Subgroups — #14907407

Order 2816.

564 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077323	C₂	1	1	yes	maximal, characteristic
2	#13070448	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070447	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070444	C₂	1	2	yes	maximal, complemented
5	#13070441	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070418	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070413	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070229	C₂	1	2	yes	maximal, complemented
10	#13070228	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66954	C₂²	1	2	yes	—
13	#10623079	C₄	1	1	yes	characteristic
14	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
15	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
16	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
17	C₂×#66876	C₂²	1	2	yes	—
18	#10622625	C₄	1	1	yes	characteristic
19	C₂×#66862	C₂²	1	2	yes	—
20	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
21	#10622451	C₄	1	1	yes	characteristic
22	#10622450	C₄	1	1	yes	characteristic
23	#10622342	C₄	1	2	yes	—
24	#10622340	C₄	1	2	yes	—
25	C₈×D₄₄	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
27	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
31	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
32	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
33	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
34	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
35	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
36	#10622140	C₂²	1	2	yes	—
37	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
38	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#4339	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
43	C₄×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
44	#66993	C₈	1	1	yes	characteristic
45	#66991	C₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
47	C₂×#4318	C₂×C₄	1	2	yes	—
48	#66954	—	4	2	no	—
49	#66954	—	4	2	no	—
50	C₂×#4313	C₂×C₄	1	2	yes	—
51	C₂×#4313	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₁	C₂³	1	2	yes	—
54	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
55	#66948	C₈	1	1	yes	characteristic
56	#66941	C₈	1	1	yes	characteristic
57	#66938	C₈	1	2	yes	—
58	#66937	C₈	1	2	yes	—
59	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₁×C₁₆	—	4	2	no	—
62	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
64	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
69	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
70	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
71	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
72	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
74	#66876	—	4	2	no	—
75	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
76	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
78	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
79	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
80	#66862	—	4	2	no	—
81	C₁₆×D₁₁	—	4	2	no	—
82	#33495	—	11	1	no	maximal, Sylow
83	C₂×D₄×C₁₁	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
85	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
86	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
87	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
88	#4339	—	4	2	no	—
89	#4339	—	4	2	no	—
90	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
91	C₂×D₄₄	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
97	C₂×C₈×C₁₁	—	2	2	no	—
98	C₂×C₈×C₁₁	—	2	2	no	—
99	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
102	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
103	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
106	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
107	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
108	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
109	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
110	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
113	#4318	—	4	2	no	—
114	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
117	#4313	—	4	2	no	—
118	#4313	—	4	2	no	—
119	C₈×D₁₁	—	2	2	no	—
120	C₈×D₁₁	—	2	2	no	—
121	C₈×D₁₁	—	2	1	no	—
122	C₈×D₁₁	—	2	1	no	—
123	C₈×D₁₁	—	4	1	no	—
124	C₁₁×C₁₆	—	2	1	no	—
125	C₁₁×C₁₆	—	2	2	no	—
126	C₁₁×C₁₆	—	2	1	no	—
127	C₁₁⋊C₁₆	—	2	2	no	—
128	C₁₁⋊C₁₆	—	2	1	no	—
129	C₁₁⋊C₁₆	—	2	1	no	—
130	#2931	—	11	1	no	—
131	#2169	—	11	1	no	—
132	#2169	—	11	1	no	—
133	#2168	—	11	2	no	—
134	#2168	—	11	2	no	—
135	D₄×C₁₆	—	11	1	no	—
136	D₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
137	#2163	—	11	1	no	—
138	#2162	—	11	1	no	—
139	#2113	—	11	2	no	—
140	#2113	—	11	2	no	—
141	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
142	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
143	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
144	D₄×C₁₁	—	2	2	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	—	2	2	no	—
147	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂²⋊D₁₁	—	4	2	no	—
152	C₂²⋊D₁₁	—	4	2	no	—
153	C₂²⋊D₁₁	—	4	2	no	—
154	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
155	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
156	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
157	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
158	D₄₄	—	2	2	no	—
159	D₄₄	—	4	2	no	—
160	C₄×D₁₁	—	2	1	no	—
161	C₄×D₁₁	—	2	2	no	—
162	C₄×D₁₁	—	2	1	no	—
163	C₄×D₁₁	—	2	2	no	—
164	C₄×D₁₁	—	2	2	no	—
165	C₄×D₁₁	—	4	1	no	—
166	Dic₂₂	—	4	2	no	—
167	C₈×C₁₁	—	2	2	no	—
168	C₈×C₁₁	—	2	1	no	—
169	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
170	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
171	C₁₁⋊C₈	—	2	1	no	—
172	C₁₁⋊C₈	—	2	2	no	—
173	C₁₁⋊C₈	—	2	1	no	—
174	C₂×#131	—	11	2	no	—
175	#532	—	11	1	no	—
176	C₂×M₃₂	—	11	2	no	—
177	C₂×M₃₂	—	11	2	no	—
178	C₂²×C₁₆	—	11	2	no	—
179	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
180	#436	—	11	1	no	—
181	#436	—	11	1	no	—
182	#436	—	11	2	no	—
183	C₈×D₄	—	11	2	no	—
184	C₈×D₄	—	11	1	no	—
185	C₈×D₄	C₂×D₁₁	1	1	yes	characteristic
186	#423	—	11	1	no	—
187	#408	—	11	2	no	—
188	C₂×C₄×C₈	—	11	1	no	—
189	C₄⋊C₁₆	—	11	1	no	—
190	C₄⋊C₁₆	—	11	1	no	—
191	C₄⋊C₁₆	—	11	1	no	—
192	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
193	C₂²⋊C₁₆	—	11	2	no	—
194	C₂²⋊C₁₆	—	11	1	no	—
195	C₂²⋊C₁₆	—	11	1	no	—
196	C₂²⋊C₁₆	—	11	2	no	—
197	C₂²⋊C₁₆	—	11	2	no	—
198	C₂²⋊C₁₆	—	11	2	no	—
199	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
200	#346	—	11	2	no	—
201	C₄×C₁₆	—	11	1	no	—
202	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
203	C₂²×C₁₁	—	2	2	no	—
204	C₂²×C₁₁	—	2	2	no	—
205	C₂²×C₁₁	—	2	2	no	—
206	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
207	C₂×D₁₁	—	2	1	no	—
208	C₂×D₁₁	—	2	2	no	—
209	C₂×D₁₁	—	2	1	no	—
210	C₂×D₁₁	—	2	2	no	—
211	C₂×D₁₁	—	2	1	no	—
212	C₂×D₁₁	—	2	2	no	—
213	C₄×C₁₁	—	2	1	no	—
214	C₄×C₁₁	—	2	2	no	—
215	C₄×C₁₁	#504	1	2	yes	—
216	C₄×C₁₁	#518	1	1	yes	characteristic
217	C₄×C₁₁	#568	1	1	yes	characteristic
218	Dic₁₁	—	2	1	no	—
219	Dic₁₁	—	2	1	no	—
220	Dic₁₁	—	2	1	no	—
221	Dic₁₁	—	2	2	no	—
222	C₂×#41	—	11	2	no	—
223	#131	—	44	2	no	—
224	#131	—	44	2	no	—
225	C₂×M₁₆	—	11	2	no	—
226	C₂×M₁₆	—	11	2	no	—
227	C₂×M₁₆	—	11	2	no	—
228	C₂²×C₈	—	11	1	no	—
229	C₂²×C₈	—	11	2	no	—
230	C₂²×C₈	—	11	1	no	—
231	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
232	C₄⋊Q₈	—	11	1	no	—
233	#127	—	11	1	no	—
234	#124	—	11	2	no	—
235	#121	—	11	1	no	—
236	#121	—	11	1	no	—
237	#121	—	11	2	no	—
238	C₄×D₄	—	11	2	no	—
239	C₄×D₄	—	11	1	no	—
240	C₄×D₄	C₄×D₁₁	1	1	yes	characteristic
241	C₂×C₄²	—	11	1	no	—
242	M₃₂	—	44	2	no	—
243	M₃₂	—	44	2	no	—
244	C₂×C₁₆	—	4	2	no	—
245	C₂×C₁₆	—	11	1	no	—
246	C₂×C₁₆	—	11	1	no	—
247	C₂×C₁₆	—	11	2	no	—
248	C₂×C₁₆	—	44	2	no	—
249	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
251	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
252	C₄⋊C₈	—	11	2	no	—
253	C₄⋊C₈	—	11	1	no	—
254	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
255	C₂²⋊C₈	—	11	2	no	—
256	C₂²⋊C₈	—	11	2	no	—
257	C₂²⋊C₈	—	11	2	no	—
258	C₂²⋊C₈	C₄×D₁₁	1	2	yes	—
259	C₄×C₈	—	11	1	no	—
260	C₄×C₈	—	11	1	no	—
261	C₄×C₈	—	11	1	no	—
262	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
263	C₂×C₁₁	—	4	2	no	—
264	C₂×C₁₁	#2972	1	1	yes	characteristic
265	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
266	C₂×C₁₁	#3407	1	1	yes	characteristic
267	D₁₁	—	4	1	no	—
268	D₁₁	—	4	2	no	—
269	#41	—	44	2	no	—
270	#41	—	44	2	no	—
271	C₂×Q₈	—	11	2	no	—
272	C₂×D₄	—	11	2	no	—
273	C₂×D₄	—	11	2	no	—
274	C₂×D₄	—	11	2	no	—
275	C₂×D₄	—	11	1	no	—
276	C₂×D₄	—	11	2	no	—
277	C₂×D₄	C₈×D₁₁	1	1	yes	characteristic
278	C₂²×C₄	—	11	1	no	—
279	C₂²×C₄	—	11	2	no	—
280	C₂²×C₄	—	11	1	no	—
281	C₂²×C₄	—	11	1	no	—
282	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
283	M₁₆	—	44	2	no	—
284	M₁₆	—	44	2	no	—
285	M₁₆	—	44	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	11	1	no	—
289	C₂×C₈	—	11	1	no	—
290	C₂×C₈	—	11	2	no	—
291	C₂×C₈	—	22	2	no	—
292	C₂×C₈	—	22	1	no	—
293	C₂×C₈	—	22	2	no	—
294	C₂×C₈	—	22	1	no	—
295	C₂×C₈	—	44	1	no	—
296	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
298	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
299	C₄⋊C₄	—	11	2	no	—
300	C₄⋊C₄	—	11	1	no	—
301	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
302	C₂²⋊C₄	—	11	2	no	—
303	C₂²⋊C₄	—	11	2	no	—
304	C₂²⋊C₄	—	11	2	no	—
305	C₂²⋊C₄	C₈×D₁₁	1	2	yes	—
306	C₄²	—	11	1	no	—
307	C₄²	—	11	1	no	—
308	C₄²	—	11	1	no	—
309	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
310	C₁₆	—	2	1	no	—
311	C₁₆	—	2	2	no	—
312	C₁₆	—	2	1	no	—
313	C₁₆	—	22	1	no	—
314	C₁₆	—	22	2	no	—
315	C₁₆	—	22	1	no	—
316	C₁₁	#33495	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	11	2	no	—
318	C₂³	—	11	1	no	—
319	C₂³	C₂×C₈×D₁₁	1	2	yes	—
320	Q₈	—	44	2	no	—
321	D₄	—	2	2	no	—
322	D₄	—	22	2	no	—
323	D₄	—	44	2	no	—
324	D₄	—	44	2	no	—
325	D₄	—	44	2	no	—
326	D₄	—	44	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	11	1	no	—
330	C₂×C₄	—	11	1	no	—
331	C₂×C₄	—	11	1	no	—
332	C₂×C₄	—	11	2	no	—
333	C₂×C₄	—	22	1	no	—
334	C₂×C₄	—	22	2	no	—
335	C₂×C₄	—	22	2	no	—
336	C₂×C₄	—	22	1	no	—
337	C₂×C₄	—	22	2	no	—
338	C₂×C₄	—	44	1	no	—
339	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
340	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
341	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	2	no	—
345	C₈	—	22	1	no	—
346	C₈	—	22	1	no	—
347	C₈	—	22	2	no	—
348	C₈	#67037	1	1	yes	characteristic
349	C₈	#67042	1	1	yes	characteristic
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	22	1	no	—
354	C₂²	—	22	2	no	—
355	C₂²	—	22	1	no	—
356	C₂²	—	22	2	no	—
357	C₂²	—	22	1	no	—
358	C₂²	—	22	2	no	—
359	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	22	1	no	—
363	C₄	—	22	1	no	—
364	C₄	—	22	1	no	—
365	C₄	—	22	2	no	—
366	C₄	#10622514	1	2	yes	—
367	C₄	#10623076	1	1	yes	characteristic
368	C₄	#10623281	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	44	1	no	—
371	C₂	—	44	2	no	—
372	C₂	#13077637	1	1	yes	characteristic
373	C₂	C₂×#10622514	1	1	yes	characteristic
374	C₂	#13081717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	D₄×C₁₁×C₁₆	C₂
prime radical	#10623079	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	D₄×C₁₆	D₁₁
hypocenter	C₁₁	#33495
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13070448	C₂	semidirect
3. #13070447	C₂	semidirect
4. #13070444	C₂	semidirect
5. #13070441	C₂	semidirect
6. C₁₆×D₄₄	C₂	semidirect
7. #13070418	C₂	semidirect
8. #13070413	C₂	semidirect
9. #13070229	C₂	semidirect
10. #13070228	C₂	semidirect
11. D₄×C₁₁×C₁₆	C₂	semidirect
136. D₄×C₁₆	D₁₁	semidirect
316. C₁₁	#33495	semidirect

Maximal subgroups

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

subgroup	index
#13077323	2
#13070448	2
#13070447	2
#13070444	2
#13070441	2
C₁₆×D₄₄	2
#13070418	2
#13070413	2
#13070229	2
#13070228	2
D₄×C₁₁×C₁₆	2
#33495	11