Subgroups — #14540915

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763500	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763481	C₂	1	1	yes	maximal, characteristic
3	#11763174	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763135	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763058	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763041	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763009	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762970	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762970	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762882	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762809	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762804	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63256	C₂²	1	1	yes	characteristic
17	C₂×#63256	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	#10587632	C₂²	1	1	yes	characteristic
23	C₂×#63230	C₂²	1	1	yes	characteristic
24	C₂×#63230	C₂²	1	1	yes	characteristic
25	C₂×#63218	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
27	#10587581	—	2	1	no	—
28	#10587581	—	2	1	no	—
29	#10587581	C₂²	1	1	yes	characteristic
30	#10587581	C₂²	1	1	yes	characteristic
31	#10587580	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	#10587575	—	2	1	no	—
35	#10587562	—	2	1	no	—
36	#10587560	—	2	1	no	—
37	#10587560	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	—	2	1	no	—
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587558	—	2	1	no	—
41	#10587556	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#97	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587491	C₂²	1	1	yes	characteristic
53	#10587491	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×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	#63256	—	2	2	no	—
62	#63256	—	2	2	no	—
63	#63256	—	4	1	no	—
64	#63256	—	4	1	no	—
65	C₂×D₆₈	—	2	1	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	—	2	1	no	—
69	C₂×D₆₈	—	2	1	no	—
70	C₂×D₆₈	—	2	1	no	—
71	C₂×D₆₈	D₄	1	1	yes	characteristic
72	C₂×D₆₈	C₂³	1	1	yes	characteristic
73	C₂×D₆₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₇	—	2	1	no	—
75	C₂×C₄×D₁₇	—	2	1	no	—
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	Q₁₆×C₁₇	—	2	2	no	—
80	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	—	2	1	no	—
84	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	C₂³	1	1	yes	characteristic
89	#63230	—	4	1	no	—
90	#63230	—	4	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	—	2	1	no	—
96	#63227	D₄	1	1	yes	characteristic
97	#63227	D₄	1	1	yes	characteristic
98	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	#63225	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	#63218	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	#41	1	2	yes	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
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	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
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	1	no	—
135	D₆₈	—	4	2	no	—
136	D₆₈	—	4	2	no	—
137	D₆₈	—	4	2	no	—
138	D₆₈	—	8	1	no	—
139	D₆₈	—	8	1	no	—
140	D₆₈	—	8	1	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	2	1	no	—
144	C₄×D₁₇	—	2	1	no	—
145	C₄×D₁₇	—	4	2	no	—
146	C₄×D₁₇	—	4	2	no	—
147	C₄×D₁₇	—	4	2	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3622	—	2	1	no	—
152	#3317	—	17	1	no	maximal, Sylow
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	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	—	4	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₄×C₁₇	—	2	2	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	—	4	1	no	—
172	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	#552	—	17	1	no	—
179	#550	—	17	1	no	—
180	#490	—	17	1	no	—
181	#482	—	17	1	no	—
182	#481	—	17	1	no	—
183	#479	—	17	1	no	—
184	#468	—	17	1	no	—
185	#465	—	17	1	no	—
186	#460	—	17	1	no	—
187	#452	—	17	1	no	—
188	#452	—	17	1	no	—
189	#441	D₁₇	1	1	yes	characteristic, complemented
190	#435	—	17	1	no	—
191	#417	—	17	1	no	—
192	#416	—	17	1	no	—
193	C₂×C₁₇	#546	1	1	yes	characteristic
194	C₂×C₁₇	#575	1	1	yes	characteristic
195	C₂×C₁₇	#577	1	1	yes	characteristic
196	D₁₇	—	4	1	no	—
197	D₁₇	—	4	1	no	—
198	D₁₇	—	8	1	no	—
199	D₁₇	—	8	1	no	—
200	C₂×#41	—	17	1	no	—
201	C₂×#41	—	17	1	no	—
202	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
203	C₂×SD₁₆	—	17	1	no	—
204	C₂×SD₁₆	—	17	1	no	—
205	C₂×M₁₆	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	#127	—	17	1	no	—
208	#127	—	34	1	no	—
209	#126	—	34	1	no	—
210	#125	—	17	1	no	—
211	#125	—	34	1	no	—
212	#123	—	34	1	no	—
213	C₂²⋊Q₈	—	17	1	no	—
214	C₂²⋊Q₈	—	17	1	no	—
215	#121	—	17	1	no	—
216	#121	—	17	1	no	—
217	#121	—	34	1	no	—
218	#121	—	34	1	no	—
219	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
220	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
221	C₄×D₄	—	17	1	no	—
222	C₄×D₄	—	34	1	no	—
223	#117	—	17	1	no	—
224	#117	—	34	1	no	—
225	C₂×C₄⋊C₄	—	17	1	no	—
226	#107	—	17	1	no	—
227	#106	—	17	1	no	—
228	#106	C₂×D₁₇	1	1	yes	characteristic
229	C₄⋊C₈	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	C₂×D₁₇	1	1	yes	characteristic
233	#103	C₂×D₁₇	1	1	yes	characteristic
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₂²⋊C₈	—	17	1	no	—
242	#97	C₂×D₁₇	1	1	yes	characteristic
243	C₁₇	#3317	1	1	yes	characteristic, complemented, Sylow
244	#41	—	34	2	no	—
245	#41	—	34	2	no	—
246	#41	—	68	1	no	—
247	#41	—	68	1	no	—
248	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	17	1	no	—
252	C₂×D₄	—	17	1	no	—
253	C₂×D₄	—	34	1	no	—
254	C₂×D₄	—	34	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂×D₄	—	34	1	no	—
259	C₂²×C₄	—	17	1	no	—
260	C₂²×C₄	—	17	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	—	34	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	Q₁₆	—	2	2	no	—
265	SD₁₆	—	68	1	no	—
266	SD₁₆	—	68	1	no	—
267	M₁₆	—	68	1	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	68	1	no	—
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	17	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	—	34	1	no	—
283	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
285	C₂²⋊C₄	—	17	1	no	—
286	C₂²⋊C₄	—	17	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	—	34	1	no	—
291	C₂²⋊C₄	—	34	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	C₂²×D₁₇	1	1	yes	characteristic
296	C₂³	—	17	1	no	—
297	C₂³	—	17	1	no	—
298	C₂³	—	34	1	no	—
299	C₂³	—	34	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	#63252	1	2	yes	—
304	D₄	—	34	1	no	—
305	D₄	—	34	1	no	—
306	D₄	—	34	1	no	—
307	D₄	—	34	1	no	—
308	D₄	—	68	1	no	—
309	D₄	—	68	2	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	136	1	no	—
313	D₄	—	136	1	no	—
314	D₄	—	136	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	17	1	no	—
320	C₂×C₄	—	17	1	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	34	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	—	68	2	no	—
330	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
331	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	34	1	no	—
336	C₈	—	34	1	no	—
337	C₂²	—	34	1	no	—
338	C₂²	—	34	1	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	34	1	no	—
342	C₂²	—	34	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	68	1	no	—
345	C₂²	—	68	1	no	—
346	C₂²	—	68	1	no	—
347	C₂²	—	68	1	no	—
348	C₂²	—	68	1	no	—
349	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	34	1	no	—
357	C₄	—	34	1	no	—
358	C₄	—	68	1	no	—
359	C₄	—	68	1	no	—
360	C₄	C₂×#63252	1	1	yes	characteristic
361	C₄	#10587696	1	1	yes	characteristic
362	C₂	—	68	1	no	—
363	C₂	—	68	1	no	—
364	C₂	—	136	1	no	—
365	C₂	—	136	1	no	—
366	C₂	#11763455	1	1	yes	characteristic
367	C₂	#11763608	1	1	yes	characteristic
368	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×#441	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763481	C₂
hypercenter	#441	D₁₇
hypocenter	C₁₇	#3317
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. #11763500	C₂	semidirect
3. #11763174	C₂	semidirect
4. #11763135	C₂	semidirect
5. #11763129	C₂	semidirect
6. #11763116	C₂	semidirect
7. #11763058	C₂	semidirect
8. #11763041	C₂	semidirect
9. #11763009	C₂	semidirect
10. #11762970	C₂	semidirect
11. #11762970	C₂	semidirect
12. #11762882	C₂	semidirect
13. #11762809	C₂	semidirect
14. #11762804	C₂	semidirect
15. C₁₇×#441	C₂	semidirect
189. #441	D₁₇	semidirect
243. C₁₇	#3317	semidirect

Maximal subgroups

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

subgroup	index
#11763500	2
#11763481	2
#11763174	2
#11763135	2
#11763129	2
#11763116	2
#11763058	2
#11763041	2
#11763009	2
#11762970	2
#11762882	2
#11762809	2
#11762804	2
C₁₇×#441	2
#3317	17