Subgroups — #30676231

Order 16768.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553935	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553795	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553640	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553631	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553600	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553595	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553536	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553529	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553457	C₂	1	1	yes	maximal, characteristic
10	#22553421	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553419	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553397	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553370	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523500	C₂²	1	1	yes	characteristic
17	C₂×#14523498	C₂²	1	1	yes	characteristic
18	#16862245	C₂²	1	1	yes	characteristic
19	#16862241	—	2	1	no	—
20	#16862235	C₂²	1	1	yes	characteristic
21	#16862232	—	2	1	no	—
22	#16862228	—	2	1	no	—
23	#16862226	—	2	1	no	—
24	#16862220	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₃₁	—	2	1	no	—
29	D₄×Dic₁₃₁	—	2	1	no	—
30	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
32	#16862208	C₂²	1	1	yes	characteristic
33	#16862205	—	2	1	no	—
34	#16862205	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃₁	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
37	#16862178	C₂²	1	1	yes	characteristic
38	#16862178	C₂²	1	1	yes	characteristic
39	#16862177	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	#16862170	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862168	C₂²	1	1	yes	characteristic
47	#16862168	C₂²	1	1	yes	characteristic
48	#16862167	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
50	#16862156	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
62	C₂²×Dic₁₃₁	—	2	1	no	—
63	C₂²×Dic₁₃₁	—	2	1	no	—
64	C₂²×Dic₁₃₁	—	2	1	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	#14523500	—	2	2	no	—
67	#14523498	—	2	2	no	—
68	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃₁	—	2	1	no	—
71	C₄⋊Dic₁₃₁	—	2	1	no	—
72	C₄⋊Dic₁₃₁	—	2	1	no	—
73	C₄⋊Dic₁₃₁	—	2	1	no	—
74	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₁₃₁	—	2	1	no	—
81	C₂²⋊Dic₁₃₁	—	2	1	no	—
82	C₂²⋊Dic₁₃₁	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₁	D₄	1	1	yes	characteristic
84	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
85	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	C₄×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	#41×C₁₃₁	—	2	2	no	—
94	#41×C₁₃₁	—	2	2	no	—
95	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₃₁	—	2	1	no	—
98	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₁	—	2	1	no	—
100	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₃₁	—	2	2	no	—
102	SD₁₆×C₁₃₁	—	2	2	no	—
103	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₁	—	2	1	no	—
106	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
108	C₂²×D₁₃₁	—	2	1	no	—
109	Dic₂₆₂	—	8	1	no	—
110	D₅₂₄	—	2	1	no	—
111	D₅₂₄	—	2	1	no	—
112	C₂²⋊D₁₃₁	—	8	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	—	2	1	no	—
116	C₂×Dic₁₃₁	—	8	1	no	—
117	C₂×Dic₁₃₁	—	8	1	no	—
118	C₂×Dic₁₃₁	—	8	1	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	C₄×D₁₃₁	—	8	1	no	—
124	C₁₃₁⋊C₈	—	2	1	no	—
125	C₁₃₁⋊C₈	—	2	1	no	—
126	C₂³×C₁₃₁	—	2	1	no	—
127	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₁	—	2	1	no	—
129	Q₈×C₁₃₁	—	2	1	no	—
130	Q₈×C₁₃₁	—	2	1	no	—
131	Q₈×C₁₃₁	#41	1	2	yes	—
132	D₄×C₁₃₁	—	2	4	no	—
133	D₄×C₁₃₁	—	2	1	no	—
134	D₄×C₁₃₁	#41	1	2	yes	—
135	C₂×C₄×C₁₃₁	—	2	4	no	—
136	C₂×C₄×C₁₃₁	—	2	1	no	—
137	C₂×C₄×C₁₃₁	—	2	1	no	—
138	C₂×C₄×C₁₃₁	—	2	1	no	—
139	C₂×C₄×C₁₃₁	—	2	1	no	—
140	C₂×C₄×C₁₃₁	—	2	1	no	—
141	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₁	—	2	1	no	—
144	C₈×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	2	2	no	—
147	C₂²×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	2	1	no	—
149	C₂²×C₁₃₁	—	4	1	no	—
150	C₂²×C₁₃₁	—	4	1	no	—
151	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₃₁	—	4	1	no	—
153	C₂×D₁₃₁	—	4	1	no	—
154	C₂×D₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	2	1	no	—
156	C₄×C₁₃₁	—	2	2	no	—
157	C₄×C₁₃₁	—	4	1	no	—
158	C₄×C₁₃₁	—	4	1	no	—
159	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
161	Dic₁₃₁	—	2	1	no	—
162	Dic₁₃₁	—	2	1	no	—
163	Dic₁₃₁	—	2	1	no	—
164	Dic₁₃₁	—	2	1	no	—
165	Dic₁₃₁	—	4	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	C₂×C₁₃₁	—	4	1	no	—
169	C₂×C₁₃₁	—	4	2	no	—
170	C₂×C₁₃₁	#525	1	1	yes	characteristic
171	C₂×C₁₃₁	#577	1	1	yes	characteristic
172	C₂×C₁₃₁	#577	1	1	yes	characteristic
173	D₁₃₁	—	8	1	no	—
174	C₁₃₁	#3125	1	1	yes	characteristic, complemented, Sylow
175	#3125	—	131	1	no	maximal, Sylow
176	#542	—	131	1	no	—
177	#520	—	131	1	no	—
178	#483	—	131	1	no	—
179	#481	—	131	1	no	—
180	#476	—	131	1	no	—
181	#475	—	131	1	no	—
182	#464	—	131	1	no	—
183	#463	—	131	1	no	—
184	#452	D₁₃₁	1	1	yes	characteristic, complemented
185	#449	—	131	1	no	—
186	#442	—	131	1	no	—
187	#441	—	131	1	no	—
188	C₄×SD₁₆	—	131	1	no	—
189	C₄×SD₁₆	—	131	1	no	—
190	#433	—	131	1	no	—
191	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
192	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
193	C₂×SD₁₆	—	131	1	no	—
194	C₂×SD₁₆	C₂×D₁₃₁	1	1	yes	characteristic
195	C₂×D₈	—	131	1	no	—
196	C₄⋊Q₈	—	131	1	no	—
197	#126	—	262	1	no	—
198	#124	—	262	1	no	—
199	#124	—	131	1	no	—
200	#123	—	262	1	no	—
201	C₂²⋊Q₈	—	262	1	no	—
202	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
203	#121	—	131	1	no	—
204	C₄×Q₈	—	131	1	no	—
205	C₄×Q₈	—	131	1	no	—
206	C₄×Q₈	—	131	1	no	—
207	C₄×D₄	—	262	1	no	—
208	C₄×D₄	—	262	1	no	—
209	C₄×D₄	—	131	1	no	—
210	C₄×D₄	—	131	1	no	—
211	C₄×D₄	—	131	1	no	—
212	#117	—	262	1	no	—
213	#117	—	131	1	no	—
214	C₂×C₄⋊C₄	—	262	1	no	—
215	C₂×C₄⋊C₄	—	131	1	no	—
216	#106	—	131	1	no	—
217	#106	—	131	1	no	—
218	#106	—	131	1	no	—
219	#106	—	131	1	no	—
220	C₄⋊C₈	—	131	1	no	—
221	C₄⋊C₈	—	131	1	no	—
222	#103	—	131	1	no	—
223	#103	—	131	1	no	—
224	#103	—	131	1	no	—
225	#103	C₂×D₁₃₁	1	1	yes	characteristic
226	#102	—	131	1	no	—
227	#102	—	131	1	no	—
228	#102	—	131	1	no	—
229	#102	C₂×D₁₃₁	1	1	yes	characteristic
230	C₂²⋊C₈	—	131	1	no	—
231	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
232	#97	—	131	1	no	—
233	C₄×C₈	—	131	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	—	262	1	no	—
237	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
238	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	262	1	no	—
241	C₂×D₄	—	131	1	no	—
242	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	262	1	no	—
245	C₂²×C₄	—	262	1	no	—
246	C₂²×C₄	—	262	1	no	—
247	C₂²×C₄	—	262	1	no	—
248	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	262	2	no	—
252	D₈	—	262	2	no	—
253	C₂×C₈	—	131	1	no	—
254	C₂×C₈	—	131	1	no	—
255	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	262	1	no	—
259	C₄⋊C₄	—	262	1	no	—
260	C₄⋊C₄	—	262	1	no	—
261	C₄⋊C₄	—	262	1	no	—
262	C₄⋊C₄	—	262	1	no	—
263	C₄⋊C₄	—	131	1	no	—
264	C₄⋊C₄	—	131	1	no	—
265	C₄⋊C₄	—	131	1	no	—
266	C₄⋊C₄	—	131	1	no	—
267	C₄⋊C₄	—	131	1	no	—
268	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	262	1	no	—
271	C₂²⋊C₄	—	262	1	no	—
272	C₂²⋊C₄	—	262	1	no	—
273	C₂²⋊C₄	—	262	1	no	—
274	C₂²⋊C₄	—	262	1	no	—
275	C₂²⋊C₄	—	262	1	no	—
276	C₂²⋊C₄	—	131	1	no	—
277	C₂²⋊C₄	—	131	1	no	—
278	C₄²	—	262	1	no	—
279	C₄²	—	262	1	no	—
280	C₄²	—	262	1	no	—
281	C₄²	—	131	1	no	—
282	C₄²	—	131	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	262	1	no	—
285	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	1048	1	no	—
290	Q₈	#14523512	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	—	262	1	no	—
294	D₄	—	262	1	no	—
295	D₄	—	1048	1	no	—
296	D₄	#14523512	1	2	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	262	1	no	—
304	C₂×C₄	—	262	1	no	—
305	C₂×C₄	—	262	1	no	—
306	C₂×C₄	—	1048	1	no	—
307	C₂×C₄	—	1048	1	no	—
308	C₂×C₄	—	1048	1	no	—
309	C₂×C₄	—	1048	1	no	—
310	C₂×C₄	—	131	1	no	—
311	C₂×C₄	—	131	1	no	—
312	C₂×C₄	—	131	1	no	—
313	C₂×C₄	—	131	1	no	—
314	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	262	1	no	—
319	C₈	—	262	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	524	1	no	—
327	C₂²	—	524	1	no	—
328	C₂²	—	524	1	no	—
329	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	262	1	no	—
335	C₄	—	262	1	no	—
336	C₄	—	262	1	no	—
337	C₄	—	262	1	no	—
338	C₄	—	524	1	no	—
339	C₄	—	524	1	no	—
340	C₄	—	524	1	no	—
341	C₄	C₂×#14523512	1	1	yes	characteristic
342	C₄	C₂×#14523512	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	1048	1	no	—
346	C₂	#22553818	1	1	yes	characteristic
347	C₂	#22554119	1	1	yes	characteristic
348	C₂	#22554121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#452×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553457	C₂
hypercenter	#452	D₁₃₁
hypocenter	C₁₃₁	#3125
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #22553935	C₂	semidirect
2. #22553795	C₂	semidirect
3. #22553640	C₂	semidirect
4. #22553631	C₂	semidirect
5. #22553600	C₂	semidirect
6. #22553595	C₂	semidirect
7. #22553536	C₂	semidirect
8. #22553529	C₂	semidirect
10. #22553421	C₂	semidirect
11. #22553419	C₂	semidirect
12. SD₁₆×Dic₁₃₁	C₂	semidirect
13. #22553397	C₂	semidirect
14. #22553370	C₂	semidirect
15. #452×C₁₃₁	C₂	semidirect
174. C₁₃₁	#3125	semidirect
184. #452	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553935	2
#22553795	2
#22553640	2
#22553631	2
#22553600	2
#22553595	2
#22553536	2
#22553529	2
#22553457	2
#22553421	2
#22553419	2
SD₁₆×Dic₁₃₁	2
#22553397	2
#22553370	2
#452×C₁₃₁	2
#3125	131