Subgroups — #30678607

Order 16768.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#22554018	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553665	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553661	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553617	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553616	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553600	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553526	C₂	1	1	yes	maximal, characteristic
9	#22553524	C₂	1	1	yes	maximal, characteristic
10	#22553516	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553397	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553392	C₂	1	1	yes	maximal, characteristic
15	#488×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523502	C₂²	1	1	yes	characteristic
17	C₂×#14523500	C₂²	1	1	yes	characteristic
18	C₂×#14523498	C₂²	1	1	yes	characteristic
19	#16862245	C₂²	1	1	yes	characteristic
20	#16862244	C₂²	1	1	yes	characteristic
21	#16862239	—	2	1	no	—
22	#16862239	—	2	1	no	—
23	#16862237	C₂²	1	1	yes	characteristic
24	#16862233	C₂²	1	1	yes	characteristic
25	#16862226	—	2	1	no	—
26	#16862224	—	2	1	no	—
27	#16862224	—	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	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
34	#16862208	C₂²	1	1	yes	characteristic
35	#16862208	C₂²	1	1	yes	characteristic
36	#16862202	—	2	1	no	—
37	C₄⋊C₄×D₁₃₁	—	2	1	no	—
38	#16862179	C₂²	1	1	yes	characteristic
39	#16862178	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
44	#16862171	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862169	C₂²	1	1	yes	characteristic
47	#16862169	C₂²	1	1	yes	characteristic
48	#16862168	C₂²	1	1	yes	characteristic
49	#16862167	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	#125×C₁₃₁	C₂²	1	1	yes	characteristic
53	#124×C₁₃₁	C₂²	1	1	yes	characteristic
54	#103×C₁₃₁	C₂²	1	1	yes	characteristic
55	#103×C₁₃₁	C₂²	1	1	yes	characteristic
56	#102×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₃₁	—	2	1	no	—
64	C₂×C₄×D₁₃₁	—	2	1	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	#14523502	—	2	2	no	—
68	#14523500	—	2	2	no	—
69	#14523498	—	2	2	no	—
70	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₃₁	—	2	1	no	—
73	C₄⋊Dic₁₃₁	—	2	1	no	—
74	C₄⋊Dic₁₃₁	D₄	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	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₁	—	2	1	no	—
99	C₄⋊C₄×C₁₃₁	—	2	1	no	—
100	C₄⋊C₄×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	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₁	—	2	1	no	—
106	Dic₂₆₂	—	2	2	no	—
107	Dic₂₆₂	—	2	1	no	—
108	Dic₂₆₂	—	2	1	no	—
109	Dic₂₆₂	—	8	1	no	—
110	D₅₂₄	—	2	1	no	—
111	D₅₂₄	#41	1	2	yes	—
112	C₂×Dic₁₃₁	—	2	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	—	4	2	no	—
116	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₃₁	—	4	2	no	—
121	C₄×D₁₃₁	—	8	1	no	—
122	C₄×D₁₃₁	—	8	1	no	—
123	C₁₃₁⋊C₈	—	2	1	no	—
124	C₁₃₁⋊C₈	—	2	1	no	—
125	C₂³×C₁₃₁	—	2	1	no	—
126	Q₈×C₁₃₁	—	2	1	no	—
127	Q₈×C₁₃₁	—	2	1	no	—
128	D₄×C₁₃₁	—	2	1	no	—
129	D₄×C₁₃₁	—	2	1	no	—
130	C₂×C₄×C₁₃₁	—	2	1	no	—
131	C₂×C₄×C₁₃₁	—	2	1	no	—
132	C₂×C₄×C₁₃₁	—	2	1	no	—
133	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₃₁	—	2	1	no	—
137	C₈×C₁₃₁	—	2	1	no	—
138	C₂²×C₁₃₁	—	4	1	no	—
139	C₂²×C₁₃₁	—	4	1	no	—
140	C₂²×C₁₃₁	—	4	1	no	—
141	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₃₁	—	2	2	no	—
143	C₂×D₁₃₁	—	4	1	no	—
144	C₂×D₁₃₁	—	4	1	no	—
145	C₄×C₁₃₁	—	2	1	no	—
146	C₄×C₁₃₁	—	2	1	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	4	1	no	—
149	C₄×C₁₃₁	—	4	1	no	—
150	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₃₁	—	2	1	no	—
153	Dic₁₃₁	—	2	1	no	—
154	Dic₁₃₁	—	2	1	no	—
155	Dic₁₃₁	—	4	1	no	—
156	Dic₁₃₁	—	4	1	no	—
157	Dic₁₃₁	—	4	1	no	—
158	Dic₁₃₁	#135	1	2	yes	—
159	C₂×C₁₃₁	—	8	1	no	—
160	C₂×C₁₃₁	#548	1	1	yes	characteristic
161	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₃₁	#577	1	1	yes	characteristic
163	D₁₃₁	—	4	2	no	—
164	C₁₃₁	#3360	1	1	yes	characteristic, complemented, Sylow
165	#3360	—	131	1	no	maximal, Sylow
166	#556	—	131	1	no	—
167	#554	—	131	1	no	—
168	#488	—	131	1	no	—
169	#488	D₁₃₁	1	1	yes	characteristic, complemented
170	#487	—	131	1	no	—
171	#479	—	131	1	no	—
172	#479	—	131	1	no	—
173	#476	—	131	1	no	—
174	#463	—	131	1	no	—
175	#463	—	131	1	no	—
176	#461	—	131	1	no	—
177	C₈×Q₈	—	131	1	no	—
178	C₄×SD₁₆	—	131	1	no	—
179	C₄×SD₁₆	—	131	1	no	—
180	C₄×D₈	—	131	1	no	—
181	C₂×SD₁₆	—	131	1	no	—
182	C₂×SD₁₆	—	131	1	no	—
183	C₂×D₈	—	131	1	no	—
184	C₄⋊Q₈	—	131	1	no	—
185	C₄⋊Q₈	—	131	1	no	—
186	#126	—	262	1	no	—
187	#126	—	262	1	no	—
188	#125	—	131	1	no	—
189	#125	C₂×D₁₃₁	1	1	yes	characteristic
190	#124	—	131	1	no	—
191	#124	C₂×D₁₃₁	1	1	yes	characteristic
192	C₂²⋊Q₈	—	262	1	no	—
193	C₂²⋊Q₈	—	262	1	no	—
194	C₂²⋊Q₈	—	262	1	no	—
195	C₂²⋊Q₈	—	262	1	no	—
196	C₄×Q₈	—	131	1	no	—
197	C₄×Q₈	—	131	1	no	—
198	C₄×Q₈	—	131	1	no	—
199	C₄×D₄	—	131	1	no	—
200	C₄×D₄	—	131	1	no	—
201	C₄×D₄	—	131	1	no	—
202	C₄×D₄	—	131	1	no	—
203	#117	—	262	1	no	—
204	C₂×C₄⋊C₄	—	262	1	no	—
205	#107	—	131	1	no	—
206	#106	—	131	1	no	—
207	#106	—	131	1	no	—
208	C₄⋊C₈	—	131	1	no	—
209	C₄⋊C₈	—	131	1	no	—
210	C₄⋊C₈	—	131	1	no	—
211	#103	—	131	1	no	—
212	#103	—	131	1	no	—
213	#103	—	131	1	no	—
214	#103	—	131	1	no	—
215	#103	C₂×D₁₃₁	1	1	yes	characteristic
216	#103	C₂×D₁₃₁	1	1	yes	characteristic
217	#102	—	131	1	no	—
218	#102	—	131	1	no	—
219	#102	C₂×D₁₃₁	1	1	yes	characteristic
220	#102	C₂×D₁₃₁	1	1	yes	characteristic
221	C₄×C₈	—	131	1	no	—
222	C₄×C₈	—	131	1	no	—
223	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
224	C₂×Q₈	—	262	1	no	—
225	C₂×Q₈	—	131	1	no	—
226	C₂×Q₈	—	131	1	no	—
227	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
228	C₂×D₄	—	131	1	no	—
229	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
230	C₂²×C₄	—	262	1	no	—
231	C₂²×C₄	—	262	1	no	—
232	C₂²×C₄	—	262	1	no	—
233	C₂²×C₄	—	262	1	no	—
234	SD₁₆	—	262	2	no	—
235	SD₁₆	—	262	2	no	—
236	D₈	—	262	2	no	—
237	C₂×C₈	—	131	1	no	—
238	C₂×C₈	—	131	1	no	—
239	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	262	1	no	—
244	C₄⋊C₄	—	262	1	no	—
245	C₄⋊C₄	—	262	1	no	—
246	C₄⋊C₄	—	262	1	no	—
247	C₄⋊C₄	—	262	1	no	—
248	C₄⋊C₄	—	262	1	no	—
249	C₄⋊C₄	—	262	1	no	—
250	C₄⋊C₄	—	131	1	no	—
251	C₄⋊C₄	—	131	1	no	—
252	C₄⋊C₄	—	131	1	no	—
253	C₄⋊C₄	—	131	1	no	—
254	C₄⋊C₄	—	131	1	no	—
255	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	262	1	no	—
260	C₂²⋊C₄	—	262	1	no	—
261	C₂²⋊C₄	—	262	1	no	—
262	C₂²⋊C₄	—	262	1	no	—
263	C₂²⋊C₄	—	262	1	no	—
264	C₂²⋊C₄	—	262	1	no	—
265	C₄²	—	262	1	no	—
266	C₄²	—	262	1	no	—
267	C₄²	—	131	1	no	—
268	C₄²	—	131	1	no	—
269	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	262	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	262	2	no	—
275	Q₈	—	262	1	no	—
276	Q₈	—	262	1	no	—
277	Q₈	—	1048	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	262	1	no	—
281	D₄	—	131	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	262	1	no	—
286	C₂×C₄	—	262	1	no	—
287	C₂×C₄	—	262	1	no	—
288	C₂×C₄	—	524	2	no	—
289	C₂×C₄	—	524	2	no	—
290	C₂×C₄	—	1048	1	no	—
291	C₂×C₄	—	1048	1	no	—
292	C₂×C₄	—	131	1	no	—
293	C₂×C₄	—	131	1	no	—
294	C₂×C₄	—	131	1	no	—
295	C₂×C₄	—	131	1	no	—
296	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	262	1	no	—
302	C₈	—	262	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	262	2	no	—
307	C₂²	—	524	1	no	—
308	C₂²	—	524	1	no	—
309	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	262	1	no	—
316	C₄	—	262	1	no	—
317	C₄	—	262	1	no	—
318	C₄	—	524	1	no	—
319	C₄	—	524	1	no	—
320	C₄	—	524	1	no	—
321	C₄	—	131	2	no	—
322	C₄	#41×D₁₃₁	1	1	yes	characteristic
323	C₄	#16862293	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	524	2	no	—
326	C₂	#22553970	1	1	yes	characteristic
327	C₂	#135×D₁₃₁	1	1	yes	characteristic
328	C₂	#22554119	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#488×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#14523498	C₂²
hypercenter	#488	D₁₃₁
hypocenter	C₁₃₁	#3360
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #22554031	C₂	semidirect
2. #22554018	C₂	semidirect
3. #22553665	C₂	semidirect
4. #22553661	C₂	semidirect
5. #22553617	C₂	semidirect
6. #22553616	C₂	semidirect
7. #22553600	C₂	semidirect
10. #22553516	C₂	semidirect
11. C₈×Dic₂₆₂	C₂	semidirect
12. C₄×#14523502	C₂	semidirect
13. #22553397	C₂	semidirect
15. #488×C₁₃₁	C₂	semidirect
164. C₁₃₁	#3360	semidirect
169. #488	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554031	2
#22554018	2
#22553665	2
#22553661	2
#22553617	2
#22553616	2
#22553600	2
#22553526	2
#22553524	2
#22553516	2
C₈×Dic₂₆₂	2
C₄×#14523502	2
#22553397	2
#22553392	2
#488×C₁₃₁	2
#3360	131