Subgroups — #14540952

Order 2176.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763512	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763179	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763056	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763039	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763015	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762907	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762879	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#125	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×#63230	C₂²	1	1	yes	characteristic
21	C₂×#63219	C₂²	1	1	yes	characteristic
22	C₂×#63219	C₂²	1	1	yes	characteristic
23	C₂×#63218	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
25	#10587584	—	2	1	no	—
26	#10587583	—	2	1	no	—
27	#10587583	—	2	1	no	—
28	#10587583	C₂²	1	1	yes	characteristic
29	#10587583	C₂²	1	1	yes	characteristic
30	#10587581	—	2	1	no	—
31	#10587581	C₂²	1	1	yes	characteristic
32	#10587581	C₂²	1	1	yes	characteristic
33	#10587580	—	2	1	no	—
34	#10587579	—	2	1	no	—
35	#10587578	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
37	C₄×D₆₈	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587555	—	2	1	no	—
41	#10587555	C₂²	1	1	yes	characteristic
42	C₁₇×#107	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×#103	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	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×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	Q₈×D₁₇	—	2	2	no	—
61	Q₈×D₁₇	—	2	2	no	—
62	Q₈×D₁₇	—	4	1	no	—
63	Q₈×D₁₇	—	4	1	no	—
64	C₂×D₆₈	—	2	1	no	—
65	C₂×D₆₈	—	2	1	no	—
66	C₂×D₆₈	D₄	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×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	—	2	1	no	—
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
88	#63230	—	2	2	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	#63227	D₄	1	1	yes	characteristic
96	#63227	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	—	2	1	no	—
101	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
102	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
103	#63225	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	#63219	—	4	1	no	—
108	#63219	—	4	1	no	—
109	#63218	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	#41	1	2	yes	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	D₆₈	—	2	2	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	4	2	no	—
132	D₆₈	—	8	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	2	1	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	4	2	no	—
139	C₄×D₁₇	—	8	1	no	—
140	C₄×D₁₇	—	8	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	2	1	no	—
145	Dic₃₄	—	4	2	no	—
146	Dic₃₄	—	4	2	no	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	#3622	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3320	—	17	1	no	maximal, Sylow
152	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₇	—	2	1	no	—
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	—	4	1	no	—
159	C₂×D₁₇	—	4	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	#137	1	2	yes	—
162	C₄×C₁₇	—	2	2	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	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₇	H(5,𝔽₂)	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	—	4	1	no	—
175	#553	—	17	1	no	—
176	D₄×Q₈	—	17	1	no	—
177	#491	—	17	1	no	—
178	#485	—	17	1	no	—
179	#484	—	17	1	no	—
180	#479	D₁₇	1	1	yes	characteristic, complemented
181	#468	—	17	1	no	—
182	#465	—	17	1	no	—
183	#461	—	17	1	no	—
184	#448	—	17	1	no	—
185	#448	—	17	1	no	—
186	#440	—	17	1	no	—
187	#435	—	17	1	no	—
188	#419	—	17	1	no	—
189	C₂×#103	—	17	1	no	—
190	C₂×C₁₇	#546	1	1	yes	characteristic
191	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
192	C₂×C₁₇	#577	1	1	yes	characteristic
193	D₁₇	—	2	2	no	—
194	D₁₇	—	4	1	no	—
195	D₁₇	—	8	1	no	—
196	C₂²×Q₈	—	17	1	no	—
197	C₂²×Q₈	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×SD₁₆	—	17	1	no	—
201	C₂×M₁₆	—	17	1	no	—
202	C₂²×C₈	—	17	1	no	—
203	C₄⋊Q₈	—	17	1	no	—
204	C₄⋊Q₈	—	34	1	no	—
205	#126	—	34	1	no	—
206	#125	C₂×D₁₇	1	1	yes	characteristic
207	#123	—	34	1	no	—
208	C₂²⋊Q₈	—	17	1	no	—
209	C₂²⋊Q₈	—	17	1	no	—
210	C₂²⋊Q₈	—	34	1	no	—
211	C₂²⋊Q₈	—	34	1	no	—
212	#121	—	17	1	no	—
213	#121	—	17	1	no	—
214	#121	—	34	1	no	—
215	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
216	C₄×D₄	—	17	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	34	1	no	—
219	C₄×D₄	—	34	1	no	—
220	#117	—	17	1	no	—
221	C₂×C₄⋊C₄	—	17	1	no	—
222	#107	—	17	1	no	—
223	#107	C₂×D₁₇	1	1	yes	characteristic
224	#106	C₂×D₁₇	1	1	yes	characteristic
225	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	C₂×D₁₇	1	1	yes	characteristic
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#102	—	17	1	no	—
236	C₂²⋊C₈	—	17	1	no	—
237	C₂²⋊C₈	—	17	1	no	—
238	#97	—	17	1	no	—
239	C₁₇	#3320	1	1	yes	characteristic, complemented, Sylow
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	17	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	34	1	no	—
244	C₂×Q₈	—	34	2	no	—
245	C₂×Q₈	—	34	2	no	—
246	C₂×Q₈	—	68	1	no	—
247	C₂×Q₈	—	68	1	no	—
248	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×D₄	—	17	1	no	—
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	34	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂²×C₄	—	17	1	no	—
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	34	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	C₂²×C₄	—	34	1	no	—
258	C₂²×C₄	—	34	1	no	—
259	SD₁₆	—	34	2	no	—
260	SD₁₆	—	68	1	no	—
261	SD₁₆	—	68	1	no	—
262	M₁₆	—	68	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	—	68	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	—	2	1	no	—
271	C₄⋊C₄	—	17	1	no	—
272	C₄⋊C₄	—	17	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
281	C₂²⋊C₄	—	17	1	no	—
282	C₂²⋊C₄	—	17	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	34	1	no	—
291	C₄²	C₂²×D₁₇	1	1	yes	characteristic
292	C₂³	—	17	1	no	—
293	C₂³	—	17	1	no	—
294	C₂³	—	34	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	34	1	no	—
298	Q₈	—	34	1	no	—
299	Q₈	—	34	1	no	—
300	Q₈	—	68	2	no	—
301	Q₈	—	68	2	no	—
302	Q₈	#63256	1	2	yes	—
303	D₄	—	34	2	no	—
304	D₄	—	34	1	no	—
305	D₄	—	34	1	no	—
306	D₄	—	68	2	no	—
307	D₄	—	136	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	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	2	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	—	136	1	no	—
324	C₂×C₄	—	136	1	no	—
325	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
326	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	34	1	no	—
331	C₈	—	34	1	no	—
332	C₂²	—	17	2	no	—
333	C₂²	—	34	1	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	34	1	no	—
336	C₂²	—	34	1	no	—
337	C₂²	—	34	1	no	—
338	C₂²	—	68	1	no	—
339	C₂²	—	68	1	no	—
340	C₂²	—	68	1	no	—
341	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	34	1	no	—
349	C₄	—	34	1	no	—
350	C₄	—	68	1	no	—
351	C₄	—	68	1	no	—
352	C₄	—	68	1	no	—
353	C₄	C₂×#63256	1	1	yes	characteristic
354	C₄	#10587696	1	1	yes	characteristic
355	C₂	—	34	2	no	—
356	C₂	—	68	1	no	—
357	C₂	—	136	1	no	—
358	C₂	#11763450	1	1	yes	characteristic
359	C₂	#137×D₁₇	1	1	yes	characteristic
360	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×#479	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763512	C₂
hypercenter	#479	D₁₇
hypocenter	C₁₇	#3320
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
2. Q₈×D₆₈	C₂	semidirect
3. #11763179	C₂	semidirect
4. #11763153	C₂	semidirect
5. #11763146	C₂	semidirect
6. #11763056	C₂	semidirect
7. #11763039	C₂	semidirect
8. #11763015	C₂	semidirect
9. #11762949	C₂	semidirect
10. #11762949	C₂	semidirect
11. #11762907	C₂	semidirect
12. #11762879	C₂	semidirect
13. #11762816	C₂	semidirect
14. #103×D₁₇	C₂	semidirect
15. C₁₇×#479	C₂	semidirect
180. #479	D₁₇	semidirect
239. C₁₇	#3320	semidirect

Maximal subgroups

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

subgroup	index
#11763512	2
Q₈×D₆₈	2
#11763179	2
#11763153	2
#11763146	2
#11763056	2
#11763039	2
#11763015	2
#11762949	2
#11762907	2
#11762879	2
#11762816	2
#103×D₁₇	2
C₁₇×#479	2
#3320	17