Subgroups — #14725232

Order 2816.

638 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	#13077567	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₄₄	C₂	1	1	yes	maximal, characteristic
3	#13067512	C₂	1	2	yes	maximal, complemented
4	#13067465	C₂	1	4	yes	maximal, complemented
5	#13067422	C₂	1	2	yes	maximal, complemented
6	#13067044	C₂	1	2	yes	maximal
7	C₈×#4318	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#1567	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₄₄	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₂	C₂²	1	2	yes	—
11	C₁₁×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622731	—	2	2	no	—
14	M₁₆×Dic₁₁	—	2	2	no	—
15	C₂×#66882	C₄	1	1	yes	characteristic
16	C₂×#66880	C₄	1	2	yes	—
17	#10622538	C₂²	1	1	yes	characteristic
18	C₄×#4318	C₂²	1	1	yes	characteristic
19	#10622444	C₂²	1	4	yes	—
20	#10622443	C₂²	1	4	yes	—
21	#10622441	—	2	4	no	—
22	C₄×Dic₄₄	C₄	1	1	yes	characteristic
23	C₄×Dic₄₄	C₄	1	2	yes	—
24	C₄×Dic₄₄	C₄	1	1	yes	characteristic
25	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₄₄	C₂²	1	2	yes	—
27	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
28	C₁₁×#335	C₂²	1	2	yes	—
29	C₁₁×#335	C₂²	1	2	yes	—
30	#10622122	C₂²	1	4	yes	—
31	#10622122	C₂²	1	4	yes	—
32	#10622092	C₂²	1	2	yes	—
33	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
34	C₂²×Dic₂₂	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	—	2	2	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
40	C₂×Dic₁₁⋊C₄	—	2	2	no	—
41	C₂×C₄×Dic₁₁	—	2	2	no	—
42	C₂×#4318	D₄	1	1	yes	characteristic
43	C₂×#4318	D₄	1	1	yes	characteristic
44	C₂×Dic₄₄	—	2	4	no	—
45	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₂₂	—	2	2	no	—
49	C₄×Dic₂₂	—	2	2	no	—
50	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₂	C₂³	1	2	yes	—
53	C₄×Dic₂₂	C₂³	1	2	yes	—
54	C₁₁×C₄⋊C₈	—	2	2	no	—
55	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
60	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
61	#66882	C₂×C₄	1	2	yes	—
62	#66882	C₂×C₄	1	1	yes	characteristic
63	#66882	C₂×C₄	1	1	yes	characteristic
64	#66880	C₂×C₄	1	2	yes	—
65	#66880	C₂×C₄	1	2	yes	—
66	#66880	C₂×C₄	1	2	yes	—
67	#66880	C₂×C₄	1	2	yes	—
68	#66879	—	2	4	no	—
69	Dic₁₁⋊C₈	—	2	4	no	—
70	Dic₁₁⋊C₈	—	2	4	no	—
71	C₈×Dic₁₁	—	2	4	no	—
72	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
73	#66867	D₄	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
76	M₁₆×Q₁₆	—	11	1	no	maximal, Sylow
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
80	C₂²×Dic₁₁	—	2	2	no	—
81	C₂²×Dic₁₁	—	2	2	no	—
82	C₂×Dic₂₂	—	2	4	no	—
83	C₂×Dic₂₂	—	2	2	no	—
84	C₂×Dic₂₂	—	4	2	no	—
85	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
87	C₁₁×M₁₆	—	4	4	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	4	no	—
90	C₂×C₈×C₁₁	—	4	1	no	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₁	#41	1	1	yes	characteristic
98	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊Dic₁₁	—	2	2	no	—
100	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
102	Dic₁₁⋊C₄	—	2	2	no	—
103	Dic₁₁⋊C₄	—	2	2	no	—
104	Dic₁₁⋊C₄	—	2	2	no	—
105	Dic₁₁⋊C₄	—	2	2	no	—
106	C₄×Dic₁₁	—	2	2	no	—
107	C₄×Dic₁₁	—	2	2	no	—
108	C₄×Dic₁₁	—	2	2	no	—
109	C₄×Dic₁₁	—	2	2	no	—
110	#4318	—	2	2	no	—
111	#4318	Q₁₆	1	4	yes	complemented
112	C₂×C₁₁⋊C₈	—	2	2	no	—
113	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
114	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
115	Dic₄₄	—	2	4	no	—
116	Dic₄₄	—	2	2	no	—
117	Dic₄₄	M₁₆	1	4	yes	complemented
118	Q₈×M₁₆	—	11	2	no	—
119	C₂×C₄×Q₁₆	—	11	1	no	—
120	#1591	—	11	2	no	—
121	#1584	—	11	4	no	—
122	C₈×Q₁₆	—	11	2	no	—
123	#1567	D₁₁	1	1	yes	characteristic
124	#1487	—	11	2	no	—
125	C₈×M₁₆	—	11	1	no	—
126	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	4	1	no	—
130	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₁₁	#135	1	2	yes	—
135	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₁	—	2	2	no	—
139	C₂×Dic₁₁	—	2	2	no	—
140	C₂×Dic₁₁	—	2	2	no	—
141	C₂×Dic₁₁	—	2	2	no	—
142	C₂×Dic₁₁	—	4	4	no	—
143	C₂×Dic₁₁	—	8	2	no	—
144	Dic₂₂	—	2	2	no	—
145	Dic₂₂	—	2	2	no	—
146	Dic₂₂	—	2	2	no	—
147	Dic₂₂	—	4	4	no	—
148	Dic₂₂	C₂×M₁₆	1	4	yes	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	—	4	4	no	—
153	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
154	C₁₁⋊C₈	—	2	2	no	—
155	C₁₁⋊C₈	—	4	2	no	—
156	C₁₁⋊C₈	C₂×Q₁₆	1	4	yes	complemented
157	C₂²×Q₁₆	—	11	1	no	—
158	C₂×C₄×Q₈	—	11	2	no	—
159	#446	—	11	4	no	—
160	#446	—	22	4	no	—
161	C₈×Q₈	—	11	4	no	—
162	C₄×Q₁₆	—	11	1	no	—
163	C₄×Q₁₆	—	11	2	no	—
164	C₄×Q₁₆	—	11	1	no	—
165	C₄×Q₁₆	—	11	1	no	—
166	C₄×Q₁₆	—	11	2	no	—
167	C₄×Q₁₆	—	11	1	no	—
168	#431	—	11	1	no	—
169	C₂×#107	—	11	1	no	—
170	#423	—	22	2	no	—
171	#423	C₂×D₁₁	1	2	yes	—
172	C₂×#103	—	11	2	no	—
173	C₄×M₁₆	—	11	1	no	—
174	C₄×M₁₆	—	22	2	no	—
175	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
176	#335	C₂×D₁₁	1	2	yes	—
177	#335	C₂×D₁₁	1	2	yes	—
178	Q₈⋊C₈	—	11	4	no	—
179	Q₈⋊C₈	—	11	4	no	—
180	#322	—	11	2	no	—
181	C₈²	—	11	2	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	C₄×Q₁₆	1	2	yes	—
185	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	C₄×Q₁₆	1	2	yes	—
190	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
191	C₄×C₁₁	#568	1	1	yes	characteristic
192	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
194	Dic₁₁	—	2	4	no	—
195	Dic₁₁	—	4	2	no	—
196	Dic₁₁	—	4	2	no	—
197	Dic₁₁	—	4	2	no	—
198	C₂²×Q₈	—	11	2	no	—
199	C₂×Q₁₆	—	11	1	no	—
200	C₂×Q₁₆	—	11	2	no	—
201	C₂×Q₁₆	—	11	1	no	—
202	C₂×Q₁₆	—	22	4	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	11	1	no	—
205	C₂×M₁₆	—	11	1	no	—
206	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
208	C₄×Q₈	—	11	2	no	—
209	C₄×Q₈	—	11	2	no	—
210	C₄×Q₈	—	11	2	no	—
211	C₄×Q₈	—	11	2	no	—
212	C₄×Q₈	—	22	2	no	—
213	C₄×Q₈	—	22	2	no	—
214	C₂×C₄⋊C₄	—	11	2	no	—
215	C₂×C₄⋊C₄	—	22	2	no	—
216	C₂×C₄²	—	22	2	no	—
217	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
218	#107	—	11	2	no	—
219	#107	—	11	1	no	—
220	#107	—	11	1	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	22	4	no	—
223	C₄⋊C₈	—	22	4	no	—
224	C₄⋊C₈	—	22	1	no	—
225	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
226	#103	—	11	2	no	—
227	#103	—	11	2	no	—
228	#103	—	11	2	no	—
229	#103	—	11	2	no	—
230	C₂²⋊C₈	—	22	1	no	—
231	#97	—	11	2	no	—
232	#97	—	22	4	no	—
233	C₄×C₈	—	11	2	no	—
234	C₄×C₈	—	11	2	no	—
235	C₄×C₈	—	22	4	no	—
236	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
240	C₂×C₁₁	—	2	2	no	—
241	C₂×C₁₁	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₁₁	#2957	1	1	yes	characteristic
244	C₂×Q₈	—	11	2	no	—
245	C₂×Q₈	—	11	2	no	—
246	C₂×Q₈	—	22	4	no	—
247	C₂×Q₈	—	22	2	no	—
248	C₂×Q₈	—	44	2	no	—
249	C₂²×C₄	—	22	2	no	—
250	C₂²×C₄	—	22	2	no	—
251	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
253	C₂²×C₄	#4343	1	1	yes	characteristic
254	Q₁₆	—	11	4	no	—
255	Q₁₆	—	22	4	no	—
256	Q₁₆	—	22	2	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	11	4	no	—
259	M₁₆	—	22	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	11	2	no	—
264	C₂×C₈	—	11	2	no	—
265	C₂×C₈	—	22	2	no	—
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	—	11	2	no	—
272	C₄⋊C₄	—	22	2	no	—
273	C₄⋊C₄	—	22	2	no	—
274	C₄⋊C₄	—	22	2	no	—
275	C₄⋊C₄	—	22	2	no	—
276	C₄⋊C₄	—	22	2	no	—
277	C₄²	—	22	2	no	—
278	C₄²	—	22	2	no	—
279	C₄²	—	22	2	no	—
280	C₄²	—	22	2	no	—
281	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
282	C₄²	D₄×D₁₁	1	1	yes	characteristic
283	C₄²	#4343	1	1	yes	characteristic
284	C₄²	C₂³×D₁₁	1	1	yes	characteristic
285	C₁₁	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	#66946	1	1	yes	characteristic
287	Q₈	—	11	4	no	—
288	Q₈	—	22	2	no	—
289	Q₈	—	22	2	no	—
290	Q₈	—	22	2	no	—
291	Q₈	—	44	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	22	2	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	22	2	no	—
299	C₂×C₄	—	44	4	no	—
300	C₂×C₄	—	88	2	no	—
301	C₂×C₄	#66946	1	1	yes	characteristic
302	C₂×C₄	#66946	1	1	yes	characteristic
303	C₂×C₄	#66946	1	1	yes	characteristic
304	C₂×C₄	#66965	1	2	yes	—
305	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#4343	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	11	4	no	—
314	C₈	—	22	2	no	—
315	C₈	—	44	2	no	—
316	C₈	M₁₆×D₁₁	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	#10622486	1	2	yes	—
320	C₂²	C₂×#66946	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	22	4	no	—
325	C₄	—	44	2	no	—
326	C₄	—	44	2	no	—
327	C₄	—	44	2	no	—
328	C₄	#10622486	1	2	yes	—
329	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
330	C₄	#10623276	1	1	yes	characteristic
331	C₄	C₂×#66965	1	1	yes	characteristic
332	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	#13077351	1	1	yes	characteristic
335	C₂	C₂×#10622486	1	1	yes	characteristic
336	C₂	#13077532	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1567	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1567	D₁₁
hypocenter	C₁₁	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #13067512	C₂	semidirect
4. #13067465	C₂	semidirect
5. #13067422	C₂	semidirect
111. #4318	Q₁₆	semidirect
117. Dic₄₄	M₁₆	semidirect
156. C₁₁⋊C₈	C₂×Q₁₆	semidirect
285. C₁₁	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13077567	2
C₂×C₄×Dic₄₄	2
#13067512	2
#13067465	2
#13067422	2
#13067044	2
C₈×#4318	2
C₁₁×#1567	2
M₁₆×Q₁₆	11