Subgroups — #30678674

Order 16768.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554031	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553671	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553670	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553661	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553608	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553605	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553590	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553525	C₂	1	1	yes	maximal, characteristic
10	#22553515	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553441	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553413	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553412	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
15	#463×C₁₃₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₂₆₂	—	2	1	no	—
17	C₂×#14523501	C₂²	1	1	yes	characteristic
18	#16862246	C₂²	1	1	yes	characteristic
19	#16862245	C₂²	1	1	yes	characteristic
20	#16862245	C₂²	1	1	yes	characteristic
21	#16862244	C₂²	1	1	yes	characteristic
22	#16862239	—	2	1	no	—
23	#16862239	—	2	1	no	—
24	#16862233	C₂²	1	1	yes	characteristic
25	#16862225	—	2	1	no	—
26	#16862223	—	2	1	no	—
27	#16862223	—	2	1	no	—
28	#16862223	—	2	1	no	—
29	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
35	#16862202	—	2	1	no	—
36	#16862180	C₂²	1	1	yes	characteristic
37	#16862180	C₂²	1	1	yes	characteristic
38	#16862179	C₂²	1	1	yes	characteristic
39	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	#16862171	C₂²	1	1	yes	characteristic
42	#16862171	C₂²	1	1	yes	characteristic
43	#16862170	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862169	C₂²	1	1	yes	characteristic
46	#16862167	C₂²	1	1	yes	characteristic
47	#16862167	C₂²	1	1	yes	characteristic
48	#16862167	C₂²	1	1	yes	characteristic
49	#16862157	C₂²	1	1	yes	characteristic
50	#16862156	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	#124×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	#102×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×Dic₂₆₂	—	4	4	no	—
63	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
64	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₃₁	—	2	1	no	—
66	C₂²×Dic₁₃₁	—	2	1	no	—
67	C₂²×Dic₁₃₁	—	2	1	no	—
68	#14523501	—	2	2	no	—
69	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₃₁	—	2	1	no	—
72	C₄⋊Dic₁₃₁	—	2	1	no	—
73	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
77	Dic₁₃₁⋊C₄	—	2	1	no	—
78	Dic₁₃₁⋊C₄	—	2	1	no	—
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₃₁	—	2	1	no	—
85	C₂²⋊Dic₁₃₁	—	2	1	no	—
86	C₂²⋊Dic₁₃₁	—	2	1	no	—
87	C₂²⋊Dic₁₃₁	—	2	1	no	—
88	C₂²⋊Dic₁₃₁	—	2	1	no	—
89	C₄×Dic₁₃₁	—	2	1	no	—
90	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₁	—	2	1	no	—
97	SD₁₆×C₁₃₁	—	2	2	no	—
98	D₈×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
104	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
106	Dic₂₆₂	—	2	2	no	—
107	Dic₂₆₂	—	2	1	no	—
108	Dic₂₆₂	—	2	1	no	—
109	Dic₂₆₂	—	4	2	no	—
110	Dic₂₆₂	—	4	2	no	—
111	Dic₂₆₂	—	4	2	no	—
112	C₂×Dic₁₃₁	—	2	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	—	2	1	no	—
116	C₂×Dic₁₃₁	—	4	2	no	—
117	C₂×Dic₁₃₁	—	4	2	no	—
118	C₂×Dic₁₃₁	—	4	2	no	—
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₁₃₁⋊C₈	—	2	1	no	—
124	C₁₃₁⋊C₈	—	2	1	no	—
125	C₂³×C₁₃₁	—	2	1	no	—
126	C₂³×C₁₃₁	—	2	1	no	—
127	Q₈×C₁₃₁	—	2	1	no	—
128	Q₈×C₁₃₁	—	2	1	no	—
129	D₄×C₁₃₁	—	2	1	no	—
130	D₄×C₁₃₁	—	2	1	no	—
131	D₄×C₁₃₁	—	2	1	no	—
132	D₄×C₁₃₁	#41	1	2	yes	—
133	C₂×C₄×C₁₃₁	—	2	1	no	—
134	C₂×C₄×C₁₃₁	—	2	1	no	—
135	C₂×C₄×C₁₃₁	—	4	2	no	—
136	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₃₁	—	2	1	no	—
140	C₈×C₁₃₁	—	2	1	no	—
141	C₂²×C₁₃₁	—	2	2	no	—
142	C₂²×C₁₃₁	—	4	1	no	—
143	C₂²×C₁₃₁	—	4	1	no	—
144	C₂²×C₁₃₁	—	4	1	no	—
145	C₂²×C₁₃₁	—	4	1	no	—
146	C₂²×C₁₃₁	—	4	1	no	—
147	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₃₁	—	2	1	no	—
149	C₄×C₁₃₁	—	2	1	no	—
150	C₄×C₁₃₁	—	4	1	no	—
151	C₄×C₁₃₁	—	4	1	no	—
152	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₃₁	—	2	1	no	—
155	Dic₁₃₁	—	2	1	no	—
156	Dic₁₃₁	—	2	1	no	—
157	Dic₁₃₁	—	4	1	no	—
158	Dic₁₃₁	—	4	1	no	—
159	Dic₁₃₁	—	4	1	no	—
160	Dic₁₃₁	—	4	1	no	—
161	Dic₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₁₃₁	—	4	2	no	—
163	C₂×C₁₃₁	—	8	1	no	—
164	C₂×C₁₃₁	#548	1	1	yes	characteristic
165	C₂×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₁₃₁	#578	1	1	yes	characteristic
167	C₁₃₁	#3367	1	1	yes	characteristic, complemented, Sylow
168	#3367	—	131	1	no	maximal, Sylow
169	#556	—	131	1	no	—
170	D₄×Q₈	—	131	1	no	—
171	#489	—	131	1	no	—
172	#489	—	131	1	no	—
173	#487	—	131	1	no	—
174	#477	—	131	1	no	—
175	#477	—	131	1	no	—
176	#474	—	131	1	no	—
177	#463	—	131	1	no	—
178	#463	D₁₃₁	1	1	yes	characteristic
179	#461	—	131	1	no	—
180	#446	—	131	1	no	—
181	#440	—	131	1	no	—
182	#440	—	131	1	no	—
183	C₄×D₈	—	131	1	no	—
184	C₂²×Q₈	—	262	1	no	—
185	C₂×SD₁₆	—	131	1	no	—
186	C₂×SD₁₆	C₂×D₁₃₁	1	1	yes	characteristic
187	C₂×D₈	C₂×D₁₃₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	131	1	no	—
189	C₄⋊Q₈	—	131	1	no	—
190	C₄⋊Q₈	—	131	1	no	—
191	C₄⋊Q₈	—	131	1	no	—
192	#126	—	262	1	no	—
193	#126	—	262	1	no	—
194	#124	—	131	1	no	—
195	#124	C₂×D₁₃₁	1	1	yes	characteristic
196	C₂²⋊Q₈	—	262	1	no	—
197	C₂²⋊Q₈	—	262	1	no	—
198	C₂²⋊Q₈	—	262	1	no	—
199	C₂²⋊Q₈	—	262	1	no	—
200	C₄×Q₈	—	131	1	no	—
201	C₄×Q₈	—	131	1	no	—
202	C₄×Q₈	—	131	1	no	—
203	C₄×D₄	—	131	1	no	—
204	C₄×D₄	—	131	1	no	—
205	C₄×D₄	—	131	1	no	—
206	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
207	#117	—	262	1	no	—
208	#107	—	131	1	no	—
209	#107	—	131	1	no	—
210	#107	—	131	1	no	—
211	C₄⋊C₈	—	131	1	no	—
212	C₄⋊C₈	—	131	1	no	—
213	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
214	#103	—	131	1	no	—
215	#103	—	131	1	no	—
216	#103	—	131	1	no	—
217	#103	—	131	1	no	—
218	#103	—	131	1	no	—
219	#103	C₂×D₁₃₁	1	1	yes	characteristic
220	#102	—	131	1	no	—
221	#102	—	131	1	no	—
222	#102	—	131	1	no	—
223	#102	C₂×D₁₃₁	1	1	yes	characteristic
224	#97	—	131	1	no	—
225	#97	—	131	1	no	—
226	C₄×C₈	—	131	1	no	—
227	C₂×Q₈	—	262	1	no	—
228	C₂×Q₈	—	262	1	no	—
229	C₂×Q₈	—	262	1	no	—
230	C₂×Q₈	—	524	4	no	—
231	C₂×Q₈	—	131	1	no	—
232	C₂×Q₈	—	131	1	no	—
233	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	262	1	no	—
238	C₂²×C₄	—	262	1	no	—
239	C₂²×C₄	—	262	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	262	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	131	1	no	—
244	C₂×C₈	—	131	1	no	—
245	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	262	1	no	—
248	C₄⋊C₄	—	262	1	no	—
249	C₄⋊C₄	—	262	1	no	—
250	C₄⋊C₄	—	262	1	no	—
251	C₄⋊C₄	—	262	1	no	—
252	C₄⋊C₄	—	262	1	no	—
253	C₄⋊C₄	—	262	1	no	—
254	C₄⋊C₄	—	131	1	no	—
255	C₄⋊C₄	—	131	1	no	—
256	C₄⋊C₄	—	131	1	no	—
257	C₄⋊C₄	—	131	1	no	—
258	C₄⋊C₄	—	131	1	no	—
259	C₄⋊C₄	—	131	1	no	—
260	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	262	1	no	—
265	C₂²⋊C₄	—	262	1	no	—
266	C₂²⋊C₄	—	262	1	no	—
267	C₂²⋊C₄	—	262	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₄²	—	262	1	no	—
270	C₄²	—	262	1	no	—
271	C₄²	—	131	1	no	—
272	C₄²	—	131	1	no	—
273	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	262	2	no	—
279	Q₈	—	262	1	no	—
280	Q₈	—	262	1	no	—
281	Q₈	—	524	2	no	—
282	Q₈	—	524	2	no	—
283	Q₈	—	524	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#14523512	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	262	1	no	—
292	C₂×C₄	—	262	1	no	—
293	C₂×C₄	—	262	1	no	—
294	C₂×C₄	—	262	1	no	—
295	C₂×C₄	—	524	2	no	—
296	C₂×C₄	—	524	2	no	—
297	C₂×C₄	—	524	2	no	—
298	C₂×C₄	—	131	1	no	—
299	C₂×C₄	—	131	1	no	—
300	C₂×C₄	—	131	1	no	—
301	C₂×C₄	—	131	1	no	—
302	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	262	1	no	—
308	C₈	—	262	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	262	1	no	—
321	C₄	—	262	1	no	—
322	C₄	—	262	1	no	—
323	C₄	—	524	1	no	—
324	C₄	—	524	1	no	—
325	C₄	—	524	1	no	—
326	C₄	—	524	1	no	—
327	C₄	—	131	2	no	—
328	C₄	C₂×#14523512	1	1	yes	characteristic
329	C₄	#16862293	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#22553968	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₁₃₁	1	1	yes	characteristic
334	C₂	#22554126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#463×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×D₈×C₁₃₁	C₂²
hypercenter	#463	D₁₃₁
hypocenter	C₁₃₁	#3367
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #22554031	C₂	semidirect
2. D₄×Dic₂₆₂	C₂	semidirect
3. #22553671	C₂	semidirect
4. #22553670	C₂	semidirect
5. #22553661	C₂	semidirect
6. #22553608	C₂	semidirect
7. #22553605	C₂	semidirect
8. #22553590	C₂	semidirect
10. #22553515	C₂	semidirect
11. #22553441	C₂	semidirect
12. #22553413	C₂	semidirect
13. #22553412	C₂	semidirect
167. C₁₃₁	#3367	semidirect

Maximal subgroups

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

subgroup	index
#22554031	2
D₄×Dic₂₆₂	2
#22553671	2
#22553670	2
#22553661	2
#22553608	2
#22553605	2
#22553590	2
#22553525	2
#22553515	2
#22553441	2
#22553413	2
#22553412	2
D₈×Dic₁₃₁	2
#463×C₁₃₁	2
#3367	131