Subgroups — #14907346

Order 2816.

552 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#4343	C₂	1	1	yes	maximal, characteristic
2	#13070525	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070443	C₂	1	2	yes	maximal, complemented
4	#13070440	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₄₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070435	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070418	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070418	C₂	1	2	yes	maximal, complemented
9	C₄×#66862	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₁×#346	C₂	1	2	yes	maximal, complemented
11	C₁₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#4343	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
14	Q₈×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	C₂×#66862	C₂²	1	1	yes	characteristic
16	C₂×#66862	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
18	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
19	#10622447	C₄	1	2	yes	—
20	#10622447	C₄	1	1	yes	characteristic
21	#10622446	C₄	1	1	yes	characteristic
22	#10622446	C₄	1	2	yes	—
23	C₈×D₄₄	C₂²	1	1	yes	characteristic
24	C₈×D₄₄	C₂²	1	2	yes	—
25	C₄×C₈×D₁₁	C₂²	1	2	yes	—
26	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
27	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₁×#346	C₂²	1	2	yes	—
30	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
31	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
34	#10622140	C₂²	1	2	yes	—
35	#10622140	C₂²	1	1	yes	characteristic
36	#10622140	C₂²	1	1	yes	characteristic
37	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
39	#10622099	C₂²	1	2	yes	—
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#4343	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₁	C₈	1	1	yes	characteristic
44	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₁	C₂³	1	2	yes	—
47	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
48	#66946	C₈	1	2	yes	—
49	#66946	C₈	1	1	yes	characteristic
50	#66945	C₈	1	2	yes	—
51	#66945	C₈	1	1	yes	characteristic
52	C₄×D₄₄	C₂×C₄	1	2	yes	—
53	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₁₁	C₂×C₄	1	2	yes	—
55	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	2	yes	—
63	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
64	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
70	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
76	#66862	—	2	2	no	—
77	#66862	—	2	2	no	—
78	C₁₆×D₁₁	—	2	4	no	—
79	C₁₆×D₁₁	—	2	4	no	—
80	#33484	—	11	1	no	maximal, Sylow
81	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
82	#4343	—	2	4	no	—
83	C₂×D₄₄	C₂×C₈	1	2	yes	—
84	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
98	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
100	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
106	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₁₁	—	2	2	no	—
112	C₈×D₁₁	—	2	2	no	—
113	C₈×D₁₁	—	2	1	no	—
114	C₈×D₁₁	—	2	1	no	—
115	C₈×D₁₁	—	2	4	no	—
116	C₈×D₁₁	—	2	2	no	—
117	C₁₁×C₁₆	—	2	2	no	—
118	C₁₁×C₁₆	—	2	1	no	—
119	C₁₁×C₁₆	#41	1	2	yes	complemented
120	C₁₁⋊C₁₆	—	2	2	no	—
121	C₁₁⋊C₁₆	—	2	1	no	—
122	C₁₁⋊C₁₆	#41	1	2	yes	complemented
123	C₈×#41	—	11	1	no	—
124	#2183	—	11	1	no	—
125	#2183	D₁₁	1	1	yes	characteristic, complemented
126	#2168	—	11	2	no	—
127	#2168	—	11	2	no	—
128	D₄×C₁₆	—	11	1	no	—
129	D₄×C₁₆	—	11	1	no	—
130	#2163	—	11	1	no	—
131	#2163	—	11	2	no	—
132	C₄×M₃₂	—	11	1	no	—
133	C₂×#346	—	11	2	no	—
134	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
135	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
136	Q₈×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
145	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
146	D₄₄	—	2	4	no	—
147	D₄₄	M₃₂	1	4	yes	complemented
148	C₄×D₁₁	—	2	4	no	—
149	C₄×D₁₁	—	2	2	no	—
150	C₄×D₁₁	—	2	1	no	—
151	C₄×D₁₁	—	2	2	no	—
152	C₄×D₁₁	—	2	1	no	—
153	C₄×D₁₁	—	2	2	no	—
154	C₄×D₁₁	—	4	2	no	—
155	C₄×D₁₁	—	4	1	no	—
156	C₈×C₁₁	—	2	1	no	—
157	C₈×C₁₁	—	2	2	no	—
158	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
160	C₁₁⋊C₈	—	2	2	no	—
161	C₁₁⋊C₈	—	2	1	no	—
162	C₁₁⋊C₈	#131	1	2	yes	—
163	C₄×#41	—	11	1	no	—
164	C₂×M₃₂	—	11	1	no	—
165	C₂×M₃₂	—	11	1	no	—
166	C₂²×C₁₆	—	11	2	no	—
167	C₂²×C₁₆	—	11	2	no	—
168	C₈×Q₈	—	11	1	no	—
169	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
170	C₈×D₄	—	11	2	no	—
171	C₈×D₄	—	11	1	no	—
172	C₈×D₄	—	11	1	no	—
173	C₈×D₄	—	11	2	no	—
174	#431	—	11	1	no	—
175	#431	—	11	2	no	—
176	C₂×C₄×C₈	—	11	2	no	—
177	C₂×C₄×C₈	—	11	1	no	—
178	C₄⋊C₁₆	—	11	1	no	—
179	C₄⋊C₁₆	—	11	2	no	—
180	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
181	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
182	C₂²⋊C₁₆	—	11	2	no	—
183	C₂²⋊C₁₆	—	11	2	no	—
184	C₂²⋊C₁₆	—	11	2	no	—
185	#346	—	11	1	no	—
186	#346	—	11	2	no	—
187	#346	—	11	1	no	—
188	#346	—	11	2	no	—
189	#346	C₂×D₁₁	1	2	yes	—
190	C₄×C₁₆	—	11	1	no	—
191	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
192	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₁₁	—	2	2	no	—
194	C₂×D₁₁	—	2	2	no	—
195	C₂×D₁₁	—	2	1	no	—
196	C₂×D₁₁	—	2	1	no	—
197	C₂×D₁₁	—	2	1	no	—
198	C₂×D₁₁	C₂×M₃₂	1	4	yes	—
199	C₄×C₁₁	—	2	2	no	—
200	C₄×C₁₁	—	2	2	no	—
201	C₄×C₁₁	—	2	1	no	—
202	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
203	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
204	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
205	Dic₁₁	—	2	1	no	—
206	Dic₁₁	—	2	1	no	—
207	Dic₁₁	—	2	2	no	—
208	Dic₁₁	—	2	1	no	—
209	C₂×#41	—	11	1	no	—
210	C₂²×C₈	—	11	2	no	—
211	C₂²×C₈	—	11	1	no	—
212	C₂²×C₈	—	11	2	no	—
213	C₂²×C₈	—	11	1	no	—
214	C₄×Q₈	—	11	1	no	—
215	C₄×Q₈	C₄×D₁₁	1	1	yes	characteristic
216	C₄×D₄	—	11	1	no	—
217	C₄×D₄	—	11	2	no	—
218	C₄×D₄	—	11	1	no	—
219	C₄×D₄	—	11	2	no	—
220	#117	—	11	2	no	—
221	#117	—	11	1	no	—
222	C₂×C₄²	—	11	2	no	—
223	C₂×C₄²	—	11	1	no	—
224	M₃₂	—	22	2	no	—
225	M₃₂	—	22	2	no	—
226	C₂×C₁₆	—	11	1	no	—
227	C₂×C₁₆	—	11	1	no	—
228	C₂×C₁₆	—	11	2	no	—
229	C₂×C₁₆	—	22	4	no	—
230	C₂×C₁₆	—	22	4	no	—
231	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
233	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
234	C₄⋊C₈	—	11	1	no	—
235	C₄⋊C₈	—	11	2	no	—
236	C₄⋊C₈	C₄×D₁₁	1	2	yes	—
237	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
238	C₂²⋊C₈	—	11	2	no	—
239	C₂²⋊C₈	—	11	2	no	—
240	C₂²⋊C₈	—	11	2	no	—
241	C₄×C₈	—	11	2	no	—
242	C₄×C₈	—	11	1	no	—
243	C₄×C₈	—	11	1	no	—
244	C₄×C₈	—	11	2	no	—
245	C₄×C₈	—	11	1	no	—
246	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₁₁	1	2	yes	—
248	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
249	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
250	C₂×C₁₁	#3407	1	1	yes	characteristic
251	D₁₁	—	2	4	no	—
252	D₁₁	—	4	1	no	—
253	#41	—	22	4	no	—
254	C₂×Q₈	C₈×D₁₁	1	1	yes	characteristic
255	C₂×D₄	—	11	1	no	—
256	C₂×D₄	—	11	2	no	—
257	C₂²×C₄	—	11	2	no	—
258	C₂²×C₄	—	11	1	no	—
259	C₂²×C₄	—	11	2	no	—
260	C₂²×C₄	—	11	1	no	—
261	C₂²×C₄	—	11	2	no	—
262	C₂²×C₄	—	11	1	no	—
263	C₂×C₈	—	11	1	no	—
264	C₂×C₈	—	11	2	no	—
265	C₂×C₈	—	11	1	no	—
266	C₂×C₈	—	22	4	no	—
267	C₂×C₈	—	22	2	no	—
268	C₂×C₈	—	22	2	no	—
269	C₂×C₈	—	22	1	no	—
270	C₂×C₈	—	22	2	no	—
271	C₂×C₈	—	22	1	no	—
272	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
274	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
275	C₄⋊C₄	—	11	2	no	—
276	C₄⋊C₄	—	11	1	no	—
277	C₄⋊C₄	C₈×D₁₁	1	2	yes	—
278	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	11	2	no	—
280	C₂²⋊C₄	—	11	2	no	—
281	C₂²⋊C₄	—	11	2	no	—
282	C₄²	—	11	2	no	—
283	C₄²	—	11	1	no	—
284	C₄²	—	11	2	no	—
285	C₄²	—	11	1	no	—
286	C₄²	—	11	1	no	—
287	C₄²	C₂×C₄×D₁₁	1	2	yes	—
288	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	11	2	no	—
292	C₁₆	—	22	2	no	—
293	C₁₆	—	22	1	no	—
294	C₁₆	#4343	1	2	yes	complemented
295	C₁₁	#33484	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	11	2	no	—
297	C₂³	—	11	1	no	—
298	Q₈	—	2	2	no	—
299	D₄	—	11	4	no	—
300	D₄	—	22	4	no	—
301	C₂×C₄	—	11	1	no	—
302	C₂×C₄	—	11	1	no	—
303	C₂×C₄	—	11	2	no	—
304	C₂×C₄	—	11	1	no	—
305	C₂×C₄	—	22	4	no	—
306	C₂×C₄	—	22	2	no	—
307	C₂×C₄	—	22	1	no	—
308	C₂×C₄	—	22	2	no	—
309	C₂×C₄	—	22	2	no	—
310	C₂×C₄	—	22	1	no	—
311	C₂×C₄	—	44	2	no	—
312	C₂×C₄	—	44	1	no	—
313	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
316	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	11	2	no	—
321	C₈	—	22	2	no	—
322	C₈	—	22	1	no	—
323	C₈	C₂×#4343	1	1	yes	characteristic
324	C₈	#41×D₁₁	1	1	yes	characteristic
325	C₂²	—	11	4	no	—
326	C₂²	—	22	2	no	—
327	C₂²	—	22	2	no	—
328	C₂²	—	22	1	no	—
329	C₂²	—	22	1	no	—
330	C₂²	—	22	1	no	—
331	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	22	2	no	—
336	C₄	—	22	1	no	—
337	C₄	—	22	1	no	—
338	C₄	—	22	1	no	—
339	C₄	D₁₁×M₃₂	1	2	yes	—
340	C₄	C₄×#4343	1	1	yes	characteristic
341	C₄	D₁₁×#131	1	1	yes	characteristic
342	C₂	—	22	4	no	—
343	C₂	—	44	1	no	—
344	C₂	C₈×#4343	1	1	yes	characteristic
345	C₂	C₂×D₁₁×M₃₂	1	1	yes	characteristic
346	C₂	#13081717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2183	C₂
prime radical	C₂×#4343	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2183	D₁₁
hypocenter	C₁₁	#33484
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #13070525	C₂	semidirect
3. #13070443	C₂	semidirect
4. #13070440	C₂	semidirect
5. C₁₆×D₄₄	C₂	semidirect
6. #13070435	C₂	semidirect
7. #13070418	C₂	semidirect
8. #13070418	C₂	semidirect
9. C₄×#66862	C₂	semidirect
10. D₁₁×#346	C₂	semidirect
11. C₁₁×#2183	C₂	semidirect
119. C₁₁×C₁₆	#41	semidirect
122. C₁₁⋊C₁₆	#41	semidirect
125. #2183	D₁₁	semidirect
147. D₄₄	M₃₂	semidirect
294. C₁₆	#4343	semidirect
295. C₁₁	#33484	semidirect

Maximal subgroups

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

subgroup	index
C₈×#4343	2
#13070525	2
#13070443	2
#13070440	2
C₁₆×D₄₄	2
#13070435	2
#13070418	2
C₄×#66862	2
D₁₁×#346	2
C₁₁×#2183	2
#33484	11