Subgroups — #14723013

Order 2816.

610 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#4339	C₂	1	1	yes	maximal, characteristic
2	#13077281	C₂	1	2	yes	maximal
3	#13067395	C₂	1	4	yes	maximal, complemented
4	#13067386	C₂	1	2	yes	maximal, complemented
5	#13067378	C₂	1	2	yes	maximal, complemented
6	#13067375	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067363	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#4339	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	#10622723	—	2	2	no	—
13	C₈×C₂²⋊D₁₁	C₄	1	2	yes	—
14	C₈×C₂²⋊D₁₁	C₄	1	2	yes	—
15	#10622714	—	2	2	no	—
16	#10622538	C₂²	1	1	yes	characteristic
17	#10622537	C₂²	1	2	yes	—
18	#10622453	—	2	4	no	—
19	#10622448	C₂²	1	4	yes	—
20	#10622446	C₂²	1	4	yes	—
21	C₈×D₄₄	C₄	1	1	yes	characteristic
22	C₈×D₄₄	C₄	1	1	yes	characteristic
23	#10622298	C₂²	1	2	yes	—
24	C₄×C₈×D₁₁	C₂²	1	2	yes	—
25	C₈×Dic₂₂	C₄	1	1	yes	characteristic
26	C₈×Dic₂₂	C₄	1	1	yes	characteristic
27	C₁₁×#335	C₂²	1	1	yes	characteristic
28	C₁₁×#335	C₂²	1	1	yes	characteristic
29	C₁₁×#335	C₂²	1	2	yes	—
30	#10622118	C₂²	1	4	yes	—
31	#10622118	C₂²	1	4	yes	—
32	#10622096	C₂²	1	2	yes	—
33	#10622096	C₂²	1	2	yes	—
34	C₂×#4339	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	—	2	2	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
40	C₄×C₂²⋊D₁₁	Q₈	1	2	yes	—
41	#66979	D₄	1	1	yes	characteristic
42	C₂×#4318	—	2	2	no	—
43	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
45	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
46	C₄×D₄₄	Q₈	1	1	yes	characteristic
47	#66925	D₄	1	2	yes	—
48	C₄²×D₁₁	C₂³	1	2	yes	—
49	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₂₂	Q₈	1	1	yes	characteristic
51	C₁₁×C₄⋊C₈	—	2	2	no	—
52	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
59	(C₂×D₁₁)⋊C₈	—	2	4	no	—
60	(C₂×D₁₁)⋊C₈	—	2	4	no	—
61	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
62	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
63	#66879	—	2	4	no	—
64	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₁	—	2	4	no	—
67	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
69	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
70	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	#17829	—	11	1	no	maximal, Sylow
75	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₁	C₄⋊C₄	1	2	yes	—
79	#4339	—	2	2	no	—
80	#4339	—	2	2	no	—
81	C₂×D₄₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₁₁	C₂×Q₈	1	2	yes	—
85	C₂×Dic₂₂	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	4	1	no	—
90	C₂×C₈×C₁₁	—	4	4	no	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
100	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
101	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
103	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
104	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
105	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
107	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
108	C₄×Dic₁₁	C₂×Q₈	1	2	yes	—
109	#4318	—	4	4	no	—
110	C₂×C₁₁⋊C₈	—	2	4	no	—
111	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
113	C₈×D₁₁	—	2	4	no	—
114	C₈×D₁₁	—	4	2	no	—
115	C₈×#41	—	11	1	no	—
116	#2925	—	11	2	no	—
117	#1572	—	11	4	no	—
118	#1570	—	11	2	no	—
119	#1568	—	11	2	no	—
120	#1567	—	11	1	no	—
121	#1564	—	11	2	no	—
122	C₂×#335	D₁₁	1	1	yes	characteristic, complemented
123	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₁₁	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₁	—	2	1	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	4	1	no	—
128	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	#135	1	2	yes	—
132	C₂×C₄×C₁₁	#135	1	2	yes	—
133	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂²⋊D₁₁	—	2	4	no	—
137	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
140	D₄₄	—	2	2	no	—
141	C₄×D₁₁	—	2	2	no	—
142	C₄×D₁₁	—	2	2	no	—
143	C₄×D₁₁	—	2	2	no	—
144	C₄×D₁₁	—	2	2	no	—
145	C₄×D₁₁	—	4	2	no	—
146	Dic₂₂	—	2	2	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	4	2	no	—
151	C₈×C₁₁	—	4	2	no	—
152	C₈×C₁₁	#131	1	2	yes	—
153	C₁₁⋊C₈	—	2	2	no	—
154	C₁₁⋊C₈	—	2	2	no	—
155	C₁₁⋊C₈	—	4	4	no	—
156	C₄×#41	—	11	1	no	—
157	C₈×Q₈	—	11	1	no	—
158	C₈×Q₈	—	11	1	no	—
159	C₈×D₄	—	11	2	no	—
160	C₈×D₄	—	11	2	no	—
161	C₈×D₄	—	11	1	no	—
162	C₈×D₄	—	11	1	no	—
163	#433	—	22	4	no	—
164	#432	—	11	4	no	—
165	#431	—	11	1	no	—
166	#431	—	11	2	no	—
167	#431	—	11	4	no	—
168	#423	—	11	2	no	—
169	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
170	#408	—	22	2	no	—
171	#405	—	22	2	no	—
172	C₂×C₄×C₈	—	11	2	no	—
173	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
174	#335	—	11	2	no	—
175	#335	—	11	2	no	—
176	#335	C₂×D₁₁	1	1	yes	characteristic
177	#335	C₂×D₁₁	1	1	yes	characteristic
178	#335	C₂×D₁₁	1	2	yes	—
179	#334	—	11	4	no	—
180	#334	—	11	4	no	—
181	C₂²×C₁₁	—	2	1	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	#427	1	2	yes	—
184	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂×D₁₁	—	2	2	no	—
186	C₂×D₁₁	—	2	2	no	—
187	C₂×D₁₁	—	2	2	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	#427	1	2	yes	—
192	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
193	C₄×C₁₁	#568	1	1	yes	characteristic
194	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
195	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
196	Dic₁₁	—	2	2	no	—
197	Dic₁₁	—	2	2	no	—
198	Dic₁₁	—	2	2	no	—
199	C₂×#41	—	11	1	no	—
200	C₂×M₁₆	—	22	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	11	2	no	—
203	C₂²×C₈	—	11	2	no	—
204	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
206	C₄×Q₈	—	11	1	no	—
207	C₄×Q₈	—	11	1	no	—
208	C₄×D₄	—	11	2	no	—
209	C₄×D₄	—	11	2	no	—
210	C₄×D₄	—	11	1	no	—
211	C₄×D₄	—	11	1	no	—
212	#117	—	11	1	no	—
213	#117	—	11	2	no	—
214	C₂×C₄²	—	11	2	no	—
215	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	11	2	no	—
218	C₄⋊C₈	—	11	2	no	—
219	C₄⋊C₈	—	11	1	no	—
220	C₄⋊C₈	—	11	1	no	—
221	C₄⋊C₈	—	11	4	no	—
222	C₄⋊C₈	—	22	2	no	—
223	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
224	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
225	C₂²⋊C₈	—	11	2	no	—
226	C₂²⋊C₈	—	11	2	no	—
227	C₂²⋊C₈	—	11	2	no	—
228	C₂²⋊C₈	—	22	4	no	—
229	C₂²⋊C₈	—	22	4	no	—
230	#97	—	22	4	no	—
231	C₄×C₈	—	11	2	no	—
232	C₄×C₈	—	11	2	no	—
233	C₄×C₈	—	11	2	no	—
234	C₄×C₈	—	22	4	no	—
235	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
236	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₂×C₁₁	—	2	2	no	—
240	C₂×C₁₁	#2906	1	1	yes	characteristic
241	C₂×C₁₁	C₂×#427	1	1	yes	characteristic
242	C₂×C₁₁	#2916	1	1	yes	characteristic
243	D₁₁	—	4	2	no	—
244	#41	—	22	2	no	—
245	#41	—	22	2	no	—
246	C₂×Q₈	—	11	1	no	—
247	C₂×D₄	—	11	2	no	—
248	C₂×D₄	—	11	1	no	—
249	C₂²×C₄	—	11	2	no	—
250	C₂²×C₄	—	11	2	no	—
251	C₂²×C₄	—	11	2	no	—
252	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
254	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
255	M₁₆	—	44	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	4	no	—
261	C₂×C₈	—	11	2	no	—
262	C₂×C₈	—	11	2	no	—
263	C₂×C₈	—	22	4	no	—
264	C₂×C₈	—	22	4	no	—
265	C₂×C₈	—	44	2	no	—
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	—	11	2	no	—
272	C₄⋊C₄	—	11	1	no	—
273	C₄⋊C₄	—	11	1	no	—
274	C₂²⋊C₄	—	11	2	no	—
275	C₂²⋊C₄	—	11	2	no	—
276	C₂²⋊C₄	—	11	2	no	—
277	C₄²	—	11	2	no	—
278	C₄²	—	11	2	no	—
279	C₄²	—	11	2	no	—
280	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
281	C₄²	D₄×D₁₁	1	1	yes	characteristic
282	C₄²	Q₈×D₁₁	1	1	yes	characteristic
283	C₄²	C₂³×D₁₁	1	1	yes	characteristic
284	C₁₁	#17829	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	11	2	no	—
286	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
287	Q₈	—	22	2	no	—
288	D₄	—	22	4	no	—
289	D₄	—	22	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	11	2	no	—
294	C₂×C₄	—	11	2	no	—
295	C₂×C₄	—	11	2	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	22	2	no	—
299	C₂×C₄	—	22	2	no	—
300	C₂×C₄	—	44	2	no	—
301	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	#66965	1	2	yes	—
305	C₂×C₄	#66972	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	22	2	no	—
315	C₈	—	22	2	no	—
316	C₈	—	44	4	no	—
317	C₈	#66954	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	22	2	no	—
321	C₂²	—	22	2	no	—
322	C₂²	—	22	2	no	—
323	C₂²	#10622493	1	2	yes	—
324	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	22	2	no	—
329	C₄	—	22	2	no	—
330	C₄	—	22	2	no	—
331	C₄	#10622493	1	2	yes	—
332	C₄	C₂×#66954	1	1	yes	characteristic
333	C₄	#10623276	1	1	yes	characteristic
334	C₄	C₂×#66965	1	1	yes	characteristic
335	C₄	C₂×#66972	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	44	2	no	—
338	C₂	#13077181	1	1	yes	characteristic
339	C₂	C₂×#10622493	1	1	yes	characteristic
340	C₂	#13077224	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×#4339	D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#17829
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #13067395	C₂	semidirect
4. #13067386	C₂	semidirect
5. #13067378	C₂	semidirect
6. #13067375	C₂	semidirect
7. #13067363	C₂	semidirect
8. C₂×C₁₁×#335	C₂	semidirect
122. C₂×#335	D₁₁	semidirect
284. C₁₁	#17829	semidirect

Maximal subgroups

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

subgroup	index
C₈×#4339	2
#13077281	2
#13067395	2
#13067386	2
#13067378	2
#13067375	2
#13067363	2
C₂×C₁₁×#335	2
#17829	11