₄	—	68	1	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	—	68	2	no	—
324	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	34	1	no	—
330	C₈	—	34	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	34	1	no	—
336	C₂²	—	34	1	no	—
337	C₂²	—	68	1	no	—
338	C₂²	—	68	1	no	—
339	C₂²	—	68	1	no	—
340	C₂²	—	68	1	no	—
341	C₂²	—	68	1	no	—
342	C₂²	—	68	1	no	—
343	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	17	2	no	—
350	C₄	—	34	1	no	—
351	C₄	—	34	1	no	—
352	C₄	—	68	1	no	—
353	C₄	—	68	1	no	—
354	C₄	#10587694	1	1	yes	characteristic
355	C₄	#10587696	1	1	yes	characteristic
356	C₂	—	8	1	no	—
357	C₂	—	68	1	no	—
358	C₂	—	136	1	no	—
359	C₂	—	136	1	no	—
360	C₂	#11763389	1	1	yes	characteristic
361	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
362	C₂	#11763606	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
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3251
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. #11763356	C₂	semidirect
2. #11763344	C₂	semidirect
3. #11763106	C₂	semidirect
4. #11763093	C₂	semidirect
5. #11763088	C₂	semidirect
6. #11763063	C₂	semidirect
7. #11763061	C₂	semidirect
8. #11763056	C₂	semidirect
9. #11763054	C₂	semidirect
10. #11763025	C₂	semidirect
11. #11763024	C₂	semidirect
12. #11763008	C₂	semidirect
13. #11763003	C₂	semidirect
14. #11762831	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
178. #478	D₁₇	semidirect
240. C₁₇	#3251	semidirect

Maximal subgroups

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

subgroup	index
#11763356	2
#11763344	2
#11763106	2
#11763093	2
#11763088	2
#11763063	2
#11763061	2
#11763056	2
#11763054	2
#11763025	2
#11763024	2
#11763008	2
#11763003	2
#11762831	2
C₁₇×#478	2
#3251	17