Subgroups — #14725563

Order 2816.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079189	C₂	1	1	yes	maximal, characteristic
2	#13077622	C₂	1	2	yes	maximal, complemented
3	#13067500	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067456	C₂	1	2	yes	maximal, complemented
5	#13067450	C₂	1	2	yes	maximal, complemented
6	Q₁₆×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067363	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067023	C₂	1	2	yes	maximal, complemented
10	C₈×#4313	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623111	C₂²	1	2	yes	—
13	C₁₁×#446	C₂²	1	2	yes	—
14	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
15	#10622804	C₄	1	1	yes	characteristic
16	#10622801	C₄	1	1	yes	characteristic
17	C₄×#4325	C₂²	1	2	yes	—
18	Q₈×C₁₁⋊C₈	C₂²	1	2	yes	—
19	#10622496	C₄	1	1	yes	characteristic
20	#10622491	C₄	1	1	yes	characteristic
21	#10622453	—	2	2	no	—
22	#10622451	—	2	2	no	—
23	#10622451	C₂²	1	2	yes	—
24	#10622446	C₂²	1	2	yes	—
25	#10622420	C₄	1	2	yes	—
26	#10622413	C₄	1	2	yes	—
27	#10622321	—	2	2	no	—
28	#10622319	—	2	2	no	—
29	C₄×D₈₈	C₂²	1	1	yes	characteristic
30	C₄×#4313	C₂²	1	1	yes	characteristic
31	C₈×D₄₄	C₂²	1	2	yes	—
32	#10622298	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
34	C₁₁×#335	C₂²	1	1	yes	characteristic
35	C₁₁×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₁	C₂²	1	1	yes	characteristic
37	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
38	#10622121	C₂²	1	2	yes	—
39	#10622118	C₂²	1	2	yes	—
40	#10622097	C₂²	1	2	yes	—
41	#10622096	C₂²	1	1	yes	characteristic
42	#10622093	C₂²	1	1	yes	characteristic
43	#10622092	C₂²	1	1	yes	characteristic
44	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
47	#67000	C₂×C₄	1	2	yes	—
48	#66999	C₂×C₄	1	2	yes	—
49	C₂×#4325	C₂×C₄	1	2	yes	—
50	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
51	C₂×#4313	—	2	2	no	—
52	C₂×#4313	D₄	1	1	yes	characteristic
53	C₂×#4313	D₄	1	1	yes	characteristic
54	C₂×C₈×D₁₁	—	2	2	no	—
55	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
57	#66951	—	2	2	no	—
58	#66948	C₂×C₄	1	2	yes	—
59	#66947	—	2	2	no	—
60	#66943	C₂×C₄	1	2	yes	—
61	C₄×D₄₄	—	2	2	no	—
62	C₄×D₄₄	C₂³	1	2	yes	—
63	#66925	—	2	2	no	—
64	C₄²×D₁₁	C₂³	1	1	yes	characteristic
65	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₁×C₄⋊C₈	—	2	2	no	—
67	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₁×#103	C₂×C₄	1	2	yes	—
69	C₁₁×#97	—	2	2	no	—
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	#66899	C₂×C₄	1	2	yes	—
74	#66885	C₂×C₄	1	2	yes	—
75	(C₂×D₁₁)⋊C₈	—	2	2	no	—
76	(C₂×D₁₁)⋊C₈	—	2	1	no	—
77	(C₂×D₁₁)⋊C₈	—	2	2	no	—
78	(C₂×D₁₁)⋊C₈	—	2	2	no	—
79	#66882	C₂×C₄	1	1	yes	characteristic
80	#66879	D₄	1	1	yes	characteristic
81	#66879	D₄	1	1	yes	characteristic
82	Dic₁₁⋊C₈	—	2	1	no	—
83	C₈×Dic₁₁	—	2	2	no	—
84	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
86	#66873	C₂×C₄	1	2	yes	—
87	#66872	C₂×C₄	1	2	yes	—
88	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
89	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
90	C₄×C₁₁⋊C₈	—	2	2	no	—
91	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
93	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
94	#18069	—	11	1	no	maximal, Sylow
95	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄₄	—	2	2	no	—
97	C₂×D₄₄	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₁	—	2	2	no	—
99	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₁₁	#41	1	1	yes	characteristic
102	C₁₁×Q₁₆	—	2	2	no	—
103	C₂×C₈×C₁₁	—	2	2	no	—
104	C₂×C₈×C₁₁	—	2	1	no	—
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₁₁×C₄⋊C₄	—	2	2	no	—
110	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₁	—	2	2	no	—
112	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
113	#4325	—	4	2	no	—
114	(C₂×D₁₁)⋊C₄	—	2	2	no	—
115	(C₂×D₁₁)⋊C₄	—	2	2	no	—
116	(C₂×D₁₁)⋊C₄	—	2	2	no	—
117	(C₂×D₁₁)⋊C₄	—	2	2	no	—
118	(C₂×D₁₁)⋊C₄	—	2	2	no	—
119	C₄⋊Dic₁₁	—	2	2	no	—
120	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
121	Dic₁₁⋊C₄	—	2	2	no	—
122	Dic₁₁⋊C₄	—	2	2	no	—
123	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₁	#41	1	1	yes	characteristic
126	C₂×C₁₁⋊C₈	—	2	2	no	—
127	C₂×C₁₁⋊C₈	—	2	1	no	—
128	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
132	D₈₈	—	2	2	no	—
133	#4313	—	2	2	no	—
134	#4313	—	2	2	no	—
135	#4313	—	4	4	no	—
136	C₈×D₁₁	—	2	2	no	—
137	C₈×D₁₁	—	4	1	no	—
138	C₈×D₁₁	—	4	4	no	—
139	#3168	—	11	1	no	—
140	#2969	—	11	2	no	—
141	#1591	D₁₁	1	1	yes	characteristic, complemented
142	#1589	—	11	1	no	—
143	#1583	—	11	2	no	—
144	#1582	—	11	2	no	—
145	C₈×Q₁₆	—	11	1	no	—
146	C₈×D₈	—	11	1	no	—
147	#1564	—	11	1	no	—
148	#1482	—	11	2	no	—
149	C₈×M₁₆	—	11	1	no	—
150	C₂²×D₁₁	—	2	2	no	—
151	C₂²×D₁₁	C₄×D₄	1	1	yes	characteristic
152	Q₈×C₁₁	—	2	2	no	—
153	Q₈×C₁₁	—	2	2	no	—
154	C₂×C₄×C₁₁	—	2	2	no	—
155	C₂×C₄×C₁₁	—	2	2	no	—
156	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₁₁	—	2	2	no	—
160	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
162	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
163	D₄₄	—	2	2	no	—
164	D₄₄	—	2	2	no	—
165	D₄₄	—	8	2	no	—
166	C₄×D₁₁	—	2	1	no	—
167	C₄×D₁₁	—	2	1	no	—
168	C₄×D₁₁	—	2	1	no	—
169	C₄×D₁₁	—	2	1	no	—
170	C₄×D₁₁	—	4	1	no	—
171	C₄×D₁₁	—	4	2	no	—
172	C₄×D₁₁	—	4	2	no	—
173	C₈×C₁₁	—	2	1	no	—
174	C₈×C₁₁	—	2	1	no	—
175	C₈×C₁₁	—	4	1	no	—
176	C₈×C₁₁	—	4	2	no	—
177	C₈×C₁₁	#131	1	2	yes	—
178	C₈×C₁₁	#135	1	2	yes	—
179	C₁₁⋊C₈	—	2	1	no	—
180	C₁₁⋊C₈	—	2	1	no	—
181	C₁₁⋊C₈	—	2	1	no	—
182	C₁₁⋊C₈	—	4	1	no	—
183	C₁₁⋊C₈	—	4	2	no	—
184	C₁₁⋊C₈	#135	1	2	yes	—
185	#529	—	11	2	no	—
186	#499	—	11	1	no	—
187	#495	—	11	1	no	—
188	#488	—	11	2	no	—
189	#467	—	11	1	no	—
190	#466	—	11	1	no	—
191	#465	—	11	2	no	—
192	#446	C₂×D₁₁	1	2	yes	—
193	C₈×Q₈	—	11	2	no	—
194	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
195	C₄×SD₁₆	—	11	2	no	—
196	C₄×D₈	—	11	1	no	—
197	#436	—	11	2	no	—
198	#436	—	22	2	no	—
199	#435	—	22	2	no	—
200	C₈×D₄	—	11	2	no	—
201	#433	—	22	2	no	—
202	#431	—	11	1	no	—
203	#431	—	11	2	no	—
204	#405	—	22	2	no	—
205	C₄×M₁₆	—	11	1	no	—
206	C₂×C₄×C₈	—	11	1	no	—
207	#335	—	11	1	no	—
208	#335	C₂×D₁₁	1	1	yes	characteristic
209	#334	—	11	2	no	—
210	Q₈⋊C₈	—	11	2	no	—
211	Q₈⋊C₈	C₂×D₁₁	1	2	yes	—
212	D₄⋊C₈	—	11	2	no	—
213	D₄⋊C₈	—	11	2	no	—
214	#322	—	11	1	no	—
215	#322	—	11	1	no	—
216	C₈²	—	11	1	no	—
217	C₈²	C₂×D₁₁	1	1	yes	characteristic
218	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂×D₁₁	—	2	1	no	—
220	C₂×D₁₁	—	2	1	no	—
221	C₂×D₁₁	—	2	1	no	—
222	C₂×D₁₁	—	4	2	no	—
223	C₂×D₁₁	—	4	2	no	—
224	C₂×D₁₁	—	4	2	no	—
225	C₄×C₁₁	—	2	1	no	—
226	C₄×C₁₁	—	4	2	no	—
227	C₄×C₁₁	—	4	2	no	—
228	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
229	C₄×C₁₁	#568	1	1	yes	characteristic
230	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
231	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
232	Dic₁₁	—	2	1	no	—
233	Dic₁₁	—	2	1	no	—
234	Dic₁₁	—	2	1	no	—
235	Dic₁₁	—	4	2	no	—
236	C₂×Q₁₆	C₄×D₁₁	1	1	yes	characteristic
237	C₂×SD₁₆	—	11	2	no	—
238	C₂×D₈	—	11	1	no	—
239	C₂×M₁₆	—	11	1	no	—
240	C₂×M₁₆	—	11	1	no	—
241	C₂×M₁₆	—	22	2	no	—
242	C₂²×C₈	—	11	1	no	—
243	C₂²×C₈	—	11	1	no	—
244	C₂²×C₈	—	22	2	no	—
245	#126	—	22	2	no	—
246	#125	—	11	2	no	—
247	#124	—	11	2	no	—
248	#123	—	22	2	no	—
249	C₂²⋊Q₈	—	11	2	no	—
250	#121	—	11	2	no	—
251	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
252	C₄×D₄	—	11	2	no	—
253	C₄×D₄	—	22	2	no	—
254	#117	—	22	2	no	—
255	C₂×C₄²	—	11	1	no	—
256	#107	—	11	1	no	—
257	#107	C₄×D₁₁	1	1	yes	characteristic
258	#106	—	11	2	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	—	11	2	no	—
261	C₄⋊C₈	—	22	2	no	—
262	C₄⋊C₈	—	22	1	no	—
263	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
264	#103	—	11	2	no	—
265	#103	C₄×D₁₁	1	2	yes	—
266	#102	—	11	2	no	—
267	#102	—	11	2	no	—
268	C₂²⋊C₈	—	22	2	no	—
269	C₂²⋊C₈	—	22	1	no	—
270	C₂²⋊C₈	—	22	2	no	—
271	C₂²⋊C₈	—	22	2	no	—
272	#97	—	2	2	no	—
273	#97	—	11	1	no	—
274	#97	—	11	1	no	—
275	C₄×C₈	—	11	1	no	—
276	C₄×C₈	—	11	1	no	—
277	C₄×C₈	—	11	1	no	—
278	C₄×C₈	—	11	1	no	—
279	C₄×C₈	—	11	1	no	—
280	C₄×C₈	—	22	2	no	—
281	C₄×C₈	—	22	2	no	—
282	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
285	C₂×C₁₁	#2935	1	1	yes	characteristic
286	C₂×C₁₁	C₄×#135	1	1	yes	characteristic
287	C₂×C₁₁	#2957	1	1	yes	characteristic
288	D₁₁	—	4	1	no	—
289	D₁₁	—	8	2	no	—
290	C₂×Q₈	C₂×C₄×D₁₁	1	2	yes	—
291	C₂×D₄	—	11	2	no	—
292	C₂×D₄	—	22	2	no	—
293	C₂²×C₄	—	11	1	no	—
294	C₂²×C₄	—	11	1	no	—
295	C₂²×C₄	—	11	1	no	—
296	C₂²×C₄	—	22	2	no	—
297	Q₁₆	—	2	2	no	—
298	SD₁₆	—	44	2	no	—
299	D₈	—	22	2	no	—
300	M₁₆	—	22	2	no	—
301	M₁₆	—	22	2	no	—
302	M₁₆	—	44	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	11	1	no	—
306	C₂×C₈	—	11	1	no	—
307	C₂×C₈	—	11	1	no	—
308	C₂×C₈	—	11	1	no	—
309	C₂×C₈	—	22	2	no	—
310	C₂×C₈	—	22	2	no	—
311	C₂×C₈	—	22	1	no	—
312	C₂×C₈	—	44	1	no	—
313	C₂×C₈	—	44	4	no	—
314	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
316	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
317	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	11	2	no	—
320	C₄⋊C₄	—	22	2	no	—
321	C₄⋊C₄	—	22	2	no	—
322	C₄⋊C₄	—	22	2	no	—
323	C₄⋊C₄	C₂×C₄×D₁₁	1	2	yes	—
324	C₂²⋊C₄	—	22	2	no	—
325	C₂²⋊C₄	—	22	2	no	—
326	C₂²⋊C₄	—	22	2	no	—
327	C₂²⋊C₄	—	22	2	no	—
328	C₂²⋊C₄	—	22	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	11	1	no	—
331	C₄²	—	11	1	no	—
332	C₄²	—	11	1	no	—
333	C₄²	C₂³×D₁₁	1	1	yes	characteristic
334	C₁₁	#18069	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	11	1	no	—
336	C₂³	—	22	2	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	22	2	no	—
340	D₄	—	22	2	no	—
341	D₄	—	88	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	11	1	no	—
345	C₂×C₄	—	11	1	no	—
346	C₂×C₄	—	11	1	no	—
347	C₂×C₄	—	22	1	no	—
348	C₂×C₄	—	22	1	no	—
349	C₂×C₄	—	22	1	no	—
350	C₂×C₄	—	22	1	no	—
351	C₂×C₄	—	22	2	no	—
352	C₂×C₄	—	44	1	no	—
353	C₂×C₄	—	44	2	no	—
354	C₂×C₄	—	44	2	no	—
355	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
357	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	11	2	no	—
363	C₈	—	22	1	no	—
364	C₈	—	22	1	no	—
365	C₈	—	22	1	no	—
366	C₈	—	44	1	no	—
367	C₈	—	44	2	no	—
368	C₈	#66960	1	2	yes	—
369	C₈	#66972	1	2	yes	—
370	C₂²	—	22	1	no	—
371	C₂²	—	22	1	no	—
372	C₂²	—	22	1	no	—
373	C₂²	—	44	2	no	—
374	C₂²	—	44	2	no	—
375	C₂²	—	44	2	no	—
376	C₂²	C₄×D₄×D₁₁	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	22	1	no	—
381	C₄	—	22	1	no	—
382	C₄	—	22	1	no	—
383	C₄	—	44	2	no	—
384	C₄	C₂×#66960	1	1	yes	characteristic
385	C₄	#10623281	1	1	yes	characteristic
386	C₄	C₂×#66972	1	1	yes	characteristic
387	C₄	D₁₁×#135	1	1	yes	characteristic
388	C₂	—	44	1	no	—
389	C₂	—	88	2	no	—
390	C₂	#13077361	1	1	yes	characteristic
391	C₂	C₄×#66972	1	1	yes	characteristic
392	C₂	#13077528	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1591	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#10622496	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1591	D₁₁
hypocenter	C₁₁	#18069
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
2. #13077622	C₂	semidirect
3. #13067500	C₂	semidirect
4. #13067456	C₂	semidirect
5. #13067450	C₂	semidirect
6. Q₁₆×C₁₁⋊C₈	C₂	semidirect
7. C₈×D₈₈	C₂	semidirect
8. #13067363	C₂	semidirect
9. #13067023	C₂	semidirect
10. C₈×#4313	C₂	semidirect
11. C₁₁×#1591	C₂	semidirect
141. #1591	D₁₁	semidirect
334. C₁₁	#18069	semidirect

Maximal subgroups

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

subgroup	index
#13079189	2
#13077622	2
#13067500	2
#13067456	2
#13067450	2
Q₁₆×C₁₁⋊C₈	2
C₈×D₈₈	2
#13067363	2
#13067023	2
C₈×#4313	2
C₁₁×#1591	2
#18069	11