Subgroups — #14907349

Order 2816.

548 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×#4341	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070518	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070448	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070446	C₂	1	2	yes	maximal, complemented
6	#13070445	C₂	1	2	yes	maximal, complemented
7	#13070418	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070416	C₂	1	2	yes	maximal, complemented
9	M₃₂×Dic₁₁	C₂	1	2	yes	maximal, complemented
10	C₄×#66862	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#4341	C₄	1	1	yes	characteristic
13	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
14	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
17	C₂×#66876	C₂²	1	2	yes	—
18	D₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
19	C₂×#66862	C₂²	1	1	yes	characteristic
20	C₂×#66862	C₂²	1	1	yes	characteristic
21	#10622446	C₄	1	1	yes	characteristic
22	#10622443	C₄	1	1	yes	characteristic
23	(C₂²⋊D₁₁)⋊C₈	C₄	1	2	yes	—
24	#10622331	C₄	1	2	yes	—
25	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
26	C₈×Dic₂₂	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	2	yes	—
32	#10622140	C₂²	1	1	yes	characteristic
33	#10622140	C₂²	1	2	yes	—
34	#10622140	C₂²	1	1	yes	characteristic
35	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
36	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
37	C₁₆×Dic₁₁	C₂²	1	2	yes	—
38	C₁₆×Dic₁₁	C₂²	1	2	yes	—
39	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#4341	C₈	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
43	C₄×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
44	D₄×Dic₁₁	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
47	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
50	#66945	C₈	1	1	yes	characteristic
51	#66940	C₈	1	1	yes	characteristic
52	(C₂²⋊D₁₁)⋊C₄	C₈	1	2	yes	—
53	#66930	C₈	1	2	yes	—
54	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
56	C₁₁×M₃₂	—	2	4	no	—
57	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
66	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₁	C₂³	1	2	yes	—
71	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
72	#66876	—	2	4	no	—
73	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
75	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
76	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
78	#66862	—	2	2	no	—
79	#66862	—	2	2	no	—
80	#33485	—	11	1	no	maximal, Sylow
81	C₂×D₄×C₁₁	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
83	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
84	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
85	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
86	#4341	—	2	4	no	—
87	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₁₁	—	2	2	no	—
92	C₂×C₈×C₁₁	—	2	2	no	—
93	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
96	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
98	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
100	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
102	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
103	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
110	C₂×C₁₁⋊C₈	—	2	4	no	—
111	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
114	C₈×D₁₁	—	2	1	no	—
115	C₈×D₁₁	—	2	1	no	—
116	C₈×D₁₁	—	2	2	no	—
117	C₁₁×C₁₆	—	2	1	no	—
118	C₁₁×C₁₆	—	2	2	no	—
119	C₁₁×C₁₆	#41	1	2	yes	complemented
120	C₁₁⋊C₁₆	—	2	1	no	—
121	C₁₁⋊C₁₆	—	2	2	no	—
122	C₁₁⋊C₁₆	#41	1	2	yes	complemented
123	C₈×#41	—	11	1	no	—
124	Q₈×C₁₆	—	11	1	no	—
125	Q₈×C₁₆	—	11	1	no	—
126	#2169	—	11	2	no	—
127	#2169	—	11	1	no	—
128	#2169	—	11	2	no	—
129	#2169	D₁₁	1	1	yes	characteristic, complemented
130	#2163	—	11	1	no	—
131	#2163	—	11	2	no	—
132	C₄×M₃₂	—	11	2	no	—
133	C₄×M₃₂	—	11	1	no	—
134	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
135	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
136	D₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	—	2	2	no	—
139	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₁₁	—	2	4	no	—
144	C₂×Dic₁₁	—	2	4	no	—
145	C₂×Dic₁₁	—	4	2	no	—
146	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
148	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
150	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
151	C₄×D₁₁	—	2	1	no	—
152	C₄×D₁₁	—	2	1	no	—
153	C₄×D₁₁	—	2	2	no	—
154	C₄×D₁₁	—	4	1	no	—
155	Dic₂₂	M₃₂	1	4	yes	complemented
156	C₈×C₁₁	—	2	2	no	—
157	C₈×C₁₁	—	2	1	no	—
158	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
160	C₁₁⋊C₈	—	2	1	no	—
161	C₁₁⋊C₈	—	2	2	no	—
162	C₁₁⋊C₈	#131	1	2	yes	—
163	C₄×#41	—	11	1	no	—
164	C₂×M₃₂	—	11	2	no	—
165	C₂×M₃₂	—	11	1	no	—
166	C₂×M₃₂	—	11	1	no	—
167	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
168	C₈×Q₈	—	11	1	no	—
169	C₈×Q₈	—	11	1	no	—
170	C₈×D₄	—	11	2	no	—
171	C₈×D₄	—	11	1	no	—
172	C₈×D₄	—	11	2	no	—
173	C₈×D₄	C₂×D₁₁	1	1	yes	characteristic
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	2	no	—
179	C₄⋊C₁₆	—	11	2	no	—
180	C₄⋊C₁₆	—	11	1	no	—
181	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
182	C₂²⋊C₁₆	—	11	2	no	—
183	C₂²⋊C₁₆	—	11	2	no	—
184	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
185	#346	—	11	1	no	—
186	#346	—	11	2	no	—
187	#346	—	11	1	no	—
188	C₄×C₁₆	—	11	2	no	—
189	C₄×C₁₆	—	11	2	no	—
190	C₄×C₁₆	—	11	1	no	—
191	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
192	C₂²×C₁₁	—	2	2	no	—
193	C₂²×C₁₁	—	2	2	no	—
194	C₂²×C₁₁	—	2	2	no	—
195	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
196	C₂×D₁₁	—	2	1	no	—
197	C₂×D₁₁	—	2	1	no	—
198	C₂×D₁₁	—	2	1	no	—
199	C₄×C₁₁	—	2	1	no	—
200	C₄×C₁₁	—	2	2	no	—
201	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
202	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
203	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
204	Dic₁₁	—	2	1	no	—
205	Dic₁₁	—	2	1	no	—
206	Dic₁₁	—	2	2	no	—
207	Dic₁₁	—	2	1	no	—
208	Dic₁₁	C₂×M₃₂	1	4	yes	—
209	C₂×#41	—	11	1	no	—
210	C₂²×C₈	—	11	2	no	—
211	C₂²×C₈	—	11	1	no	—
212	C₂²×C₈	—	11	1	no	—
213	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
214	C₄×Q₈	—	11	1	no	—
215	C₄×Q₈	—	11	1	no	—
216	C₄×D₄	—	11	1	no	—
217	C₄×D₄	—	11	2	no	—
218	C₄×D₄	—	11	2	no	—
219	C₄×D₄	C₄×D₁₁	1	1	yes	characteristic
220	#117	—	11	1	no	—
221	#117	—	11	2	no	—
222	C₂×C₄²	—	11	2	no	—
223	C₂×C₄²	—	11	1	no	—
224	M₃₂	—	2	4	no	—
225	M₃₂	—	22	2	no	—
226	M₃₂	—	22	4	no	—
227	M₃₂	—	22	2	no	—
228	C₂×C₁₆	—	11	1	no	—
229	C₂×C₁₆	—	11	1	no	—
230	C₂×C₁₆	—	11	2	no	—
231	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
232	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
234	C₄⋊C₈	—	11	2	no	—
235	C₄⋊C₈	—	11	2	no	—
236	C₄⋊C₈	—	11	1	no	—
237	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
238	C₂²⋊C₈	—	11	2	no	—
239	C₂²⋊C₈	—	11	2	no	—
240	C₂²⋊C₈	C₄×D₁₁	1	2	yes	—
241	C₄×C₈	—	11	2	no	—
242	C₄×C₈	—	11	1	no	—
243	C₄×C₈	—	11	2	no	—
244	C₄×C₈	—	11	2	no	—
245	C₄×C₈	—	11	1	no	—
246	C₄×C₈	—	11	1	no	—
247	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
248	C₂×C₁₁	—	4	2	no	—
249	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
250	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
251	C₂×C₁₁	#3407	1	1	yes	characteristic
252	D₁₁	—	4	1	no	—
253	#41	—	22	4	no	—
254	C₂×Q₈	—	11	1	no	—
255	C₂×D₄	—	11	2	no	—
256	C₂×D₄	C₈×D₁₁	1	1	yes	characteristic
257	C₂²×C₄	—	11	2	no	—
258	C₂²×C₄	—	11	2	no	—
259	C₂²×C₄	—	11	1	no	—
260	C₂²×C₄	—	11	1	no	—
261	C₂²×C₄	—	11	1	no	—
262	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	11	1	no	—
266	C₂×C₈	—	11	2	no	—
267	C₂×C₈	—	11	1	no	—
268	C₂×C₈	—	22	2	no	—
269	C₂×C₈	—	22	4	no	—
270	C₂×C₈	—	22	1	no	—
271	C₂×C₈	—	22	1	no	—
272	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
273	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
275	C₄⋊C₄	—	11	2	no	—
276	C₄⋊C₄	—	11	2	no	—
277	C₄⋊C₄	—	11	1	no	—
278	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	11	2	no	—
280	C₂²⋊C₄	—	11	2	no	—
281	C₂²⋊C₄	C₈×D₁₁	1	2	yes	—
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	—	11	2	no	—
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	1	no	—
293	C₁₆	—	22	2	no	—
294	C₁₆	#4341	1	2	yes	complemented
295	C₁₁	#33485	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	11	1	no	—
297	C₂³	C₂×C₈×D₁₁	1	2	yes	—
298	Q₈	—	11	4	no	—
299	D₄	—	2	2	no	—
300	D₄	—	22	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	11	2	no	—
304	C₂×C₄	—	11	1	no	—
305	C₂×C₄	—	11	1	no	—
306	C₂×C₄	—	11	1	no	—
307	C₂×C₄	—	11	2	no	—
308	C₂×C₄	—	22	4	no	—
309	C₂×C₄	—	22	1	no	—
310	C₂×C₄	—	22	2	no	—
311	C₂×C₄	—	22	1	no	—
312	C₂×C₄	—	44	1	no	—
313	C₂×C₄	—	44	2	no	—
314	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
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×#4341	1	1	yes	characteristic
324	C₈	#41×D₁₁	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	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	1	no	—
334	C₄	—	11	4	no	—
335	C₄	—	22	1	no	—
336	C₄	—	22	1	no	—
337	C₄	—	22	1	no	—
338	C₄	—	22	2	no	—
339	C₄	D₁₁×M₃₂	1	2	yes	—
340	C₄	C₄×#4341	1	1	yes	characteristic
341	C₄	D₁₁×#131	1	1	yes	characteristic
342	C₂	—	4	2	no	—
343	C₂	—	44	1	no	—
344	C₂	C₈×#4341	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×#2169	C₂
prime radical	C₂×#4341	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2169	D₁₁
hypocenter	C₁₁	#33485
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₂₂	C₂	semidirect
3. #13070518	C₂	semidirect
4. #13070448	C₂	semidirect
5. #13070446	C₂	semidirect
6. #13070445	C₂	semidirect
7. #13070418	C₂	semidirect
8. #13070416	C₂	semidirect
9. M₃₂×Dic₁₁	C₂	semidirect
10. C₄×#66862	C₂	semidirect
11. C₁₁×#2169	C₂	semidirect
119. C₁₁×C₁₆	#41	semidirect
122. C₁₁⋊C₁₆	#41	semidirect
129. #2169	D₁₁	semidirect
155. Dic₂₂	M₃₂	semidirect
294. C₁₆	#4341	semidirect
295. C₁₁	#33485	semidirect

Maximal subgroups

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

subgroup	index
C₈×#4341	2
C₁₆×Dic₂₂	2
#13070518	2
#13070448	2
#13070446	2
#13070445	2
#13070418	2
#13070416	2
M₃₂×Dic₁₁	2
C₄×#66862	2
C₁₁×#2169	2
#33485	11