Subgroups — #14758196

Order 2816.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077418	C₂	1	1	yes	maximal, characteristic
2	#13077258	C₂	1	2	yes	maximal, complemented
3	#13069509	C₂	1	1	yes	maximal, characteristic
4	#13069504	C₂	1	1	yes	maximal, characteristic
5	#13069369	C₂	1	1	yes	maximal, characteristic, complemented
6	#13069364	C₂	1	1	yes	maximal, characteristic, complemented
7	#13069327	C₂	1	2	yes	maximal, complemented
8	#13069320	C₂	1	2	yes	maximal, complemented
9	#13068522	C₂	1	1	yes	maximal, characteristic, complemented
10	#13068427	C₂	1	2	yes	maximal, complemented
11	C₁₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66957	C₂²	1	1	yes	characteristic
13	C₄×#4339	C₂²	1	2	yes	—
14	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₁₁×#376	C₂²	1	1	yes	characteristic
21	#10622726	C₂²	1	2	yes	—
22	#10622723	C₂²	1	2	yes	—
23	#10622723	C₂²	1	2	yes	—
24	C₂×#66883	C₂²	1	1	yes	characteristic
25	C₂×#66883	C₂²	1	1	yes	characteristic
26	#10622719	C₂²	1	1	yes	characteristic
27	#10622715	C₂²	1	2	yes	—
28	#10622715	C₂²	1	2	yes	—
29	C₂×#66869	C₂²	1	2	yes	—
30	C₂×#66869	C₂²	1	2	yes	—
31	C₄×#4318	C₂²	1	2	yes	—
32	#10622329	C₄	1	2	yes	complemented
33	#10622324	C₄	1	2	yes	complemented
34	#10622323	C₄	1	4	yes	complemented
35	#10622303	—	2	4	no	—
36	#10622298	—	2	4	no	—
37	#10622190	C₂²	1	2	yes	—
38	#10622161	C₂²	1	2	yes	—
39	#10622161	C₂²	1	2	yes	—
40	C₂×#4339	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
48	C₄×C₂²⋊D₁₁	—	2	4	no	—
49	#66979	C₂³	1	2	yes	—
50	C₂×#4318	C₂³	1	2	yes	—
51	C₂×#4318	C₂³	1	2	yes	—
52	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
53	#66957	—	2	2	no	—
54	#66957	—	2	2	no	—
55	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
56	C₂×#4314	C₂×C₄	1	2	yes	—
57	C₄×D₄₄	C₂×C₄	1	4	yes	—
58	#66925	—	2	2	no	—
59	C₄²×D₁₁	—	2	2	no	—
60	C₄×Dic₂₂	C₂×C₄	1	4	yes	—
61	C₁₁×#97	—	2	2	no	—
62	C₁₁×#97	—	2	2	no	—
63	C₁₁×#97	C₂×C₄	1	2	yes	—
64	C₁₁×#97	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₁	—	2	2	no	—
66	C₄×C₈×C₁₁	—	2	2	no	—
67	C₄×C₈×C₁₁	D₄	1	4	yes	—
68	C₁₁×#95	—	2	2	no	—
69	#66885	C₂×C₄	1	4	yes	—
70	(C₂×D₁₁)⋊C₈	—	2	4	no	—
71	(C₂×D₁₁)⋊C₈	—	2	4	no	—
72	#66883	—	2	2	no	—
73	#66883	—	2	2	no	—
74	#66882	C₂×C₄	1	2	yes	—
75	#66881	C₂×C₄	1	2	yes	—
76	#66880	C₂×C₄	1	4	yes	—
77	Dic₁₁⋊C₈	—	2	4	no	—
78	Dic₁₁⋊C₈	—	2	4	no	—
79	#66869	—	2	4	no	—
80	#66869	—	2	4	no	—
81	#66867	—	2	2	no	—
82	C₄×C₁₁⋊C₈	—	2	2	no	—
83	#21296	—	11	1	no	maximal, Sylow
84	C₂²×C₄×C₁₁	—	2	1	no	—
85	C₂²×C₄×C₁₁	—	2	1	no	—
86	C₂²×C₄×C₁₁	#41	1	2	yes	—
87	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₁	—	2	2	no	—
89	#4339	—	2	2	no	—
90	#4339	—	2	2	no	—
91	#4339	—	2	2	no	—
92	#4339	—	2	2	no	—
93	C₂×D₄₄	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₁	—	2	4	no	—
95	C₂×C₄×D₁₁	—	2	2	no	—
96	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₁	—	2	1	no	—
98	C₂×C₈×C₁₁	—	2	1	no	—
99	C₂×C₈×C₁₁	—	2	1	no	—
100	C₂×C₈×C₁₁	—	2	1	no	—
101	C₂×C₈×C₁₁	—	2	1	no	—
102	C₂×C₈×C₁₁	—	2	1	no	—
103	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₁	#41	1	2	yes	—
109	C₂×C₈×C₁₁	#41	1	2	yes	—
110	C₄²×C₁₁	—	2	1	no	—
111	C₄²×C₁₁	—	2	1	no	—
112	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₁	C₂×D₄	1	2	yes	—
114	C₄²×C₁₁	#41	1	2	yes	—
115	C₄²×C₁₁	#41	1	2	yes	—
116	C₂²⋊Dic₁₁	—	2	2	no	—
117	(C₂×D₁₁)⋊C₄	—	2	4	no	—
118	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
119	Dic₁₁⋊C₄	—	2	4	no	—
120	C₄×Dic₁₁	—	2	4	no	—
121	C₄×Dic₁₁	—	2	2	no	—
122	#4318	—	2	2	no	—
123	#4318	—	2	2	no	—
124	#4318	—	2	2	no	—
125	#4318	—	2	2	no	—
126	C₂×C₁₁⋊C₈	—	2	2	no	—
127	C₂×C₁₁⋊C₈	—	2	2	no	—
128	Dic₄₄	—	2	2	no	—
129	D₈₈	—	2	2	no	—
130	#4314	—	2	4	no	—
131	#2943	—	11	1	no	—
132	#2922	—	11	2	no	—
133	#1975	—	11	1	no	—
134	#1974	—	11	1	no	—
135	#1946	—	11	1	no	—
136	#1945	—	11	1	no	—
137	#1938	—	11	2	no	—
138	#1937	—	11	2	no	—
139	#1778	—	11	1	no	—
140	#1759	—	11	2	no	—
141	#1726	D₁₁	1	1	yes	characteristic, complemented
142	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
143	C₂²×D₁₁	—	2	2	no	—
144	C₂×C₄×C₁₁	—	2	2	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	—	4	2	no	—
147	C₂×C₄×C₁₁	—	4	2	no	—
148	C₂×C₄×C₁₁	—	4	2	no	—
149	C₂×C₄×C₁₁	C₄×D₄	1	4	yes	—
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
156	C₂²⋊D₁₁	—	4	2	no	—
157	C₂²⋊D₁₁	—	4	2	no	—
158	C₂×Dic₁₁	—	2	2	no	—
159	C₂×Dic₁₁	—	2	4	no	—
160	D₄₄	—	2	2	no	—
161	D₄₄	—	2	2	no	—
162	C₄×D₁₁	—	4	8	no	—
163	C₄×D₁₁	—	4	2	no	—
164	C₄×D₁₁	—	4	2	no	—
165	Dic₂₂	—	2	2	no	—
166	Dic₂₂	—	2	2	no	—
167	C₈×C₁₁	—	2	2	no	—
168	C₈×C₁₁	—	2	2	no	—
169	C₈×C₁₁	—	2	2	no	—
170	C₈×C₁₁	#117	1	2	yes	—
171	C₈×C₁₁	#117	1	2	yes	—
172	C₁₁⋊C₈	—	4	2	no	—
173	C₁₁⋊C₈	—	4	2	no	—
174	C₂×#135	—	11	1	no	—
175	C₄×#41	—	11	2	no	—
176	#442	—	11	2	no	—
177	#441	—	11	2	no	—
178	#440	—	11	4	no	—
179	#433	—	22	4	no	—
180	#431	—	22	4	no	—
181	C₂×#108	—	11	1	no	—
182	C₂×#108	—	11	1	no	—
183	#427	—	11	1	no	—
184	#424	—	11	2	no	—
185	#424	—	11	2	no	—
186	C₂×#104	—	11	2	no	—
187	C₂×#104	—	11	2	no	—
188	#416	—	11	2	no	—
189	#408	—	11	2	no	—
190	#408	—	11	2	no	—
191	C₄×M₁₆	—	11	2	no	—
192	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
193	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
194	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
198	#376	C₂×D₁₁	1	1	yes	characteristic
199	#341	—	11	2	no	—
200	#339	—	11	2	no	—
201	#339	—	11	2	no	—
202	C₂²×C₁₁	—	2	2	no	—
203	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₁₁	#529	1	1	yes	characteristic
205	C₂²×C₁₁	#529	1	1	yes	characteristic
206	C₂×D₁₁	—	4	4	no	—
207	C₂×D₁₁	—	4	2	no	—
208	C₄×C₁₁	—	2	4	no	—
209	C₄×C₁₁	—	4	2	no	—
210	C₄×C₁₁	—	4	2	no	—
211	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
212	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
213	C₄×C₁₁	#532	1	1	yes	characteristic
214	C₄×C₁₁	#532	1	1	yes	characteristic
215	Dic₁₁	—	4	4	no	—
216	Dic₁₁	—	4	2	no	—
217	C₂×#41	—	11	2	no	—
218	#135	—	22	2	no	—
219	#135	—	22	2	no	—
220	C₂×Q₁₆	—	11	1	no	—
221	C₂×SD₁₆	—	11	2	no	—
222	C₂×D₈	—	11	1	no	—
223	C₂×M₁₆	—	11	2	no	—
224	C₂×M₁₆	—	11	2	no	—
225	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
229	C₄×Q₈	—	11	4	no	—
230	C₄×D₄	—	11	4	no	—
231	C₄×D₄	—	22	4	no	—
232	#117	—	11	2	no	—
233	#117	—	22	2	no	—
234	C₂×C₄²	—	22	2	no	—
235	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
238	#108	—	22	2	no	—
239	#108	—	22	2	no	—
240	#107	—	11	2	no	—
241	#106	—	11	2	no	—
242	C₄⋊C₈	—	22	4	no	—
243	C₄⋊C₈	—	22	4	no	—
244	#104	—	22	4	no	—
245	#104	—	22	4	no	—
246	#103	—	11	4	no	—
247	#102	—	11	4	no	—
248	C₂²⋊C₈	—	22	4	no	—
249	C₂²⋊C₈	—	22	4	no	—
250	#97	—	2	2	no	—
251	#97	—	2	2	no	—
252	#97	—	22	2	no	—
253	#97	C₄×D₁₁	1	2	yes	—
254	#97	C₄×D₁₁	1	2	yes	—
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	—	2	2	no	—
257	C₄×C₈	—	22	2	no	—
258	C₄×C₈	D₄₄	1	4	yes	—
259	#95	—	2	2	no	—
260	C₂×C₁₁	—	2	1	no	—
261	C₂×C₁₁	—	2	1	no	—
262	C₂×C₁₁	#2293	1	1	yes	characteristic
263	C₂×C₁₁	#2955	1	2	yes	—
264	D₁₁	—	8	2	no	—
265	#41	—	22	2	no	—
266	#41	—	22	2	no	—
267	#41	—	22	2	no	—
268	#41	—	22	2	no	—
269	C₂×Q₈	—	11	2	no	—
270	C₂×D₄	—	11	2	no	—
271	C₂×D₄	—	22	2	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	22	4	no	—
275	C₂²×C₄	—	22	2	no	—
276	C₂²×C₄	#4339	1	2	yes	—
277	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
278	Q₁₆	—	22	2	no	—
279	SD₁₆	—	22	4	no	—
280	D₈	—	22	2	no	—
281	M₁₆	—	22	2	no	—
282	M₁₆	—	22	2	no	—
283	M₁₆	—	22	2	no	—
284	M₁₆	—	22	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	22	2	no	—
292	C₂×C₈	—	22	2	no	—
293	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄₄	1	2	yes	—
297	C₂×C₈	C₂×D₄₄	1	2	yes	—
298	C₂×C₈	#4339	1	2	yes	—
299	C₂×C₈	#4339	1	2	yes	—
300	C₄⋊C₄	—	11	4	no	—
301	C₄⋊C₄	—	22	4	no	—
302	C₂²⋊C₄	—	22	2	no	—
303	C₂²⋊C₄	—	22	4	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	22	4	no	—
307	C₄²	—	22	2	no	—
308	C₄²	C₂×C₄×D₁₁	1	2	yes	—
309	C₄²	C₂×D₄₄	1	2	yes	—
310	C₄²	#4339	1	2	yes	—
311	C₄²	#4339	1	2	yes	—
312	C₁₁	#21296	1	1	yes	characteristic, complemented, Sylow
313	C₂³	—	22	2	no	—
314	C₂³	C₂×#4339	1	1	yes	characteristic
315	Q₈	—	22	2	no	—
316	Q₈	—	22	2	no	—
317	D₄	—	22	2	no	—
318	D₄	—	22	2	no	—
319	D₄	—	44	2	no	—
320	D₄	—	44	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	22	4	no	—
327	C₂×C₄	—	22	2	no	—
328	C₂×C₄	—	44	8	no	—
329	C₂×C₄	—	44	2	no	—
330	C₂×C₄	—	44	2	no	—
331	C₂×C₄	C₄×D₄₄	1	4	yes	—
332	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
334	C₂×C₄	C₂×#4339	1	1	yes	characteristic
335	C₂×C₄	C₂×#4339	1	1	yes	characteristic
336	C₂×C₄	C₂×#4339	1	1	yes	characteristic
337	C₂×C₄	C₂×#4339	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	44	2	no	—
342	C₈	—	44	2	no	—
343	C₈	#66925	1	2	yes	—
344	C₈	#66925	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	44	4	no	—
347	C₂²	—	44	2	no	—
348	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
349	C₂²	#10623010	1	1	yes	characteristic
350	C₂²	#10623010	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	44	4	no	—
355	C₄	—	44	2	no	—
356	C₄	C₂×#66925	1	1	yes	characteristic
357	C₄	C₄×#4339	1	1	yes	characteristic
358	C₄	#10623008	1	1	yes	characteristic
359	C₄	#10623008	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	88	2	no	—
363	C₂	#13071080	1	1	yes	characteristic
364	C₂	#13077520	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#1726	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×#66957	C₂²
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#1726	D₁₁
hypocenter	C₁₁	#21296
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #13077258	C₂	semidirect
5. #13069369	C₂	semidirect
6. #13069364	C₂	semidirect
7. #13069327	C₂	semidirect
8. #13069320	C₂	semidirect
9. #13068522	C₂	semidirect
10. #13068427	C₂	semidirect
11. C₁₁×#1726	C₂	semidirect
32. #10622329	C₄	semidirect
33. #10622324	C₄	semidirect
34. #10622323	C₄	semidirect
141. #1726	D₁₁	semidirect
312. C₁₁	#21296	semidirect

Maximal subgroups

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

subgroup	index
#13077418	2
#13077258	2
#13069509	2
#13069504	2
#13069369	2
#13069364	2
#13069327	2
#13069320	2
#13068522	2
#13068427	2
C₁₁×#1726	2
#21296	11