Subgroups — #14754134

Order 2816.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077038	C₂	1	1	yes	maximal, characteristic
2	#13068764	C₂	1	2	yes	maximal, complemented
3	#13068740	C₂	1	1	yes	maximal, characteristic, complemented
4	#13068718	C₂	1	2	yes	maximal
5	#13068663	C₂	1	1	yes	maximal, characteristic
6	#13068659	C₂	1	1	yes	maximal, characteristic
7	#13068498	C₂	1	1	yes	maximal, characteristic, complemented
8	#13068497	C₂	1	1	yes	maximal, characteristic, complemented
9	#13068415	C₂	1	2	yes	maximal
10	#13068370	C₂	1	2	yes	maximal, complemented
11	C₁₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66960	C₂²	1	1	yes	characteristic
13	C₂×#66954	C₂²	1	1	yes	characteristic
14	#10623053	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
16	C₁₁×#427	C₂²	1	1	yes	characteristic
17	C₁₁×#405	C₂²	1	1	yes	characteristic
18	#10622745	C₂²	1	2	yes	—
19	#10622735	C₂²	1	2	yes	—
20	M₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
21	C₂×#66883	C₂²	1	1	yes	characteristic
22	#10622719	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
24	#10622619	C₂²	1	2	yes	—
25	#10622610	C₂²	1	2	yes	—
26	#10622610	C₂²	1	1	yes	characteristic
27	#10622321	C₄	1	2	yes	complemented
28	#10622320	C₄	1	2	yes	complemented
29	#10622302	C₄	1	2	yes	complemented
30	C₄×C₈×D₁₁	C₄	1	2	yes	complemented
31	C₁₁×#344	C₂²	1	2	yes	—
32	C₁₁×H(3,ℤ/4)	C₂²	1	2	yes	—
33	C₁₁⋊#338	C₂²	1	2	yes	—
34	#10622194	C₂²	1	2	yes	—
35	#10622193	C₂²	1	2	yes	—
36	#10622170	C₂²	1	2	yes	—
37	#10622169	C₂²	1	2	yes	—
38	#10622169	C₂²	1	2	yes	—
39	C₂×#4339	C₂³	1	1	yes	characteristic
40	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
41	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₁₁×#117	C₂³	1	2	yes	—
44	C₁₁×#117	C₂³	1	1	yes	characteristic
45	#66979	C₂³	1	2	yes	—
46	C₂×C₄×Dic₁₁	C₂³	1	1	yes	characteristic
47	C₂×#4318	C₂³	1	2	yes	—
48	C₂×#4318	C₂³	1	1	yes	characteristic
49	C₂²×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
50	#66960	—	2	1	no	—
51	#66960	—	2	1	no	—
52	#66960	—	2	1	no	—
53	#66960	—	2	1	no	—
54	#66954	—	2	1	no	—
55	#66954	—	2	1	no	—
56	#66954	—	2	1	no	—
57	#66954	—	2	1	no	—
58	C₂×#4313	C₂×C₄	1	2	yes	—
59	C₂×#4313	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
61	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
62	#66946	—	2	1	no	—
63	#66940	—	2	1	no	—
64	(C₂²⋊D₁₁)⋊C₄	—	2	2	no	—
65	#66925	C₂×C₄	1	2	yes	—
66	C₄²×D₁₁	C₂×C₄	1	2	yes	—
67	C₁₁×#108	—	2	1	no	—
68	C₁₁×#108	—	2	1	no	—
69	C₁₁×#107	—	2	1	no	—
70	C₁₁×#106	—	2	1	no	—
71	C₁₁×#97	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
73	#66888	—	2	2	no	—
74	#66887	—	2	2	no	—
75	#66883	—	2	1	no	—
76	#66883	—	2	1	no	—
77	#66882	—	2	1	no	—
78	#66881	—	2	1	no	—
79	#66879	—	2	2	no	—
80	#66879	C₂×C₄	1	2	yes	—
81	C₈×Dic₁₁	—	2	2	no	—
82	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
83	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
87	#66870	—	2	4	no	—
88	#66867	—	2	2	no	—
89	#66867	C₂×C₄	1	2	yes	—
90	C₄×C₁₁⋊C₈	—	2	2	no	—
91	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
92	#20787	—	11	1	no	maximal, Sylow
93	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₂×C₂²⋊D₁₁	#41	1	2	yes	—
95	C₂²×Dic₁₁	—	2	1	no	—
96	#4339	—	2	1	no	—
97	#4339	—	2	1	no	—
98	#4339	—	2	1	no	—
99	#4339	—	2	1	no	—
100	C₂×D₄₄	#41	1	1	yes	characteristic
101	C₂×C₄×D₁₁	—	2	2	no	—
102	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₂₂	#41	1	1	yes	characteristic
104	C₁₁×M₁₆	—	2	1	no	—
105	C₁₁×M₁₆	—	2	1	no	—
106	C₁₁×M₁₆	—	2	2	no	—
107	C₁₁×M₁₆	—	2	2	no	—
108	C₂×C₈×C₁₁	—	2	2	no	—
109	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
114	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
115	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
116	C₁₁×C₄⋊C₄	—	2	1	no	—
117	C₁₁×C₄⋊C₄	—	2	2	no	—
118	C₁₁×C₂²⋊C₄	—	2	1	no	—
119	C₁₁×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₁₁	—	2	2	no	—
121	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
122	C₂²⋊Dic₁₁	—	2	2	no	—
123	(C₂×D₁₁)⋊C₄	—	2	2	no	—
124	C₄⋊Dic₁₁	—	2	2	no	—
125	Dic₁₁⋊C₄	—	2	2	no	—
126	C₄×Dic₁₁	—	2	2	no	—
127	C₄×Dic₁₁	—	2	1	no	—
128	C₄×Dic₁₁	—	2	2	no	—
129	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
131	#4318	—	2	1	no	—
132	#4318	—	2	1	no	—
133	#4318	—	2	2	no	—
134	#4318	—	2	2	no	—
135	C₂×C₁₁⋊C₈	—	2	2	no	—
136	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₁₁⋊C₈	#41	1	1	yes	characteristic
140	C₂×C₁₁⋊C₈	#41	1	1	yes	characteristic
141	C₂×C₁₁⋊C₈	#41	1	1	yes	characteristic
142	C₂×C₁₁⋊C₈	#41	1	1	yes	characteristic
143	#4313	—	2	2	no	—
144	#4313	—	2	2	no	—
145	#4313	—	2	2	no	—
146	#4313	—	2	2	no	—
147	C₈×D₁₁	—	2	2	no	—
148	C₈×D₁₁	—	2	2	no	—
149	C₈×D₁₁	—	2	2	no	—
150	C₈×D₁₁	—	2	2	no	—
151	C₄×#131	—	11	1	no	—
152	#1841	—	11	2	no	—
153	#1833	—	11	1	no	—
154	#1833	D₁₁	1	1	yes	characteristic, complemented
155	#1827	—	11	2	no	—
156	#1813	—	11	1	no	—
157	#1812	—	11	1	no	—
158	#1772	—	11	1	no	—
159	#1772	—	11	1	no	—
160	#1757	—	11	2	no	—
161	#1747	—	11	2	no	—
162	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
163	C₂²×D₁₁	#117	1	2	yes	—
164	C₂×C₄×C₁₁	—	2	2	no	—
165	C₂×C₄×C₁₁	—	4	2	no	—
166	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
172	C₂²⋊D₁₁	—	2	2	no	—
173	C₂²⋊D₁₁	—	2	2	no	—
174	C₂×Dic₁₁	—	2	1	no	—
175	C₂×Dic₁₁	—	2	1	no	—
176	C₂×Dic₁₁	—	4	2	no	—
177	C₂×Dic₁₁	—	4	2	no	—
178	C₂×Dic₁₁	#117	1	2	yes	—
179	D₄₄	—	2	1	no	—
180	D₄₄	—	2	1	no	—
181	C₄×D₁₁	—	2	2	no	—
182	C₄×D₁₁	—	2	2	no	—
183	C₄×D₁₁	—	4	4	no	—
184	Dic₂₂	—	2	1	no	—
185	Dic₂₂	—	2	1	no	—
186	C₈×C₁₁	—	2	2	no	—
187	C₈×C₁₁	—	4	2	no	—
188	C₈×C₁₁	#117	1	1	yes	characteristic
189	C₈×C₁₁	#117	1	1	yes	characteristic
190	C₈×C₁₁	#117	1	1	yes	characteristic
191	C₈×C₁₁	#117	1	1	yes	characteristic
192	C₁₁⋊C₈	—	2	2	no	—
193	C₁₁⋊C₈	—	4	2	no	—
194	C₁₁⋊C₈	#117	1	1	yes	characteristic
195	C₁₁⋊C₈	#117	1	1	yes	characteristic
196	C₁₁⋊C₈	#117	1	1	yes	characteristic
197	C₁₁⋊C₈	#117	1	1	yes	characteristic
198	C₂×#131	—	11	1	no	—
199	C₂×#131	—	11	1	no	—
200	C₄×#41	—	11	1	no	—
201	C₂×#108	—	11	1	no	—
202	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
203	#427	—	11	1	no	—
204	#427	C₂×D₁₁	1	1	yes	characteristic
205	#413	—	11	2	no	—
206	#410	—	11	2	no	—
207	#405	—	11	1	no	—
208	#405	—	11	2	no	—
209	#405	—	11	2	no	—
210	#405	—	11	2	no	—
211	#405	—	11	2	no	—
212	#405	C₂×D₁₁	1	1	yes	characteristic
213	C₄×M₁₆	—	11	1	no	—
214	C₄×M₁₆	—	11	2	no	—
215	C₂×C₄×C₈	—	11	1	no	—
216	C₂×C₄×C₈	—	11	2	no	—
217	#344	—	11	2	no	—
218	#344	C₂×D₁₁	1	2	yes	—
219	#339	—	11	2	no	—
220	#339	—	11	2	no	—
221	#338	—	11	2	no	—
222	H(3,ℤ/4)	—	11	2	no	—
223	H(3,ℤ/4)	—	11	2	no	—
224	H(3,ℤ/4)	C₂×D₁₁	1	2	yes	—
225	C₂²×C₁₁	—	4	1	no	—
226	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
227	C₂²×C₁₁	#529	1	1	yes	characteristic
228	C₂²×C₁₁	#529	1	1	yes	characteristic
229	C₂×D₁₁	—	2	2	no	—
230	C₂×D₁₁	—	4	2	no	—
231	C₄×C₁₁	—	4	2	no	—
232	C₄×C₁₁	—	8	2	no	—
233	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
234	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
235	C₄×C₁₁	#529	1	1	yes	characteristic
236	C₄×C₁₁	#529	1	1	yes	characteristic
237	Dic₁₁	—	2	2	no	—
238	Dic₁₁	—	2	2	no	—
239	Dic₁₁	—	4	1	no	—
240	Dic₁₁	—	8	2	no	—
241	C₂×#41	—	11	1	no	—
242	#131	—	22	1	no	—
243	#131	—	22	1	no	—
244	#131	—	22	1	no	—
245	#131	—	22	1	no	—
246	#131	—	22	1	no	—
247	#131	—	22	1	no	—
248	#131	—	22	1	no	—
249	#131	—	22	1	no	—
250	C₂×M₁₆	—	11	1	no	—
251	C₂×M₁₆	—	11	2	no	—
252	C₂×M₁₆	—	11	2	no	—
253	C₂×M₁₆	—	11	2	no	—
254	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
255	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
256	C₂²×C₈	—	11	1	no	—
257	C₂²×C₈	—	11	2	no	—
258	C₂²×C₈	—	11	2	no	—
259	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
260	C₄×Q₈	—	22	1	no	—
261	C₄×D₄	—	22	2	no	—
262	C₄×D₄	—	22	1	no	—
263	#117	—	11	2	no	—
264	#117	—	11	2	no	—
265	#117	C₂²×D₁₁	1	2	yes	—
266	#117	C₂²×D₁₁	1	1	yes	characteristic
267	C₂×C₄²	—	11	1	no	—
268	C₂×C₄²	—	11	2	no	—
269	#108	—	2	1	no	—
270	#108	—	2	1	no	—
271	#108	—	22	1	no	—
272	#108	—	22	1	no	—
273	#107	—	2	1	no	—
274	#107	—	22	1	no	—
275	#106	—	2	1	no	—
276	#106	—	22	1	no	—
277	#101	—	22	2	no	—
278	#100	—	22	2	no	—
279	#99	—	22	4	no	—
280	#97	—	11	2	no	—
281	#97	—	11	2	no	—
282	#97	—	22	2	no	—
283	#97	—	22	2	no	—
284	#97	C₄×D₁₁	1	2	yes	—
285	C₄×C₈	—	11	1	no	—
286	C₄×C₈	—	11	1	no	—
287	C₄×C₈	—	11	2	no	—
288	C₄×C₈	—	11	1	no	—
289	C₄×C₈	—	11	1	no	—
290	C₄×C₈	—	11	2	no	—
291	C₄×C₈	—	22	2	no	—
292	C₄×C₈	—	22	2	no	—
293	C₄×C₈	C₄×D₁₁	1	2	yes	—
294	C₂×C₁₁	—	2	1	no	—
295	C₂×C₁₁	—	2	1	no	—
296	C₂×C₁₁	—	2	1	no	—
297	C₂×C₁₁	#2283	1	1	yes	characteristic
298	D₁₁	—	4	2	no	—
299	#41	—	22	1	no	—
300	#41	—	22	1	no	—
301	#41	—	22	1	no	—
302	#41	—	22	1	no	—
303	C₂×Q₈	—	11	1	no	—
304	C₂×D₄	—	11	2	no	—
305	C₂×D₄	—	11	1	no	—
306	C₂²×C₄	—	11	2	no	—
307	C₂²×C₄	—	22	2	no	—
308	C₂²×C₄	—	22	1	no	—
309	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
310	M₁₆	—	2	1	no	—
311	M₁₆	—	2	1	no	—
312	M₁₆	—	2	2	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	22	2	no	—
315	M₁₆	—	22	2	no	—
316	M₁₆	—	22	1	no	—
317	M₁₆	—	22	1	no	—
318	M₁₆	—	22	2	no	—
319	M₁₆	—	22	2	no	—
320	M₁₆	—	22	2	no	—
321	M₁₆	—	22	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	11	1	no	—
324	C₂×C₈	—	11	1	no	—
325	C₂×C₈	—	11	1	no	—
326	C₂×C₈	—	11	1	no	—
327	C₂×C₈	—	11	1	no	—
328	C₂×C₈	—	11	1	no	—
329	C₂×C₈	—	11	2	no	—
330	C₂×C₈	—	22	2	no	—
331	C₂×C₈	—	22	2	no	—
332	C₂×C₈	—	22	2	no	—
333	C₂×C₈	—	22	2	no	—
334	C₂×C₈	—	22	2	no	—
335	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
336	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
337	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
338	C₂×C₈	#4341	1	1	yes	characteristic
339	C₂×C₈	#4341	1	1	yes	characteristic
340	C₂×C₈	#4343	1	1	yes	characteristic
341	C₂×C₈	#4343	1	1	yes	characteristic
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	22	2	no	—
345	C₄⋊C₄	—	22	2	no	—
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	22	2	no	—
349	C₂²⋊C₄	—	22	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	11	1	no	—
352	C₄²	—	11	1	no	—
353	C₄²	—	22	2	no	—
354	C₄²	—	22	1	no	—
355	C₄²	—	22	2	no	—
356	C₄²	C₂×C₄×D₁₁	1	2	yes	—
357	C₁₁	#20787	1	1	yes	characteristic, complemented, Sylow
358	C₂³	—	11	2	no	—
359	C₂³	#41×D₁₁	1	1	yes	characteristic
360	Q₈	—	22	1	no	—
361	Q₈	—	22	1	no	—
362	D₄	—	22	2	no	—
363	D₄	—	22	2	no	—
364	D₄	—	22	1	no	—
365	D₄	—	22	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	11	2	no	—
369	C₂×C₄	—	22	1	no	—
370	C₂×C₄	—	22	2	no	—
371	C₂×C₄	—	22	2	no	—
372	C₂×C₄	—	22	1	no	—
373	C₂×C₄	—	44	4	no	—
374	C₂×C₄	—	44	2	no	—
375	C₂×C₄	—	44	2	no	—
376	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
377	C₂×C₄	C₂×#4341	1	1	yes	characteristic
378	C₂×C₄	C₂×#4343	1	1	yes	characteristic
379	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
380	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
381	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	—	11	1	no	—
385	C₈	—	11	1	no	—
386	C₈	—	11	1	no	—
387	C₈	—	11	1	no	—
388	C₈	—	22	2	no	—
389	C₈	—	44	2	no	—
390	C₈	#66945	1	1	yes	characteristic
391	C₈	#66945	1	1	yes	characteristic
392	C₈	#66945	1	1	yes	characteristic
393	C₈	#66945	1	1	yes	characteristic
394	C₂²	—	4	1	no	—
395	C₂²	—	22	2	no	—
396	C₂²	—	44	2	no	—
397	C₂²	D₁₁×#117	1	1	yes	characteristic
398	C₂²	#10623211	1	1	yes	characteristic
399	C₂²	#10623222	1	1	yes	characteristic
400	C₄	—	4	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	22	2	no	—
403	C₄	—	22	2	no	—
404	C₄	—	44	1	no	—
405	C₄	—	88	2	no	—
406	C₄	C₂×#66945	1	1	yes	characteristic
407	C₄	D₁₁×#117	1	1	yes	characteristic
408	C₄	#10623211	1	1	yes	characteristic
409	C₄	#10623222	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	44	2	no	—
414	C₂	#13071009	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₁×#117
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#1833	C₂
socle	C₂×C₁₁	#2283
prime radical	C₂×#4339	C₂³
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#1833	D₁₁
hypocenter	C₁₁	#20787
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #13068764	C₂	semidirect
3. #13068740	C₂	semidirect
7. #13068498	C₂	semidirect
8. #13068497	C₂	semidirect
10. #13068370	C₂	semidirect
11. C₁₁×#1833	C₂	semidirect
27. #10622321	C₄	semidirect
28. #10622320	C₄	semidirect
29. #10622302	C₄	semidirect
30. C₄×C₈×D₁₁	C₄	semidirect
154. #1833	D₁₁	semidirect
357. C₁₁	#20787	semidirect

Maximal subgroups

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

subgroup	index
#13077038	2
#13068764	2
#13068740	2
#13068718	2
#13068663	2
#13068659	2
#13068498	2
#13068497	2
#13068415	2
#13068370	2
C₁₁×#1833	2
#20787	11