Subgroups — #14907367

Order 2816.

626 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₄	C₂	1	1	yes	maximal, characteristic
2	#13070449	C₂	1	2	yes	maximal, complemented
3	#13070440	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₄₄	C₂	1	2	yes	maximal, complemented
5	#13070419	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070225	C₂	1	2	yes	maximal, complemented
7	D₁₁×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	C₁₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄₄	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₁₁)⋊C₈	C₄	1	2	yes	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×#66862	C₂²	1	4	yes	complemented
15	C₂×C₁₆×D₁₁	C₂²	1	4	yes	complemented
16	C₈×D₄₄	C₄	1	1	yes	characteristic
17	C₈×D₄₄	C₄	1	2	yes	—
18	C₈×D₄₄	C₄	1	1	yes	characteristic
19	C₈×D₄₄	C₂²	1	1	yes	characteristic
20	C₈×D₄₄	C₂²	1	2	yes	—
21	C₈×D₄₄	C₂²	1	1	yes	characteristic
22	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
23	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₁₁×C₁₆	C₂²	1	2	yes	complemented
26	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
27	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	complemented
28	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	complemented
29	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	complemented
30	#10622099	C₂²	1	2	yes	complemented
31	C₂²×D₄₄	C₈	1	1	yes	characteristic
32	C₂²×C₄×D₁₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂×(C₂×D₁₁)⋊C₄	C₈	1	2	yes	—
37	C₂×C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
38	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₁	—	4	4	no	—
40	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₁	C₂³	1	2	yes	—
43	C₂×C₈×D₁₁	C₂³	1	2	yes	—
44	C₄×D₄₄	C₈	1	2	yes	—
45	C₄×D₄₄	C₈	1	1	yes	characteristic
46	C₄×D₄₄	C₈	1	1	yes	characteristic
47	C₄×D₄₄	C₂×C₄	1	2	yes	—
48	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
49	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
56	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
57	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
58	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
59	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
61	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
62	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
63	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
64	#66862	—	4	4	no	—
65	C₁₆×D₁₁	—	4	4	no	—
66	#33488	—	11	1	no	maximal, Sylow
67	C₂³×D₁₁	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
72	C₂×D₄₄	—	2	4	no	—
73	C₂×D₄₄	C₂×C₈	1	2	yes	—
74	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₁	—	4	2	no	—
77	C₂×C₄×D₁₁	—	4	2	no	—
78	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	4	1	no	—
84	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
88	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
93	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
94	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
95	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
97	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
98	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
99	C₂×C₁₁⋊C₈	—	4	2	no	—
100	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
102	C₈×D₁₁	—	2	4	no	—
103	C₈×D₁₁	—	2	4	no	—
104	C₈×D₁₁	—	4	2	no	—
105	C₈×D₁₁	—	4	2	no	—
106	C₁₁×C₁₆	—	2	2	no	—
107	C₁₁×C₁₆	—	2	2	no	—
108	C₁₁⋊C₁₆	—	2	4	no	—
109	C₂×C₈×D₄	—	11	1	no	—
110	#2169	—	11	2	no	—
111	#2168	—	11	4	no	—
112	D₄×C₁₆	—	11	2	no	—
113	#2163	—	11	1	no	—
114	#2162	D₁₁	1	1	yes	characteristic, complemented
115	#2112	—	11	2	no	—
116	C₂×C₂²⋊C₁₆	—	11	2	no	—
117	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
118	C₂²×D₁₁	—	2	4	no	—
119	C₂²×D₁₁	—	2	4	no	—
120	C₂²×D₁₁	—	4	2	no	—
121	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
122	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₁₁	—	2	2	no	—
124	C₂×C₄×C₁₁	—	2	1	no	—
125	C₂×C₄×C₁₁	—	2	1	no	—
126	C₂×C₄×C₁₁	—	2	2	no	—
127	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×Dic₁₁	—	4	2	no	—
134	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
136	D₄₄	—	2	4	no	—
137	D₄₄	—	2	2	no	—
138	D₄₄	M₃₂	1	4	yes	complemented
139	C₄×D₁₁	—	2	2	no	—
140	C₄×D₁₁	—	2	2	no	—
141	C₄×D₁₁	—	2	2	no	—
142	C₄×D₁₁	—	2	2	no	—
143	C₄×D₁₁	—	4	2	no	—
144	C₄×D₁₁	—	4	2	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
149	C₁₁⋊C₈	—	2	2	no	—
150	C₁₁⋊C₈	—	2	2	no	—
151	C₂³×C₈	—	11	2	no	—
152	C₂×C₄×D₄	—	11	1	no	—
153	C₂×M₃₂	—	11	4	no	—
154	C₂²×C₁₆	—	11	4	no	—
155	C₈×D₄	—	11	1	no	—
156	C₈×D₄	—	11	2	no	—
157	C₈×D₄	—	11	1	no	—
158	C₈×D₄	—	11	2	no	—
159	C₈×D₄	—	11	1	no	—
160	C₈×D₄	—	11	1	no	—
161	C₂×C₄⋊C₈	—	11	1	no	—
162	C₂×C₂²⋊C₈	—	11	2	no	—
163	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
164	C₄⋊C₁₆	—	11	2	no	—
165	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	complemented
166	C₂²⋊C₁₆	—	11	2	no	—
167	C₂²⋊C₁₆	—	11	4	no	—
168	C₂²⋊C₁₆	—	11	4	no	—
169	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
170	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
171	#346	—	11	2	no	—
172	C₄×C₁₆	C₂×D₁₁	1	2	yes	complemented
173	C₂²×C₁₁	—	2	1	no	—
174	C₂²×C₁₁	—	2	1	no	—
175	C₂²×C₁₁	#504	1	2	yes	—
176	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
177	C₂×D₁₁	—	2	2	no	—
178	C₂×D₁₁	—	2	2	no	—
179	C₂×D₁₁	—	2	2	no	—
180	C₂×D₁₁	—	2	2	no	—
181	C₂×D₁₁	—	2	2	no	—
182	C₂×D₁₁	—	4	8	no	—
183	C₂×D₁₁	C₂×M₃₂	1	4	yes	—
184	C₄×C₁₁	—	2	1	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
189	C₄×C₁₁	#518	1	1	yes	characteristic
190	C₄×C₁₁	#568	1	1	yes	characteristic
191	Dic₁₁	—	2	2	no	—
192	Dic₁₁	—	2	2	no	—
193	C₂²×D₄	—	11	1	no	—
194	C₂³×C₄	—	11	2	no	—
195	C₂²×C₈	—	11	2	no	—
196	C₂²×C₈	—	11	2	no	—
197	C₂²×C₈	—	11	2	no	—
198	C₂²×C₈	—	11	2	no	—
199	C₂²×C₈	—	11	2	no	—
200	C₂²×C₈	—	44	4	no	—
201	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₄×D₄	—	11	1	no	—
204	C₄×D₄	—	11	2	no	—
205	C₄×D₄	—	11	1	no	—
206	C₄×D₄	—	11	1	no	—
207	C₄×D₄	—	11	2	no	—
208	C₄×D₄	—	11	1	no	—
209	C₂×C₄⋊C₄	—	11	1	no	—
210	C₂×C₂²⋊C₄	—	11	2	no	—
211	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
212	M₃₂	—	44	4	no	—
213	C₂×C₁₆	—	11	4	no	—
214	C₂×C₁₆	—	44	4	no	—
215	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
216	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
217	C₄⋊C₈	—	11	1	no	—
218	C₄⋊C₈	—	11	2	no	—
219	C₄⋊C₈	—	11	1	no	—
220	C₂²⋊C₈	—	11	2	no	—
221	C₂²⋊C₈	—	11	2	no	—
222	C₂²⋊C₈	—	11	2	no	—
223	C₂²⋊C₈	—	11	2	no	—
224	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
225	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
228	C₂×C₁₁	—	2	2	no	—
229	C₂×C₁₁	#2972	1	1	yes	characteristic
230	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
231	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
232	D₁₁	—	2	4	no	—
233	D₁₁	—	4	2	no	—
234	C₂⁴	—	11	2	no	—
235	C₂×D₄	—	11	1	no	—
236	C₂×D₄	—	11	2	no	—
237	C₂×D₄	—	11	1	no	—
238	C₂×D₄	—	22	4	no	—
239	C₂²×C₄	—	11	2	no	—
240	C₂²×C₄	—	11	2	no	—
241	C₂²×C₄	—	11	2	no	—
242	C₂²×C₄	—	11	2	no	—
243	C₂²×C₄	—	11	2	no	—
244	C₂²×C₄	—	44	2	no	—
245	C₂²×C₄	—	44	2	no	—
246	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
247	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	11	2	no	—
252	C₂×C₈	—	11	2	no	—
253	C₂×C₈	—	22	4	no	—
254	C₂×C₈	—	22	4	no	—
255	C₂×C₈	—	44	2	no	—
256	C₂×C₈	—	44	2	no	—
257	C₂×C₈	—	44	2	no	—
258	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
262	C₄⋊C₄	—	11	2	no	—
263	C₄⋊C₄	—	11	1	no	—
264	C₄⋊C₄	—	11	1	no	—
265	C₂²⋊C₄	—	11	2	no	—
266	C₂²⋊C₄	—	11	2	no	—
267	C₂²⋊C₄	—	11	2	no	—
268	C₂²⋊C₄	—	11	2	no	—
269	C₄²	C₈×D₁₁	1	1	yes	characteristic
270	C₄²	C₈×D₁₁	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
272	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	22	4	no	—
276	C₁₁	#33488	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	11	2	no	—
278	C₂³	—	11	2	no	—
279	C₂³	—	22	4	no	—
280	C₂³	—	22	4	no	—
281	C₂³	—	44	2	no	—
282	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
283	D₄	—	11	4	no	—
284	D₄	—	22	4	no	—
285	D₄	—	22	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	11	2	no	—
291	C₂×C₄	—	11	2	no	—
292	C₂×C₄	—	22	2	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	22	2	no	—
295	C₂×C₄	—	22	2	no	—
296	C₂×C₄	—	44	2	no	—
297	C₂×C₄	—	44	2	no	—
298	C₂×C₄	—	44	2	no	—
299	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
300	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
303	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	22	2	no	—
309	C₈	—	22	2	no	—
310	C₈	#67042	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	11	4	no	—
314	C₂²	—	22	2	no	—
315	C₂²	—	22	2	no	—
316	C₂²	—	22	2	no	—
317	C₂²	—	22	2	no	—
318	C₂²	—	22	2	no	—
319	C₂²	—	44	8	no	—
320	C₂²	#10622514	1	2	yes	—
321	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	22	2	no	—
327	C₄	—	22	2	no	—
328	C₄	D₁₁×M₃₂	1	2	yes	—
329	C₄	#10623052	1	1	yes	characteristic
330	C₄	#10623276	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	22	4	no	—
333	C₂	—	44	2	no	—
334	C₂	#13077639	1	1	yes	characteristic
335	C₂	C₂×D₁₁×M₃₂	1	1	yes	characteristic
336	C₂	C₂×#10622514	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2162	C₂
prime radical	C₂²×D₄₄	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2162	D₁₁
hypocenter	C₁₁	#33488
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #13070449	C₂	semidirect
3. #13070440	C₂	semidirect
4. C₁₆×D₄₄	C₂	semidirect
5. #13070419	C₂	semidirect
6. #13070225	C₂	semidirect
7. D₁₁×C₂²⋊C₁₆	C₂	semidirect
8. C₁₁×#2162	C₂	semidirect
14. C₂×#66862	C₂²	semidirect
15. C₂×C₁₆×D₁₁	C₂²	semidirect
22. C₁₁×C₄⋊C₁₆	C₂²	semidirect
25. C₄×C₁₁×C₁₆	C₂²	semidirect
27. (C₂×D₁₁)⋊C₁₆	C₂²	semidirect
28. (C₂×D₁₁)⋊C₁₆	C₂²	semidirect
29. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
30. #10622099	C₂²	semidirect
114. #2162	D₁₁	semidirect
138. D₄₄	M₃₂	semidirect
165. C₄⋊C₁₆	C₂×D₁₁	semidirect
172. C₄×C₁₆	C₂×D₁₁	semidirect
276. C₁₁	#33488	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄₄	2
#13070449	2
#13070440	2
C₁₆×D₄₄	2
#13070419	2
#13070225	2
D₁₁×C₂²⋊C₁₆	2
C₁₁×#2162	2
#33488	11