Subgroups — #14909488

Order 2816.

516 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079188	C₂	1	2	yes	maximal, complemented
2	#13070877	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070850	C₂	1	2	yes	maximal, complemented
4	#13070730	C₂	1	2	yes	maximal, complemented
5	#13070725	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070720	C₂	1	1	yes	maximal, characteristic
7	#13070491	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070486	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070479	C₂	1	2	yes	maximal, complemented
10	C₄×#66862	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623222	—	2	2	no	—
13	C₁₁×#499	C₂²	1	2	yes	—
14	#10622681	—	2	2	no	—
15	C₂×Dic₈₈	C₂²	1	1	yes	characteristic
16	C₂×#66864	C₂²	1	2	yes	—
17	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
18	C₂×#66862	C₂²	1	1	yes	characteristic
19	C₂×#66862	C₂²	1	1	yes	characteristic
20	#10622500	C₂²	1	2	yes	—
21	#10622496	C₂²	1	2	yes	—
22	#10622486	C₂²	1	2	yes	—
23	#10622485	C₂²	1	2	yes	—
24	#10622477	—	2	2	no	—
25	#10622465	—	2	2	no	—
26	#10622309	C₂²	1	2	yes	—
27	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
28	C₁₁×#367	C₂²	1	1	yes	characteristic
29	C₁₁×#367	C₂²	1	1	yes	characteristic
30	C₁₁×#366	C₂²	1	2	yes	—
31	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
32	#10622140	C₂²	1	1	yes	characteristic
33	#10622140	C₂²	1	1	yes	characteristic
34	#10622130	C₂²	1	2	yes	—
35	#10622130	C₂²	1	2	yes	—
36	#10622129	C₂²	1	2	yes	—
37	#10622129	C₂²	1	2	yes	—
38	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₁₁×#125	—	2	2	no	—
40	C₂×Dic₄₄	C₂³	1	2	yes	—
41	C₂×D₈₈	C₂³	1	2	yes	—
42	C₂×#4314	—	2	2	no	—
43	C₂×C₈×D₁₁	D₄	1	1	yes	characteristic
44	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
45	#66951	—	4	2	no	—
46	#66950	—	2	2	no	—
47	#66948	—	2	2	no	—
48	#66947	—	4	2	no	—
49	#66946	—	2	2	no	—
50	#66946	—	4	2	no	—
51	#66945	—	4	2	no	—
52	#66940	—	2	2	no	—
53	#66928	—	2	2	no	—
54	C₄²×D₁₁	D₄	1	1	yes	characteristic
55	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
57	C₁₁×#107	C₂³	1	2	yes	—
58	C₁₁×#107	C₂³	1	2	yes	—
59	C₁₁×#106	—	2	2	no	—
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₈×Dic₁₁	D₄	1	1	yes	characteristic
62	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
65	#66874	—	2	2	no	—
66	#66874	—	2	2	no	—
67	#66873	—	2	2	no	—
68	#66873	—	2	2	no	—
69	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
70	Dic₈₈	—	2	2	no	—
71	#66864	—	2	4	no	—
72	D₁₇₆	—	2	2	no	—
73	#66862	—	2	2	no	—
74	#66862	—	2	2	no	—
75	#33750	—	11	1	no	maximal, Sylow
76	C₂×D₄₄	—	2	2	no	—
77	C₂×D₄₄	—	4	2	no	—
78	C₂×C₄×D₁₁	—	4	2	no	—
79	C₂×C₄×D₁₁	D₈	1	2	yes	—
80	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₂₂	—	2	2	no	—
82	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₄⋊C₄	—	4	2	no	—
87	C₁₁×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
89	(C₂×D₁₁)⋊C₄	—	4	2	no	—
90	(C₂×D₁₁)⋊C₄	—	4	2	no	—
91	(C₂×D₁₁)⋊C₄	—	4	2	no	—
92	(C₂×D₁₁)⋊C₄	—	4	2	no	—
93	(C₂×D₁₁)⋊C₄	—	4	2	no	—
94	C₄⋊Dic₁₁	—	4	2	no	—
95	Dic₁₁⋊C₄	—	4	2	no	—
96	C₄×Dic₁₁	—	4	2	no	—
97	C₄×Dic₁₁	D₈	1	2	yes	—
98	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
101	Dic₄₄	—	2	2	no	—
102	Dic₄₄	—	2	2	no	—
103	D₈₈	—	2	2	no	—
104	D₈₈	—	2	2	no	—
105	#4314	—	4	4	no	—
106	C₈×D₁₁	—	2	2	no	—
107	C₈×D₁₁	—	2	1	no	—
108	C₈×D₁₁	—	2	1	no	—
109	C₁₁×C₁₆	#41	1	2	yes	—
110	C₁₁×C₁₆	#41	1	2	yes	—
111	C₁₁⋊C₁₆	—	2	1	no	—
112	C₁₁⋊C₁₆	—	2	1	no	—
113	#3168	—	11	2	no	—
114	#2253	D₁₁	1	1	yes	characteristic, complemented
115	#2249	—	11	1	no	—
116	#2242	—	11	2	no	—
117	#2220	—	11	2	no	—
118	#2219	—	11	1	no	—
119	#2218	—	11	1	no	—
120	#2177	—	11	1	no	—
121	#2176	—	11	1	no	—
122	#2175	—	11	2	no	—
123	C₄×M₃₂	—	11	1	no	—
124	C₂²×D₁₁	—	4	2	no	—
125	C₂²×D₁₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	—	4	2	no	—
127	C₂×C₄×C₁₁	—	4	2	no	—
128	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₁₁	—	4	2	no	—
131	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
132	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
133	D₄₄	—	4	2	no	—
134	D₄₄	—	4	2	no	—
135	D₄₄	—	8	4	no	—
136	C₄×D₁₁	—	2	1	no	—
137	C₄×D₁₁	—	2	1	no	—
138	C₄×D₁₁	—	4	2	no	—
139	C₄×D₁₁	—	8	4	no	—
140	Dic₂₂	—	4	2	no	—
141	Dic₂₂	—	4	2	no	—
142	C₈×C₁₁	#135	1	2	yes	—
143	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
145	C₁₁⋊C₈	—	2	1	no	—
146	C₁₁⋊C₈	—	2	1	no	—
147	#529	—	22	2	no	—
148	C₂×Q₃₂	—	11	1	no	—
149	C₂×SD₃₂	—	11	2	no	—
150	C₂×D₁₆	—	11	1	no	—
151	C₂×M₃₂	—	11	1	no	—
152	C₂×M₃₂	—	11	1	no	—
153	#499	C₂×D₁₁	1	2	yes	—
154	#495	—	11	2	no	—
155	#488	—	22	2	no	—
156	#467	—	11	2	no	—
157	#466	—	11	2	no	—
158	#465	—	22	2	no	—
159	C₄×Q₁₆	—	11	2	no	—
160	C₄×SD₁₆	—	22	2	no	—
161	C₄×D₈	—	11	2	no	—
162	C₂×C₄×C₈	—	11	1	no	—
163	#367	C₂×D₁₁	1	1	yes	characteristic
164	#367	C₂×D₁₁	1	1	yes	characteristic
165	#366	C₂×D₁₁	1	2	yes	—
166	#358	—	11	2	no	—
167	#358	—	11	2	no	—
168	#357	—	11	2	no	—
169	#357	—	11	2	no	—
170	#346	—	11	1	no	—
171	#346	—	11	1	no	—
172	C₄×C₁₆	—	11	1	no	—
173	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
174	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₁₁	—	2	2	no	—
176	C₂×D₁₁	—	2	1	no	—
177	C₂×D₁₁	—	8	2	no	—
178	C₂×D₁₁	—	8	2	no	—
179	C₂×D₁₁	—	8	2	no	—
180	C₄×C₁₁	—	2	2	no	—
181	C₄×C₁₁	—	8	2	no	—
182	C₄×C₁₁	—	8	2	no	—
183	C₄×C₁₁	#532	1	1	yes	characteristic
184	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
185	Dic₁₁	—	2	2	no	—
186	Dic₁₁	—	2	1	no	—
187	Dic₁₁	—	8	2	no	—
188	C₂×Q₁₆	—	11	2	no	—
189	C₂×SD₁₆	—	22	2	no	—
190	C₂×D₈	—	11	2	no	—
191	C₂²×C₈	—	11	1	no	—
192	C₂²×C₈	—	11	1	no	—
193	#126	—	44	2	no	—
194	#125	—	2	2	no	—
195	#124	—	22	2	no	—
196	#123	—	44	2	no	—
197	C₂²⋊Q₈	—	22	2	no	—
198	#121	—	22	2	no	—
199	C₄×Q₈	—	22	2	no	—
200	C₄×D₄	—	22	2	no	—
201	C₄×D₄	—	44	2	no	—
202	#117	—	44	2	no	—
203	C₂×C₄²	—	11	1	no	—
204	Q₃₂	—	22	2	no	—
205	SD₃₂	—	22	4	no	—
206	D₁₆	—	22	2	no	—
207	M₃₂	—	22	2	no	—
208	M₃₂	—	22	2	no	—
209	C₂×C₁₆	—	11	1	no	—
210	C₂×C₁₆	—	11	1	no	—
211	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
212	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
213	#107	C₂²×D₁₁	1	2	yes	—
214	#107	C₂²×D₁₁	1	2	yes	—
215	#106	—	2	2	no	—
216	#103	—	22	2	no	—
217	#103	—	22	2	no	—
218	#102	—	22	2	no	—
219	#102	—	22	2	no	—
220	C₄×C₈	—	11	1	no	—
221	C₄×C₈	—	11	1	no	—
222	C₄×C₈	—	11	1	no	—
223	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
224	C₂×C₁₁	#3169	1	1	yes	characteristic
225	C₂×C₁₁	#3414	1	2	yes	—
226	D₁₁	—	4	1	no	—
227	D₁₁	—	16	2	no	—
228	C₂×Q₈	—	22	2	no	—
229	C₂×D₄	—	22	2	no	—
230	C₂×D₄	—	44	2	no	—
231	C₂²×C₄	—	11	1	no	—
232	C₂²×C₄	—	11	2	no	—
233	C₂²×C₄	—	44	2	no	—
234	Q₁₆	—	22	2	no	—
235	Q₁₆	—	22	2	no	—
236	SD₁₆	—	44	4	no	—
237	D₈	—	22	2	no	—
238	D₈	—	22	2	no	—
239	C₂×C₈	—	11	1	no	—
240	C₂×C₈	—	11	1	no	—
241	C₂×C₈	—	22	2	no	—
242	C₂×C₈	—	22	1	no	—
243	C₂×C₈	—	22	1	no	—
244	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
245	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	44	2	no	—
251	C₄⋊C₄	—	44	2	no	—
252	C₂²⋊C₄	—	44	2	no	—
253	C₂²⋊C₄	—	44	2	no	—
254	C₂²⋊C₄	—	44	2	no	—
255	C₂²⋊C₄	—	44	2	no	—
256	C₂²⋊C₄	—	44	2	no	—
257	C₄²	—	11	2	no	—
258	C₄²	—	11	1	no	—
259	C₄²	—	44	2	no	—
260	C₄²	D₄×D₁₁	1	1	yes	characteristic
261	C₁₆	—	22	1	no	—
262	C₁₆	—	22	1	no	—
263	C₁₆	#4343	1	2	yes	—
264	C₁₆	#4343	1	2	yes	—
265	C₁₁	#33750	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	11	1	no	—
267	C₂³	—	44	2	no	—
268	Q₈	—	44	2	no	—
269	Q₈	—	44	2	no	—
270	D₄	—	44	2	no	—
271	D₄	—	44	2	no	—
272	D₄	—	88	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	11	2	no	—
276	C₂×C₄	—	11	1	no	—
277	C₂×C₄	—	22	1	no	—
278	C₂×C₄	—	22	1	no	—
279	C₂×C₄	—	44	2	no	—
280	C₂×C₄	—	44	2	no	—
281	C₂×C₄	—	88	4	no	—
282	C₂×C₄	D₈×D₁₁	1	2	yes	—
283	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
284	C₈	—	22	1	no	—
285	C₈	—	22	1	no	—
286	C₈	#66972	1	2	yes	—
287	C₈	C₂×#4343	1	1	yes	characteristic
288	C₈	C₂×#4343	1	1	yes	characteristic
289	C₂²	—	22	2	no	—
290	C₂²	—	22	1	no	—
291	C₂²	—	88	2	no	—
292	C₂²	—	88	2	no	—
293	C₂²	—	88	2	no	—
294	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	22	2	no	—
299	C₄	—	22	1	no	—
300	C₄	—	88	2	no	—
301	C₄	#10623256	1	1	yes	characteristic
302	C₄	C₂×#66972	1	1	yes	characteristic
303	C₂	—	44	1	no	—
304	C₂	—	176	2	no	—
305	C₂	#13079196	1	1	yes	characteristic
306	C₂	#13081749	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₁
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2253	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	#13070720	C₂
hypercenter	#2253	D₁₁
hypocenter	C₁₁	#33750
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
1. #13079188	C₂	semidirect
2. #13070877	C₂	semidirect
3. #13070850	C₂	semidirect
4. #13070730	C₂	semidirect
5. #13070725	C₂	semidirect
7. #13070491	C₂	semidirect
8. #13070486	C₂	semidirect
9. #13070479	C₂	semidirect
10. C₄×#66862	C₂	semidirect
11. C₁₁×#2253	C₂	semidirect
114. #2253	D₁₁	semidirect
265. C₁₁	#33750	semidirect

Maximal subgroups

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

subgroup	index
#13079188	2
#13070877	2
#13070850	2
#13070730	2
#13070725	2
#13070720	2
#13070491	2
#13070486	2
#13070479	2
C₄×#66862	2
C₁₁×#2253	2
#33750	11