Subgroups — #14907016

Order 2816.

642 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	#13081717	C₂	1	2	yes	maximal, complemented
2	#13077514	C₂	1	1	yes	maximal, characteristic
3	#13070458	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070457	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070456	C₂	1	2	yes	maximal, complemented
6	#13070454	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070450	C₂	1	2	yes	maximal, complemented
8	#13070380	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₁×#350	C₂	1	2	yes	maximal, complemented
10	#13070211	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623299	C₄	1	1	yes	characteristic
13	D₁₁×#131	C₂²	1	2	yes	—
14	C₁₁×#504	C₂²	1	2	yes	complemented
15	C₁₁×#443	C₂²	1	1	yes	characteristic
16	#10622810	C₄	1	1	yes	characteristic
17	#10622757	C₂²	1	2	yes	—
18	#10622753	C₂²	1	2	yes	complemented
19	#10622520	—	2	2	no	—
20	#10622520	C₂²	1	2	yes	complemented
21	D₁₁×M₃₂	—	2	2	no	—
22	D₁₁×M₃₂	C₂²	1	2	yes	—
23	#10622514	—	2	2	no	—
24	#10622514	C₂²	1	2	yes	complemented
25	C₂×#66862	—	2	2	no	—
26	C₂×#66862	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₁₁	C₂²	1	1	yes	characteristic
28	#10622507	C₄	1	1	yes	characteristic
29	#10622506	C₄	1	1	yes	characteristic
30	#10622436	C₄	1	2	yes	—
31	#10622434	C₄	1	2	yes	—
32	#10622313	C₂²	1	1	yes	characteristic
33	#10622302	C₂²	1	1	yes	characteristic
34	C₁₁×#364	C₂²	1	1	yes	characteristic, complemented
35	C₁₁×#350	C₂²	1	2	yes	complemented
36	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic, complemented
37	#10622199	C₂²	1	2	yes	complemented
38	#10622153	C₂²	1	2	yes	complemented
39	#10622149	C₂²	1	2	yes	complemented
40	#10622146	C₂²	1	2	yes	complemented
41	#10622140	C₂²	1	1	yes	characteristic, complemented
42	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic, complemented
43	#10622101	C₂²	1	1	yes	characteristic, complemented
44	#10622099	C₂²	1	1	yes	characteristic, complemented
45	#41×D₁₁	C₂×C₄	1	2	yes	—
46	C₁₁×#135	C₂×C₄	1	1	yes	characteristic
47	C₁₁×#131	C₂³	1	2	yes	—
48	#67003	C₂×C₄	1	2	yes	—
49	#67001	C₂×C₄	1	2	yes	—
50	#66971	C₈	1	2	yes	—
51	#66968	C₈	1	2	yes	—
52	#66967	C₈	1	2	yes	—
53	#66964	C₈	1	2	yes	—
54	#66960	—	2	2	no	—
55	#66960	C₂×C₄	1	2	yes	—
56	M₁₆×D₁₁	—	2	2	no	—
57	M₁₆×D₁₁	C₂×C₄	1	2	yes	—
58	#66957	C₂×C₄	1	1	yes	characteristic
59	#66954	—	2	2	no	—
60	#66954	C₂³	1	2	yes	—
61	C₂×#4313	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₁₁	—	2	2	no	—
63	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
64	#66925	C₂×C₄	1	1	yes	characteristic
65	C₁₁×M₃₂	—	2	2	no	—
66	C₁₁×M₃₂	C₂³	1	2	yes	—
67	C₂×C₁₁×C₁₆	—	2	2	no	—
68	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₁×#108	C₂×C₄	1	1	yes	characteristic
71	C₁₁×#104	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	#66901	C₂×C₄	1	2	yes	—
74	#66889	C₂×C₄	1	2	yes	—
75	#66886	C₂×C₄	1	2	yes	—
76	#66883	C₂×C₄	1	1	yes	characteristic
77	#66879	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
79	#66876	—	2	2	no	—
80	#66876	C₂³	1	2	yes	—
81	C₂×C₁₁⋊C₁₆	—	2	2	no	—
82	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
83	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
84	#66869	C₂×C₄	1	2	yes	—
85	#66867	C₂×C₄	1	1	yes	characteristic
86	#66862	—	2	2	no	—
87	#66862	—	2	2	no	—
88	#66862	—	2	2	no	—
89	#66862	D₄	1	2	yes	—
90	#66862	D₄	1	2	yes	—
91	C₁₆×D₁₁	—	2	2	no	—
92	C₁₆×D₁₁	—	2	2	no	—
93	C₁₆×D₁₁	—	2	2	no	—
94	C₁₆×D₁₁	—	2	2	no	—
95	#33442	—	11	1	no	maximal, Sylow
96	C₁₁×#41	C₂²×C₄	1	2	yes	—
97	#4343	C₂×C₈	1	2	yes	—
98	Q₈×D₁₁	C₂×C₈	1	2	yes	—
99	#4341	—	2	2	no	—
100	#4341	C₂×C₈	1	2	yes	—
101	D₄×D₁₁	—	2	2	no	—
102	D₄×D₁₁	C₂×C₈	1	2	yes	—
103	#4339	—	2	2	no	—
104	#4339	C₂²×C₄	1	2	yes	—
105	C₂×C₄×D₁₁	—	2	2	no	—
106	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₁₁×Q₁₆	C₂×C₈	1	1	yes	characteristic
108	C₁₁×SD₁₆	C₂×C₈	1	2	yes	—
109	C₁₁×D₈	C₂×C₈	1	1	yes	characteristic
110	C₁₁×M₁₆	—	2	2	no	—
111	C₁₁×M₁₆	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₁	—	2	2	no	—
113	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
115	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
116	#4326	C₂×C₈	1	2	yes	—
117	#4325	C₂×C₈	1	2	yes	—
118	#4324	C₂×C₈	1	2	yes	—
119	#4323	C₂×C₈	1	2	yes	—
120	(C₂×D₁₁)⋊C₄	—	2	1	no	—
121	Dic₁₁⋊C₄	—	2	1	no	—
122	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
123	#4318	—	2	2	no	—
124	#4318	C₂²×C₄	1	2	yes	—
125	C₂×C₁₁⋊C₈	—	2	2	no	—
126	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	Dic₄₄	C₂×C₈	1	1	yes	characteristic
129	D₈₈	C₂×C₈	1	1	yes	characteristic
130	#4314	C₂×C₈	1	2	yes	—
131	#4313	—	2	2	no	—
132	#4313	—	2	2	no	—
133	#4313	—	2	2	no	—
134	#4313	C₂×C₈	1	2	yes	—
135	#4313	C₂×C₈	1	2	yes	—
136	C₈×D₁₁	—	2	2	no	—
137	C₈×D₁₁	—	2	2	no	—
138	C₈×D₁₁	—	2	2	no	—
139	C₈×D₁₁	C₂×D₄	1	2	yes	—
140	C₈×D₁₁	#41	1	2	yes	—
141	C₁₁×C₁₆	—	2	2	no	—
142	C₁₁×C₁₆	—	2	1	no	—
143	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
144	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
145	C₁₁⋊C₁₆	—	2	2	no	—
146	C₁₁⋊C₁₆	—	2	1	no	—
147	C₁₁⋊C₁₆	C₂×D₄	1	2	yes	—
148	#3407	—	11	2	no	—
149	#2955	—	11	1	no	—
150	#2171	—	11	2	no	—
151	#2171	—	11	1	no	—
152	#2171	—	11	1	no	—
153	#2170	—	11	2	no	—
154	#2170	—	11	1	no	—
155	#2170	D₁₁	1	1	yes	characteristic, complemented
156	#2153	—	11	1	no	—
157	C₂×#350	—	11	2	no	—
158	#2108	—	11	1	no	—
159	C₂²×D₁₁	—	2	2	no	—
160	C₂²×D₁₁	#131	1	1	yes	characteristic
161	Q₈×C₁₁	C₂²×C₈	1	2	yes	—
162	D₄×C₁₁	—	2	2	no	—
163	D₄×C₁₁	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₁₁	—	2	2	no	—
165	C₂×C₄×C₁₁	—	2	1	no	—
166	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
167	C₂²⋊D₁₁	—	2	2	no	—
168	C₂²⋊D₁₁	—	2	2	no	—
169	C₂²⋊D₁₁	—	2	2	no	—
170	C₂×Dic₁₁	—	2	2	no	—
171	C₂×Dic₁₁	—	2	1	no	—
172	C₂×Dic₁₁	#131	1	1	yes	characteristic
173	D₄₄	—	2	2	no	—
174	D₄₄	C₂²×C₈	1	2	yes	—
175	C₄×D₁₁	—	2	2	no	—
176	C₄×D₁₁	—	2	2	no	—
177	C₄×D₁₁	—	2	2	no	—
178	C₄×D₁₁	C₄×D₄	1	2	yes	—
179	C₄×D₁₁	C₂²×C₈	1	2	yes	—
180	Dic₂₂	—	2	2	no	—
181	Dic₂₂	C₂²×C₈	1	2	yes	—
182	C₈×C₁₁	—	2	2	no	—
183	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
184	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
185	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
187	C₁₁⋊C₈	—	2	2	no	—
188	C₁₁⋊C₈	C₄×D₄	1	2	yes	—
189	C₁₁⋊C₈	C₂²×C₈	1	2	yes	—
190	#575	—	11	1	no	—
191	C₂×#131	—	11	2	no	—
192	#504	—	11	2	no	—
193	#504	—	11	2	no	—
194	#504	—	11	2	no	—
195	#504	—	22	2	no	—
196	#504	—	22	2	no	—
197	#504	C₂×D₁₁	1	2	yes	complemented
198	C₂×M₃₂	—	11	1	no	—
199	C₂×M₃₂	—	11	2	no	—
200	C₂×M₃₂	—	22	2	no	—
201	C₂×M₃₂	—	22	2	no	—
202	C₂²×C₁₆	—	11	1	no	—
203	#444	—	11	1	no	—
204	#444	—	11	1	no	—
205	#444	—	11	2	no	—
206	#443	—	11	2	no	—
207	#443	—	11	1	no	—
208	#443	C₂×D₁₁	1	1	yes	characteristic
209	#430	—	11	1	no	—
210	#421	—	11	2	no	—
211	#405	—	11	1	no	—
212	#364	—	11	2	no	—
213	#364	—	11	1	no	—
214	#364	C₂×D₁₁	1	1	yes	characteristic, complemented
215	#350	—	11	2	no	—
216	#350	—	11	2	no	—
217	#350	—	11	2	no	—
218	#350	C₂×D₁₁	1	2	yes	complemented
219	#346	—	11	1	no	—
220	#346	—	11	1	no	—
221	C₄×C₁₆	—	11	1	no	—
222	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic, complemented
223	C₂²×C₁₁	—	2	2	no	—
224	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
225	C₂×D₁₁	—	2	2	no	—
226	C₂×D₁₁	—	4	1	no	—
227	C₂×D₁₁	—	8	2	no	—
228	C₂×D₁₁	C₈×D₄	1	2	yes	—
229	C₄×C₁₁	—	2	2	no	—
230	C₄×C₁₁	—	4	1	no	—
231	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
232	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
233	Dic₁₁	—	2	2	no	—
234	Dic₁₁	—	4	1	no	—
235	Dic₁₁	C₈×D₄	1	2	yes	—
236	C₂×#41	—	11	2	no	—
237	#137	—	11	2	no	—
238	#137	—	11	2	no	—
239	AGL(1,ℤ/8)	—	11	2	no	—
240	AGL(1,ℤ/8)	—	11	2	no	—
241	#135	—	11	2	no	—
242	#135	—	11	1	no	—
243	#135	C₄×D₁₁	1	1	yes	characteristic
244	#131	—	11	2	no	—
245	#131	—	11	2	no	—
246	#131	—	11	2	no	—
247	#131	—	22	2	no	—
248	#131	—	22	2	no	—
249	#131	C₂²×D₁₁	1	2	yes	—
250	C₂×M₁₆	—	11	1	no	—
251	C₂×M₁₆	—	11	2	no	—
252	C₂×M₁₆	—	22	2	no	—
253	C₂²×C₈	—	11	1	no	—
254	C₂²×C₈	—	22	2	no	—
255	#117	—	11	1	no	—
256	M₃₂	—	2	2	no	—
257	M₃₂	—	11	2	no	—
258	M₃₂	—	11	2	no	—
259	M₃₂	—	11	2	no	—
260	M₃₂	—	22	2	no	—
261	M₃₂	—	22	2	no	—
262	M₃₂	—	22	2	no	—
263	M₃₂	—	22	2	no	—
264	M₃₂	C₂²×D₁₁	1	2	yes	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	11	1	no	—
267	C₂×C₁₆	—	11	1	no	—
268	C₂×C₁₆	—	22	2	no	—
269	C₂×C₁₆	—	22	2	no	—
270	C₂×C₁₆	—	22	2	no	—
271	C₂×C₁₆	—	22	2	no	—
272	C₂×C₁₆	—	22	2	no	—
273	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
274	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
275	#108	—	11	2	no	—
276	#108	—	11	1	no	—
277	#108	C₄×D₁₁	1	1	yes	characteristic
278	#104	—	11	2	no	—
279	#104	—	11	2	no	—
280	#104	—	11	2	no	—
281	#104	C₄×D₁₁	1	2	yes	—
282	#97	—	11	1	no	—
283	#97	—	11	1	no	—
284	C₄×C₈	—	11	1	no	—
285	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
286	C₂×C₁₁	—	2	1	no	—
287	C₂×C₁₁	—	4	2	no	—
288	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
289	D₁₁	—	2	2	no	—
290	D₁₁	—	4	2	no	—
291	#41	—	11	2	no	—
292	#41	—	11	2	no	—
293	#41	—	11	2	no	—
294	#41	—	22	2	no	—
295	#41	—	22	2	no	—
296	#41	C₂×C₄×D₁₁	1	2	yes	—
297	C₂×Q₈	—	11	2	no	—
298	C₂×D₄	—	11	2	no	—
299	C₂×D₄	—	22	2	no	—
300	C₂²×C₄	—	11	1	no	—
301	C₂²×C₄	—	22	2	no	—
302	Q₁₆	—	11	2	no	—
303	Q₁₆	—	11	1	no	—
304	Q₁₆	C₈×D₁₁	1	1	yes	characteristic
305	SD₁₆	—	11	2	no	—
306	SD₁₆	—	11	2	no	—
307	SD₁₆	—	11	2	no	—
308	SD₁₆	C₈×D₁₁	1	2	yes	—
309	D₈	—	11	2	no	—
310	D₈	—	11	1	no	—
311	D₈	C₈×D₁₁	1	1	yes	characteristic
312	M₁₆	—	2	2	no	—
313	M₁₆	—	11	2	no	—
314	M₁₆	—	11	2	no	—
315	M₁₆	—	11	2	no	—
316	M₁₆	—	22	2	no	—
317	M₁₆	—	22	2	no	—
318	M₁₆	—	22	2	no	—
319	M₁₆	—	22	2	no	—
320	M₁₆	C₂×C₄×D₁₁	1	2	yes	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	11	1	no	—
323	C₂×C₈	—	11	1	no	—
324	C₂×C₈	—	11	2	no	—
325	C₂×C₈	—	11	2	no	—
326	C₂×C₈	—	22	2	no	—
327	C₂×C₈	—	22	2	no	—
328	C₂×C₈	—	22	2	no	—
329	C₂×C₈	—	22	2	no	—
330	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
331	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
332	C₄⋊C₄	—	22	1	no	—
333	C₂²⋊C₄	—	22	1	no	—
334	C₄²	—	11	1	no	—
335	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	—	2	1	no	—
338	C₁₆	—	11	2	no	—
339	C₁₆	—	22	2	no	—
340	C₁₆	—	22	1	no	—
341	C₁₆	D₄×D₁₁	1	1	yes	characteristic
342	C₁₆	D₄×D₁₁	1	1	yes	characteristic
343	C₁₁	#33442	1	1	yes	characteristic, complemented, Sylow
344	C₂³	—	11	1	no	—
345	C₂³	—	22	2	no	—
346	Q₈	—	11	2	no	—
347	Q₈	—	22	2	no	—
348	Q₈	C₂×C₈×D₁₁	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	11	2	no	—
351	D₄	—	22	2	no	—
352	D₄	—	22	2	no	—
353	D₄	—	22	2	no	—
354	D₄	—	22	2	no	—
355	D₄	C₂×C₈×D₁₁	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	11	1	no	—
359	C₂×C₄	—	11	2	no	—
360	C₂×C₄	—	11	2	no	—
361	C₂×C₄	—	22	2	no	—
362	C₂×C₄	—	22	2	no	—
363	C₂×C₄	—	22	2	no	—
364	C₂×C₄	—	22	2	no	—
365	C₂×C₄	—	22	1	no	—
366	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	11	2	no	—
369	C₈	—	11	2	no	—
370	C₈	—	22	2	no	—
371	C₈	C₂×C₈×D₁₁	1	1	yes	characteristic
372	C₈	C₂×C₈×D₁₁	1	1	yes	characteristic
373	C₈	C₂×D₄×D₁₁	1	1	yes	characteristic
374	C₈	#41×D₁₁	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	11	2	no	—
377	C₂²	—	22	2	no	—
378	C₂²	—	44	1	no	—
379	C₂²	—	88	2	no	—
380	C₂²	D₁₁×#131	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	11	2	no	—
384	C₄	—	22	2	no	—
385	C₄	—	44	1	no	—
386	C₄	C₄×D₄×D₁₁	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	22	2	no	—
391	C₂	—	44	2	no	—
392	C₂	C₈×D₄×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2170	C₂
socle	C₂×C₁₁	C₂×C₈×D₄
prime radical	#10623299	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2170	D₁₁
hypocenter	C₁₁	#33442
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #13081717	C₂	semidirect
3. #13070458	C₂	semidirect
4. #13070457	C₂	semidirect
5. #13070456	C₂	semidirect
6. #13070454	C₂	semidirect
7. #13070450	C₂	semidirect
8. #13070380	C₂	semidirect
9. D₁₁×#350	C₂	semidirect
10. #13070211	C₂	semidirect
11. C₁₁×#2170	C₂	semidirect
14. C₁₁×#504	C₂²	semidirect
18. #10622753	C₂²	semidirect
20. #10622520	C₂²	semidirect
24. #10622514	C₂²	semidirect
34. C₁₁×#364	C₂²	semidirect
35. C₁₁×#350	C₂²	semidirect
36. C₄×C₁₁×C₁₆	C₂²	semidirect
37. #10622199	C₂²	semidirect
38. #10622153	C₂²	semidirect
39. #10622149	C₂²	semidirect
40. #10622146	C₂²	semidirect
41. #10622140	C₂²	semidirect
42. C₁₆×Dic₁₁	C₂²	semidirect
43. #10622101	C₂²	semidirect
44. #10622099	C₂²	semidirect
155. #2170	D₁₁	semidirect
197. #504	C₂×D₁₁	semidirect
214. #364	C₂×D₁₁	semidirect
218. #350	C₂×D₁₁	semidirect
222. C₄×C₁₆	C₂×D₁₁	semidirect
343. C₁₁	#33442	semidirect

Maximal subgroups

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

subgroup	index
#13081717	2
#13077514	2
#13070458	2
#13070457	2
#13070456	2
#13070454	2
#13070450	2
#13070380	2
D₁₁×#350	2
#13070211	2
C₁₁×#2170	2
#33442	11