Subgroups — #14754136

Order 2816.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₁₁	C₂	1	1	yes	maximal, characteristic
2	#13068763	C₂	1	2	yes	maximal, complemented
3	#13068739	C₂	1	2	yes	maximal, complemented
4	#13068717	C₂	1	2	yes	maximal
5	#13068666	C₂	1	1	yes	maximal, characteristic
6	#13068497	C₂	1	2	yes	maximal, complemented
7	#13068418	C₂	1	2	yes	maximal
8	#13068370	C₂	1	2	yes	maximal, complemented
9	C₁₁×#1812	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₁×#131	C₂²	1	1	yes	characteristic
11	#41×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂×#67001	C₂²	1	1	yes	characteristic
13	C₁₁×#427	C₂²	1	2	yes	—
14	C₁₁×#410	C₂²	1	2	yes	—
15	#10622850	C₂²	1	2	yes	—
16	#10622735	C₄	1	2	yes	complemented
17	#10622735	C₂²	1	1	yes	characteristic
18	M₁₆×Dic₁₁	C₄	1	2	yes	complemented
19	M₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66883	C₂²	1	2	yes	—
21	#10622714	C₄	1	2	yes	complemented
22	#10622714	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₁₁	C₄	1	2	yes	complemented
24	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
25	#10622610	C₂²	1	4	yes	—
26	C₁₁×H(3,ℤ/4)	C₂²	1	2	yes	—
27	C₁₁⋊#338	C₂²	1	2	yes	—
28	#10622171	C₂²	1	4	yes	—
29	#10622170	C₂²	1	4	yes	—
30	#10622169	C₂²	1	4	yes	—
31	C₂×C₁₁×#41	C₂³	1	1	yes	characteristic
32	C₁₁×#131	—	2	2	no	—
33	C₁₁×#131	—	2	2	no	—
34	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₁₁×#117	C₂³	1	4	yes	—
39	#67001	—	2	2	no	—
40	#67001	—	2	2	no	—
41	Q₈×Dic₁₁	—	2	1	no	—
42	D₄×Dic₁₁	—	2	2	no	—
43	D₄×Dic₁₁	—	2	1	no	—
44	#66979	C₂×C₄	1	2	yes	—
45	#66979	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₁₁	C₂³	1	1	yes	characteristic
48	C₂×#4318	C₂×C₄	1	2	yes	—
49	C₂×#4318	C₂³	1	4	yes	—
50	C₂×#4318	C₂³	1	1	yes	characteristic
51	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
52	C₂²×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₁₁×#107	—	2	2	no	—
54	C₁₁×#106	—	2	2	no	—
55	C₁₁×#99	—	2	4	no	—
56	#66900	—	2	2	no	—
57	#66897	—	2	2	no	—
58	#66883	—	2	2	no	—
59	#66883	—	2	2	no	—
60	#66879	C₂×C₄	1	2	yes	—
61	#66879	C₂×C₄	1	2	yes	—
62	#66879	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
65	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₁₁	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	1	yes	characteristic
70	#66867	—	2	4	no	—
71	C₄×C₁₁⋊C₈	—	2	4	no	—
72	#20787	—	11	1	no	maximal, Sylow
73	C₁₁×#41	—	2	2	no	—
74	C₁₁×#41	—	2	2	no	—
75	C₂×Q₈×C₁₁	#41	1	1	yes	characteristic
76	C₂×D₄×C₁₁	#41	1	2	yes	—
77	C₂×D₄×C₁₁	#41	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₁₁	—	2	2	no	—
81	C₂²×Dic₁₁	—	2	1	no	—
82	C₁₁×M₁₆	—	2	4	no	—
83	C₁₁×M₁₆	—	2	2	no	—
84	C₂×C₈×C₁₁	—	2	2	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	#41	1	2	yes	—
90	C₂×C₈×C₁₁	#41	1	2	yes	—
91	C₁₁×C₄⋊C₄	—	2	4	no	—
92	C₁₁×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₁₁	—	2	4	no	—
94	C₂²⋊Dic₁₁	—	2	2	no	—
95	C₂²⋊Dic₁₁	—	2	1	no	—
96	C₄⋊Dic₁₁	—	2	2	no	—
97	C₄⋊Dic₁₁	—	2	1	no	—
98	C₄×Dic₁₁	—	2	2	no	—
99	C₄×Dic₁₁	—	2	1	no	—
100	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
102	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
103	#4318	—	2	4	no	—
104	#4318	—	2	2	no	—
105	#4318	—	2	4	no	—
106	#4318	—	2	4	no	—
107	C₂×C₁₁⋊C₈	—	2	2	no	—
108	C₂×C₁₁⋊C₈	—	2	2	no	—
109	C₂×C₁₁⋊C₈	—	2	4	no	—
110	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₁₁⋊C₈	#41	1	2	yes	—
114	C₂×C₁₁⋊C₈	#41	1	2	yes	—
115	C₄×#131	—	11	1	no	—
116	#1841	—	11	2	no	—
117	#1833	—	11	2	no	—
118	#1827	—	11	2	no	—
119	#1813	—	11	1	no	—
120	#1812	D₁₁	1	1	yes	characteristic
121	#1772	—	11	2	no	—
122	#1757	—	11	2	no	—
123	#1747	—	11	2	no	—
124	C₂³×C₁₁	#117	1	2	yes	—
125	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
126	Q₈×C₁₁	—	2	2	no	—
127	D₄×C₁₁	—	2	4	no	—
128	D₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	4	4	no	—
132	C₂×C₄×C₁₁	#117	1	2	yes	—
133	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
137	C₂×Dic₁₁	—	2	1	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	C₂×Dic₁₁	—	2	1	no	—
140	C₂×Dic₁₁	—	4	2	no	—
141	C₂×Dic₁₁	—	4	4	no	—
142	C₈×C₁₁	—	2	2	no	—
143	C₈×C₁₁	#117	1	1	yes	characteristic
144	C₈×C₁₁	#117	1	1	yes	characteristic
145	C₈×C₁₁	#117	1	2	yes	—
146	C₁₁⋊C₈	—	2	2	no	—
147	C₁₁⋊C₈	—	4	4	no	—
148	C₁₁⋊C₈	#117	1	1	yes	characteristic
149	C₁₁⋊C₈	#117	1	1	yes	characteristic
150	C₁₁⋊C₈	#117	1	2	yes	—
151	C₂×#131	—	11	1	no	—
152	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
153	C₄×#41	—	11	1	no	—
154	C₂×#108	—	11	2	no	—
155	#427	C₂×D₁₁	1	2	yes	—
156	#413	—	11	2	no	—
157	#410	C₂×D₁₁	1	2	yes	—
158	#405	—	11	1	no	—
159	#405	—	11	2	no	—
160	#405	—	11	1	no	—
161	#405	—	11	2	no	—
162	#405	—	11	4	no	—
163	C₄×M₁₆	—	11	1	no	—
164	C₄×M₁₆	—	11	2	no	—
165	C₂×C₄×C₈	—	11	1	no	—
166	C₂×C₄×C₈	—	11	2	no	—
167	#344	—	11	4	no	—
168	#339	—	11	4	no	—
169	#338	—	11	2	no	—
170	H(3,ℤ/4)	—	11	4	no	—
171	H(3,ℤ/4)	C₂×D₁₁	1	2	yes	—
172	C₂²×C₁₁	—	2	2	no	—
173	C₂²×C₁₁	—	4	1	no	—
174	C₂²×C₁₁	—	4	2	no	—
175	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
176	C₂²×C₁₁	#529	1	2	yes	—
177	C₄×C₁₁	—	2	2	no	—
178	C₄×C₁₁	—	8	4	no	—
179	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
180	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
181	C₄×C₁₁	#529	1	2	yes	—
182	Dic₁₁	—	2	2	no	—
183	Dic₁₁	—	4	2	no	—
184	Dic₁₁	—	4	1	no	—
185	C₂×#41	C₂²×D₁₁	1	1	yes	characteristic
186	#131	—	2	2	no	—
187	#131	—	2	2	no	—
188	#131	—	22	2	no	—
189	#131	—	22	2	no	—
190	C₂×M₁₆	—	11	2	no	—
191	C₂×M₁₆	—	11	1	no	—
192	C₂×M₁₆	—	11	4	no	—
193	C₂×M₁₆	C₄×D₁₁	1	2	yes	—
194	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
195	C₂²×C₈	—	11	1	no	—
196	C₂²×C₈	—	11	2	no	—
197	C₂²×C₈	C₄×D₁₁	1	2	yes	—
198	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
199	C₄×Q₈	—	22	1	no	—
200	C₄×D₄	—	22	2	no	—
201	C₄×D₄	—	22	1	no	—
202	#117	—	11	2	no	—
203	#117	—	11	1	no	—
204	#117	C₂²×D₁₁	1	4	yes	—
205	C₂×C₄²	—	11	2	no	—
206	C₂×C₄²	—	11	1	no	—
207	#108	—	22	2	no	—
208	#108	—	22	2	no	—
209	#107	—	2	2	no	—
210	#106	—	2	2	no	—
211	#101	—	22	2	no	—
212	#100	—	22	2	no	—
213	#99	—	2	4	no	—
214	#97	—	11	2	no	—
215	#97	—	11	2	no	—
216	#97	—	11	2	no	—
217	#97	—	22	4	no	—
218	C₄×C₈	—	11	1	no	—
219	C₄×C₈	—	11	1	no	—
220	C₄×C₈	—	11	2	no	—
221	C₄×C₈	—	11	2	no	—
222	C₄×C₈	—	11	1	no	—
223	C₄×C₈	—	11	1	no	—
224	C₄×C₈	—	11	2	no	—
225	C₄×C₈	—	22	4	no	—
226	C₂×C₁₁	—	2	1	no	—
227	C₂×C₁₁	—	2	2	no	—
228	C₂×C₁₁	—	4	2	no	—
229	C₂×C₁₁	#2283	1	1	yes	characteristic
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	C₂×Q₈	#4343	1	1	yes	characteristic
233	C₂×D₄	#4341	1	2	yes	—
234	C₂×D₄	#4343	1	1	yes	characteristic
235	C₂²×C₄	—	22	2	no	—
236	C₂²×C₄	—	22	1	no	—
237	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
238	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	22	4	no	—
242	M₁₆	—	22	2	no	—
243	M₁₆	—	22	4	no	—
244	M₁₆	—	22	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	11	1	no	—
248	C₂×C₈	—	11	1	no	—
249	C₂×C₈	—	11	2	no	—
250	C₂×C₈	—	11	2	no	—
251	C₂×C₈	—	11	2	no	—
252	C₂×C₈	—	22	2	no	—
253	C₂×C₈	—	22	2	no	—
254	C₂×C₈	—	22	4	no	—
255	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
258	C₂×C₈	#4341	1	2	yes	—
259	C₂×C₈	#4341	1	2	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	22	2	no	—
262	C₄⋊C₄	—	22	1	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	22	2	no	—
265	C₂²⋊C₄	—	22	1	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	11	1	no	—
268	C₄²	—	11	2	no	—
269	C₄²	—	11	1	no	—
270	C₄²	—	22	2	no	—
271	C₄²	—	22	1	no	—
272	C₁₁	#20787	1	1	yes	characteristic, complemented, Sylow
273	C₂³	#66945	1	2	yes	—
274	C₂³	C₂×#4341	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	22	1	no	—
282	C₂×C₄	—	22	2	no	—
283	C₂×C₄	—	22	1	no	—
284	C₂×C₄	—	44	2	no	—
285	C₂×C₄	—	44	4	no	—
286	C₂×C₄	#66945	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#4341	1	2	yes	—
289	C₂×C₄	C₂×#4343	1	1	yes	characteristic
290	C₂×C₄	#41×D₁₁	1	2	yes	—
291	C₈	—	2	2	no	—
292	C₈	—	11	1	no	—
293	C₈	—	11	1	no	—
294	C₈	—	11	2	no	—
295	C₈	—	22	2	no	—
296	C₈	—	44	4	no	—
297	C₈	#66930	1	1	yes	characteristic
298	C₈	#66930	1	1	yes	characteristic
299	C₈	#66930	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#66945	1	1	yes	characteristic
304	C₂²	#10623131	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	8	4	no	—
307	C₄	—	22	2	no	—
308	C₄	—	44	2	no	—
309	C₄	—	44	1	no	—
310	C₄	C₂×#66930	1	1	yes	characteristic
311	C₄	D₁₁×#117	1	1	yes	characteristic
312	C₄	#10623155	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#13071011	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₁×#117
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#1812	C₂
socle	C₂×C₁₁	#2283
prime radical	C₂×C₁₁×#41	C₂³
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#1812	D₁₁
hypocenter	C₁₁	#20787
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1812

Complements

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

N	H ≅ G/N	type
2. #13068763	C₂	semidirect
3. #13068739	C₂	semidirect
6. #13068497	C₂	semidirect
8. #13068370	C₂	semidirect
16. #10622735	C₄	semidirect
18. M₁₆×Dic₁₁	C₄	semidirect
21. #10622714	C₄	semidirect
23. C₂×C₈×Dic₁₁	C₄	semidirect
272. C₁₁	#20787	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₁₁	2
#13068763	2
#13068739	2
#13068717	2
#13068666	2
#13068497	2
#13068418	2
#13068370	2
C₁₁×#1812	2
#20787	11