Subgroups — #14725813

Order 2816.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079195	C₂	1	1	yes	maximal, characteristic
2	#13077682	C₂	1	2	yes	maximal, complemented
3	#13067531	C₂	1	2	yes	maximal, complemented
4	#13067447	C₂	1	4	yes	maximal, complemented
5	#13067441	C₂	1	2	yes	maximal, complemented
6	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067034	C₂	1	2	yes	maximal, complemented
8	C₁₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	#10623008	C₂²	1	2	yes	—
10	C₁₁×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622728	C₄	1	2	yes	—
14	#10622728	C₄	1	2	yes	—
15	#10622538	C₂²	1	2	yes	—
16	#10622451	—	2	2	no	—
17	#10622447	—	2	2	no	—
18	#10622340	—	2	4	no	—
19	#10622308	C₄	1	1	yes	characteristic
20	#10622307	C₄	1	2	yes	—
21	C₄×D₈₈	C₂²	1	2	yes	—
22	C₄×D₈₈	C₂²	1	2	yes	—
23	#10622301	C₂²	1	4	yes	—
24	C₈×D₄₄	C₂²	1	4	yes	—
25	#10622295	C₄	1	1	yes	characteristic
26	C₁₁×#322	C₂²	1	2	yes	—
27	C₁₁×#322	C₂²	1	2	yes	—
28	#10622097	C₂²	1	4	yes	—
29	#10622097	C₂²	1	4	yes	—
30	#10622096	C₂²	1	2	yes	—
31	#10622095	C₂²	1	1	yes	characteristic
32	#10622095	C₂²	1	1	yes	characteristic
33	C₂×#4339	—	2	2	no	—
34	C₂×C₁₁×M₁₆	—	2	1	no	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	#66983	—	2	2	no	—
39	#66983	C₂×C₄	1	4	yes	—
40	C₂×D₈₈	C₂×C₄	1	2	yes	—
41	C₂×D₈₈	C₂×C₄	1	2	yes	—
42	C₂×#4313	—	2	4	no	—
43	C₂×C₈×D₁₁	—	2	4	no	—
44	#66928	—	2	2	no	—
45	#66927	C₂×C₄	1	2	yes	—
46	C₄×D₄₄	—	2	2	no	—
47	C₄×D₄₄	C₂³	1	4	yes	—
48	#66922	C₂×C₄	1	2	yes	—
49	C₁₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₁₁×C₂²⋊C₈	D₄	1	2	yes	—
51	C₁₁×#97	—	2	1	no	—
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	2	yes	—
55	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	2	yes	—
58	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
59	#66885	C₂×C₄	1	4	yes	—
60	#66885	C₂×C₄	1	4	yes	—
61	(C₂×D₁₁)⋊C₈	—	2	4	no	—
62	(C₂×D₁₁)⋊C₈	—	2	4	no	—
63	#66882	C₂×C₄	1	2	yes	—
64	#66882	C₂×C₄	1	2	yes	—
65	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
67	C₄×C₁₁⋊C₈	—	2	2	no	—
68	#18091	—	11	1	no	maximal, Sylow
69	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₁	—	2	4	no	—
73	#4339	—	8	2	no	—
74	#4339	—	8	2	no	—
75	C₂×D₄₄	—	2	2	no	—
76	C₂×D₄₄	—	2	2	no	—
77	C₂×D₄₄	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₁₁	—	2	4	no	—
79	C₂×Dic₂₂	—	2	2	no	—
80	C₁₁×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	4	1	no	—
84	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
89	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₁	#41	1	1	yes	characteristic
92	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
93	(C₂×D₁₁)⋊C₄	—	2	4	no	—
94	(C₂×D₁₁)⋊C₄	—	2	4	no	—
95	C₄⋊Dic₁₁	—	2	2	no	—
96	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
97	C₂×C₁₁⋊C₈	—	2	2	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	D₈₈	—	2	4	no	—
100	D₈₈	—	4	2	no	—
101	#4313	—	8	4	no	—
102	C₈×D₁₁	—	8	4	no	—
103	#3169	—	11	1	no	—
104	#2979	—	11	2	no	—
105	#1594	—	11	2	no	—
106	#1581	—	11	4	no	—
107	#1580	—	11	2	no	—
108	#1564	—	11	1	no	—
109	#1484	—	11	2	no	—
110	#1451	D₁₁	1	1	yes	characteristic, complemented
111	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
112	C₂²×D₁₁	—	2	4	no	—
113	C₂×C₄×C₁₁	—	2	1	no	—
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
123	C₂²⋊D₁₁	—	8	4	no	—
124	C₂×Dic₁₁	—	2	4	no	—
125	D₄₄	—	2	4	no	—
126	D₄₄	—	2	4	no	—
127	D₄₄	—	4	4	no	—
128	D₄₄	—	8	2	no	—
129	C₄×D₁₁	—	4	4	no	—
130	C₄×D₁₁	—	4	4	no	—
131	Dic₂₂	—	8	2	no	—
132	C₈×C₁₁	—	2	1	no	—
133	C₈×C₁₁	—	2	1	no	—
134	C₈×C₁₁	—	2	1	no	—
135	C₈×C₁₁	—	2	4	no	—
136	C₈×C₁₁	—	4	2	no	—
137	C₈×C₁₁	#131	1	2	yes	—
138	C₁₁⋊C₈	—	4	2	no	—
139	C₁₁⋊C₈	—	4	2	no	—
140	#532	—	11	2	no	—
141	#500	—	11	1	no	—
142	#493	—	11	1	no	—
143	#486	—	11	2	no	—
144	#466	—	11	2	no	—
145	#466	—	11	2	no	—
146	C₄×D₈	—	11	2	no	—
147	C₄×D₈	—	11	2	no	—
148	#436	—	11	4	no	—
149	#436	—	22	2	no	—
150	C₈×D₄	—	11	4	no	—
151	C₈×D₄	—	22	2	no	—
152	#431	—	11	2	no	—
153	#431	C₂×D₁₁	1	1	yes	characteristic
154	#408	—	22	4	no	—
155	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
157	#335	—	11	2	no	—
158	#334	—	11	1	no	—
159	#334	—	11	1	no	—
160	D₄⋊C₈	—	11	4	no	—
161	D₄⋊C₈	—	11	4	no	—
162	#322	C₂×D₁₁	1	2	yes	—
163	#322	C₂×D₁₁	1	2	yes	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	—	2	1	no	—
167	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₁₁	—	4	4	no	—
169	C₂×D₁₁	—	4	4	no	—
170	C₂×D₁₁	—	4	4	no	—
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	—	2	1	no	—
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	#444	1	2	yes	—
175	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
176	C₄×C₁₁	#568	1	1	yes	characteristic
177	C₄×C₁₁	#576	1	2	yes	—
178	Dic₁₁	—	4	4	no	—
179	C₂×#41	—	22	2	no	—
180	C₂×D₈	—	11	2	no	—
181	C₂×D₈	—	11	2	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	22	4	no	—
184	C₂²×C₈	—	22	4	no	—
185	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
187	C₄⋊Q₈	—	11	2	no	—
188	#127	—	11	2	no	—
189	#124	—	22	2	no	—
190	#121	—	11	4	no	—
191	#121	—	22	2	no	—
192	C₄×D₄	—	11	4	no	—
193	C₄×D₄	—	22	2	no	—
194	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
195	#107	—	11	2	no	—
196	#107	—	11	2	no	—
197	C₄⋊C₈	—	11	2	no	—
198	C₄⋊C₈	—	11	2	no	—
199	C₄⋊C₈	D₄₄	1	2	yes	—
200	#102	—	11	4	no	—
201	#102	—	11	4	no	—
202	C₂²⋊C₈	—	22	2	no	—
203	C₂²⋊C₈	—	22	4	no	—
204	C₂²⋊C₈	—	22	4	no	—
205	C₂²⋊C₈	D₄₄	1	2	yes	—
206	#97	—	2	1	no	—
207	C₄×C₈	—	22	2	no	—
208	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
209	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₁	1	2	yes	—
211	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₁	1	2	yes	—
214	C₂×C₁₁	—	4	1	no	—
215	C₂×C₁₁	#2935	1	1	yes	characteristic
216	C₂×C₁₁	#2948	1	1	yes	characteristic
217	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
218	D₁₁	—	8	4	no	—
219	#41	—	88	2	no	—
220	#41	—	88	2	no	—
221	C₂×Q₈	—	22	2	no	—
222	C₂×D₄	—	11	4	no	—
223	C₂×D₄	—	22	4	no	—
224	C₂×D₄	—	22	2	no	—
225	C₂×D₄	—	22	2	no	—
226	C₂²×C₄	—	22	4	no	—
227	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
229	C₂²×C₄	#4339	1	1	yes	characteristic
230	D₈	—	22	4	no	—
231	D₈	—	44	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	88	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	22	2	no	—
238	C₂×C₈	—	22	2	no	—
239	C₂×C₈	—	88	4	no	—
240	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄₄	1	4	yes	—
245	C₄⋊C₄	—	11	4	no	—
246	C₄⋊C₄	—	22	2	no	—
247	C₂²⋊C₄	—	22	4	no	—
248	C₂²⋊C₄	—	22	4	no	—
249	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
250	C₄²	C₂×D₄₄	1	1	yes	characteristic
251	C₄²	#4339	1	1	yes	characteristic
252	C₄²	C₂³×D₁₁	1	1	yes	characteristic
253	C₁₁	#18091	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	22	4	no	—
255	C₂³	C₄×D₄₄	1	1	yes	characteristic
256	Q₈	—	88	2	no	—
257	D₄	—	22	4	no	—
258	D₄	—	22	4	no	—
259	D₄	—	44	4	no	—
260	D₄	—	88	2	no	—
261	D₄	—	88	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	22	4	no	—
267	C₂×C₄	—	44	4	no	—
268	C₂×C₄	—	44	4	no	—
269	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
274	C₂×C₄	C₂×#4339	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	44	2	no	—
281	C₈	—	44	2	no	—
282	C₈	#66960	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	44	4	no	—
287	C₂²	—	44	4	no	—
288	C₂²	—	44	4	no	—
289	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	44	4	no	—
294	C₄	#10622330	1	2	yes	—
295	C₄	C₂×#66960	1	1	yes	characteristic
296	C₄	#10623276	1	1	yes	characteristic
297	C₄	#10623283	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	88	4	no	—
300	C₂	#13077362	1	1	yes	characteristic
301	C₂	#13077469	1	1	yes	characteristic
302	C₂	C₂×#10622330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1451	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#13079195	C₂
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1451	D₁₁
hypocenter	C₁₁	#18091
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #13077682	C₂	semidirect
3. #13067531	C₂	semidirect
4. #13067447	C₂	semidirect
5. #13067441	C₂	semidirect
6. #13067364	C₂	semidirect
7. #13067034	C₂	semidirect
8. C₁₁×#1451	C₂	semidirect
110. #1451	D₁₁	semidirect
253. C₁₁	#18091	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13077682	2
#13067531	2
#13067447	2
#13067441	2
#13067364	2
#13067034	2
C₁₁×#1451	2
#18091	11