Subgroups — #14725772

Order 2816.

606 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	#13079186	C₂	1	1	yes	maximal, characteristic
2	#13077772	C₂	1	2	yes	maximal, complemented
3	#13067512	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067500	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067493	C₂	1	2	yes	maximal, complemented
6	#13067472	C₂	1	2	yes	maximal, complemented
7	#13067422	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067410	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067027	C₂	1	2	yes	maximal, complemented
10	C₈×#4318	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623010	C₂²	1	2	yes	—
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	#10622729	C₄	1	1	yes	characteristic
16	#10622728	C₄	1	1	yes	characteristic
17	#10622727	C₄	1	2	yes	—
18	#10622724	—	2	2	no	—
19	C₈×C₂²⋊D₁₁	—	2	2	no	—
20	#10622538	C₂²	1	1	yes	characteristic
21	C₄×#4318	C₂²	1	1	yes	characteristic
22	#10622453	—	2	2	no	—
23	#10622452	—	2	2	no	—
24	#10622451	C₂²	1	2	yes	—
25	#10622447	C₂²	1	2	yes	—
26	#10622444	C₂²	1	2	yes	—
27	#10622443	C₂²	1	2	yes	—
28	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
29	#10622310	C₄	1	2	yes	—
30	#10622309	C₄	1	1	yes	characteristic
31	C₄×D₈₈	C₂²	1	1	yes	characteristic
32	C₄×#4314	C₂²	1	2	yes	—
33	#10622296	C₄	1	1	yes	characteristic
34	C₁₁×#335	C₂²	1	1	yes	characteristic
35	C₁₁×#335	C₂²	1	2	yes	—
36	C₁₁×#335	C₂²	1	1	yes	characteristic
37	#10622122	C₂²	1	2	yes	—
38	#10622122	C₂²	1	2	yes	—
39	#10622121	C₂²	1	2	yes	—
40	#10622121	C₂²	1	2	yes	—
41	#10622092	C₂²	1	1	yes	characteristic
42	#10622092	C₂²	1	1	yes	characteristic
43	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
44	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	—	2	2	no	—
46	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	#66983	C₂×C₄	1	2	yes	—
50	#66982	—	2	2	no	—
51	C₄×C₂²⋊D₁₁	—	2	2	no	—
52	#66977	C₂×C₄	1	2	yes	—
53	C₂×#4318	—	2	1	no	—
54	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
55	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
56	C₂×#4314	C₂×C₄	1	2	yes	—
57	C₂×#4313	—	2	2	no	—
58	C₂×C₈×D₁₁	—	2	2	no	—
59	#66929	—	2	2	no	—
60	#66928	C₂×C₄	1	2	yes	—
61	C₄×D₄₄	C₂³	1	2	yes	—
62	#66925	—	2	2	no	—
63	#66923	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₂	C₂³	1	2	yes	—
65	C₁₁×C₄⋊C₈	—	2	2	no	—
66	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	C₂²⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
73	C₂²⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
74	#66885	C₂×C₄	1	2	yes	—
75	#66885	C₂×C₄	1	2	yes	—
76	(C₂×D₁₁)⋊C₈	—	2	2	no	—
77	(C₂×D₁₁)⋊C₈	—	2	2	no	—
78	#66882	C₂×C₄	1	1	yes	characteristic
79	#66882	C₂×C₄	1	1	yes	characteristic
80	#66881	C₂×C₄	1	2	yes	—
81	#66880	C₂×C₄	1	2	yes	—
82	#66880	C₂×C₄	1	2	yes	—
83	#66879	—	2	2	no	—
84	Dic₁₁⋊C₈	—	2	2	no	—
85	Dic₁₁⋊C₈	—	2	2	no	—
86	C₈×Dic₁₁	—	2	2	no	—
87	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
88	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
89	#66867	—	2	1	no	—
90	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
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	#18087	—	11	1	no	maximal, Sylow
95	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₁	—	2	2	no	—
99	C₂×D₄₄	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₁₁	—	2	2	no	—
101	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₁	—	2	2	no	—
103	C₂×C₈×C₁₁	—	2	2	no	—
104	C₂×C₈×C₁₁	—	2	2	no	—
105	C₂×C₈×C₁₁	—	4	1	no	—
106	C₂×C₈×C₁₁	—	8	2	no	—
107	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₁	#41	1	1	yes	characteristic
114	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₁₁	—	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	(C₂×D₁₁)⋊C₄	—	2	2	no	—
120	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
121	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
122	Dic₁₁⋊C₄	—	2	2	no	—
123	Dic₁₁⋊C₄	—	2	2	no	—
124	Dic₁₁⋊C₄	—	2	2	no	—
125	Dic₁₁⋊C₄	—	2	2	no	—
126	C₄×Dic₁₁	—	2	2	no	—
127	#4318	—	2	4	no	—
128	C₂×C₁₁⋊C₈	—	2	1	no	—
129	C₂×C₁₁⋊C₈	—	2	1	no	—
130	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
134	Dic₄₄	—	2	2	no	—
135	D₈₈	—	2	2	no	—
136	#4314	—	4	2	no	—
137	#4313	—	8	2	no	—
138	C₈×D₁₁	—	8	2	no	—
139	#3168	—	11	1	no	—
140	#2994	—	11	2	no	—
141	#1591	—	11	1	no	—
142	#1589	—	11	1	no	—
143	#1588	—	11	2	no	—
144	#1585	—	11	2	no	—
145	C₈×Q₁₆	—	11	1	no	—
146	C₈×D₈	—	11	1	no	—
147	C₂×#335	D₁₁	1	1	yes	characteristic, complemented
148	#1483	—	11	2	no	—
149	C₈×M₁₆	—	11	1	no	—
150	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
151	C₂²×D₁₁	—	2	2	no	—
152	C₂×C₄×C₁₁	—	2	2	no	—
153	C₂×C₄×C₁₁	—	2	1	no	—
154	C₂×C₄×C₁₁	—	2	1	no	—
155	C₂×C₄×C₁₁	—	2	1	no	—
156	C₂×C₄×C₁₁	—	2	1	no	—
157	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₁₁	—	4	4	no	—
164	C₂×Dic₁₁	—	2	2	no	—
165	C₂×Dic₁₁	—	2	2	no	—
166	C₂×Dic₁₁	—	2	2	no	—
167	D₄₄	—	2	2	no	—
168	D₄₄	—	2	2	no	—
169	C₄×D₁₁	—	4	2	no	—
170	C₄×D₁₁	—	4	2	no	—
171	Dic₂₂	—	2	2	no	—
172	Dic₂₂	—	2	2	no	—
173	C₈×C₁₁	—	2	1	no	—
174	C₈×C₁₁	—	2	1	no	—
175	C₈×C₁₁	—	2	1	no	—
176	C₈×C₁₁	—	4	2	no	—
177	C₈×C₁₁	—	4	2	no	—
178	C₈×C₁₁	#131	1	2	yes	—
179	C₁₁⋊C₈	—	2	2	no	—
180	C₁₁⋊C₈	—	2	2	no	—
181	C₁₁⋊C₈	—	2	1	no	—
182	C₁₁⋊C₈	—	2	1	no	—
183	C₁₁⋊C₈	—	2	1	no	—
184	C₁₁⋊C₈	—	2	1	no	—
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	—	11	2	no	—
193	C₈×Q₈	—	11	2	no	—
194	C₄×Q₁₆	—	11	1	no	—
195	C₄×SD₁₆	—	11	2	no	—
196	C₄×D₈	—	11	1	no	—
197	#436	—	11	2	no	—
198	#435	—	22	2	no	—
199	C₈×D₄	—	11	2	no	—
200	C₈×D₄	—	22	2	no	—
201	#433	—	22	2	no	—
202	#431	—	11	1	no	—
203	#424	—	22	2	no	—
204	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
205	C₄×M₁₆	—	11	1	no	—
206	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
207	#335	C₂×D₁₁	1	1	yes	characteristic
208	#335	C₂×D₁₁	1	2	yes	—
209	#335	C₂×D₁₁	1	1	yes	characteristic
210	Q₈⋊C₈	—	11	2	no	—
211	Q₈⋊C₈	—	11	2	no	—
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	—	11	1	no	—
218	C₂²×C₁₁	—	2	1	no	—
219	C₂²×C₁₁	—	2	1	no	—
220	C₂²×C₁₁	#443	1	2	yes	—
221	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
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	—	2	1	no	—
227	C₄×C₁₁	—	2	1	no	—
228	C₄×C₁₁	—	2	1	no	—
229	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
230	C₄×C₁₁	#568	1	1	yes	characteristic
231	C₄×C₁₁	#576	1	1	yes	characteristic
232	C₄×C₁₁	#578	1	1	yes	characteristic
233	Dic₁₁	—	4	2	no	—
234	Dic₁₁	—	4	2	no	—
235	Dic₁₁	—	4	2	no	—
236	C₂×Q₁₆	—	11	1	no	—
237	C₂×SD₁₆	—	11	2	no	—
238	C₂×D₈	—	11	1	no	—
239	C₂×M₁₆	—	22	2	no	—
240	C₂×M₁₆	—	22	1	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	22	2	no	—
243	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
244	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
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	—	11	2	no	—
252	C₄×D₄	—	11	2	no	—
253	C₄×D₄	—	22	2	no	—
254	#117	—	22	2	no	—
255	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
256	#107	—	11	1	no	—
257	#107	—	11	1	no	—
258	#106	—	11	2	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	—	11	1	no	—
261	C₄⋊C₈	—	11	1	no	—
262	C₄⋊C₈	—	22	2	no	—
263	C₄⋊C₈	—	22	2	no	—
264	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
265	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
266	#103	—	11	2	no	—
267	#103	—	11	2	no	—
268	#102	—	11	2	no	—
269	#102	—	11	2	no	—
270	C₂²⋊C₈	—	11	1	no	—
271	C₂²⋊C₈	—	11	1	no	—
272	C₂²⋊C₈	—	22	2	no	—
273	C₂²⋊C₈	—	22	2	no	—
274	#97	—	22	2	no	—
275	#97	—	22	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	C₄×D₁₁	1	1	yes	characteristic
282	C₄×C₈	C₄×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	—	2	2	no	—
286	C₂×C₁₁	#2935	1	1	yes	characteristic
287	C₂×C₁₁	#2951	1	1	yes	characteristic
288	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
289	D₁₁	—	8	2	no	—
290	C₂×Q₈	—	11	2	no	—
291	C₂×D₄	—	11	2	no	—
292	C₂×D₄	—	22	2	no	—
293	C₂²×C₄	—	22	2	no	—
294	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
296	C₂²×C₄	#4343	1	1	yes	characteristic
297	Q₁₆	—	22	2	no	—
298	SD₁₆	—	44	2	no	—
299	D₈	—	22	2	no	—
300	M₁₆	—	22	4	no	—
301	M₁₆	—	88	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	8	2	no	—
307	C₂×C₈	—	11	1	no	—
308	C₂×C₈	—	11	1	no	—
309	C₂×C₈	—	11	1	no	—
310	C₂×C₈	—	11	1	no	—
311	C₂×C₈	—	22	1	no	—
312	C₂×C₈	—	22	1	no	—
313	C₂×C₈	—	88	2	no	—
314	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
318	C₄⋊C₄	—	11	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	—	22	2	no	—
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	—	22	2	no	—
330	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
331	C₄²	D₄×D₁₁	1	1	yes	characteristic
332	C₄²	#4343	1	1	yes	characteristic
333	C₄²	C₂³×D₁₁	1	1	yes	characteristic
334	C₁₁	#18087	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	22	2	no	—
336	C₂³	#66946	1	1	yes	characteristic
337	Q₈	—	22	2	no	—
338	Q₈	—	22	2	no	—
339	D₄	—	22	2	no	—
340	D₄	—	22	2	no	—
341	D₄	—	44	4	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	22	2	no	—
348	C₂×C₄	—	22	2	no	—
349	C₂×C₄	—	22	2	no	—
350	C₂×C₄	—	44	2	no	—
351	C₂×C₄	—	44	2	no	—
352	C₂×C₄	#66946	1	1	yes	characteristic
353	C₂×C₄	#66946	1	1	yes	characteristic
354	C₂×C₄	#66946	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
357	C₂×C₄	C₂×#4343	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	22	2	no	—
364	C₈	—	22	2	no	—
365	C₈	—	22	1	no	—
366	C₈	—	22	1	no	—
367	C₈	—	22	1	no	—
368	C₈	—	22	1	no	—
369	C₈	#66954	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	44	2	no	—
373	C₂²	—	44	2	no	—
374	C₂²	—	44	2	no	—
375	C₂²	#10622508	1	2	yes	—
376	C₂²	C₂×#66946	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	44	2	no	—
382	C₄	—	44	2	no	—
383	C₄	—	44	2	no	—
384	C₄	C₂×#66954	1	1	yes	characteristic
385	C₄	#10623276	1	1	yes	characteristic
386	C₄	#10623288	1	1	yes	characteristic
387	C₄	#10623297	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	88	2	no	—
390	C₂	#13077359	1	1	yes	characteristic
391	C₂	#13077490	1	1	yes	characteristic
392	C₂	C₂×#10622508	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	C₂×C₄×D₄
prime radical	#10622728	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#18087
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
2. #13077772	C₂	semidirect
3. #13067512	C₂	semidirect
4. #13067500	C₂	semidirect
5. #13067493	C₂	semidirect
6. #13067472	C₂	semidirect
7. #13067422	C₂	semidirect
8. #13067410	C₂	semidirect
9. #13067027	C₂	semidirect
10. C₈×#4318	C₂	semidirect
11. C₂×C₁₁×#335	C₂	semidirect
147. C₂×#335	D₁₁	semidirect
334. C₁₁	#18087	semidirect

Maximal subgroups

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

subgroup	index
#13079186	2
#13077772	2
#13067512	2
#13067500	2
#13067493	2
#13067472	2
#13067422	2
#13067410	2
#13067027	2
C₈×#4318	2
C₂×C₁₁×#335	2
#18087	11