Subgroups — #14725588

Order 2816.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079202	C₂	1	1	yes	maximal, characteristic
2	#13077559	C₂	1	2	yes	maximal
3	#13067524	C₂	1	2	yes	maximal, complemented
4	#13067486	C₂	1	4	yes	maximal, complemented
5	#13067434	C₂	1	2	yes	maximal, complemented
6	#13067018	C₂	1	2	yes	maximal
7	#13066905	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic
9	#10623001	C₂²	1	2	yes	—
10	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	#10622729	C₄	1	2	yes	—
13	#10622729	C₄	1	2	yes	—
14	#10622715	—	2	2	no	—
15	#10622714	—	2	2	no	—
16	#10622538	C₂²	1	1	yes	characteristic
17	C₄×#4318	C₂²	1	1	yes	characteristic
18	#10622444	C₂²	1	4	yes	—
19	#10622443	C₂²	1	4	yes	—
20	#10622441	—	2	4	no	—
21	#10622312	C₄	1	1	yes	characteristic
22	C₄×Dic₄₄	C₂²	1	2	yes	—
23	C₄×Dic₄₄	C₂²	1	2	yes	—
24	#10622295	C₄	1	1	yes	characteristic
25	#10622294	C₄	1	2	yes	—
26	C₁₁×#334	C₂²	1	1	yes	characteristic
27	C₁₁×#334	C₂²	1	1	yes	characteristic
28	C₁₁×#334	C₂²	1	2	yes	—
29	#10622122	C₂²	1	4	yes	—
30	#10622122	C₂²	1	4	yes	—
31	#10622093	C₂²	1	2	yes	—
32	#10622092	C₂²	1	2	yes	—
33	C₂²×C₈×C₁₁	—	2	2	no	—
34	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
37	#66979	—	2	2	no	—
38	#66977	—	2	2	no	—
39	#66977	C₂×C₄	1	4	yes	—
40	C₂×#4318	D₄	1	2	yes	—
41	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
43	#66923	—	2	2	no	—
44	#66922	C₂×C₄	1	2	yes	—
45	#66922	C₂×C₄	1	2	yes	—
46	C₄×Dic₂₂	—	2	2	no	—
47	C₄×Dic₂₂	C₂³	1	4	yes	—
48	C₁₁×C₄⋊C₈	—	2	2	no	—
49	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
50	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
56	#66882	C₂×C₄	1	2	yes	—
57	#66882	C₂×C₄	1	2	yes	—
58	#66880	C₂×C₄	1	4	yes	—
59	#66880	C₂×C₄	1	4	yes	—
60	#66879	—	2	4	no	—
61	Dic₁₁⋊C₈	—	2	4	no	—
62	Dic₁₁⋊C₈	—	2	4	no	—
63	C₈×Dic₁₁	—	2	4	no	—
64	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
65	#66867	D₄	1	2	yes	—
66	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
67	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
68	#18072	—	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×Dic₂₂	—	2	2	no	—
73	C₂×Dic₂₂	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	4	1	no	—
78	C₂×C₈×C₁₁	—	4	4	no	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₁	#41	1	1	yes	characteristic
86	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
87	C₂²⋊Dic₁₁	—	2	4	no	—
88	C₄⋊Dic₁₁	—	2	2	no	—
89	C₄⋊Dic₁₁	—	2	2	no	—
90	C₄⋊Dic₁₁	—	2	2	no	—
91	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
92	Dic₁₁⋊C₄	—	2	4	no	—
93	Dic₁₁⋊C₄	—	2	4	no	—
94	C₄×Dic₁₁	—	2	4	no	—
95	#4318	—	2	4	no	—
96	C₂×C₁₁⋊C₈	—	2	2	no	—
97	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
98	Dic₄₄	—	2	4	no	—
99	Dic₄₄	—	4	2	no	—
100	#3170	—	11	1	no	—
101	#2962	—	11	2	no	—
102	#1593	—	11	2	no	—
103	#1587	—	11	4	no	—
104	#1579	—	11	2	no	—
105	C₂×#334	D₁₁	1	1	yes	characteristic
106	#1481	—	11	2	no	—
107	#1451	—	11	1	no	—
108	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	—	2	1	no	—
110	C₂×C₄×C₁₁	—	2	1	no	—
111	C₂×C₄×C₁₁	—	4	1	no	—
112	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#137	1	4	yes	—
116	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₁₁	—	2	4	no	—
120	C₂×Dic₁₁	—	2	4	no	—
121	Dic₂₂	—	2	4	no	—
122	Dic₂₂	—	2	4	no	—
123	Dic₂₂	—	8	2	no	—
124	C₈×C₁₁	—	2	1	no	—
125	C₈×C₁₁	—	2	1	no	—
126	C₈×C₁₁	—	2	1	no	—
127	C₈×C₁₁	—	4	2	no	—
128	C₈×C₁₁	—	4	2	no	—
129	C₈×C₁₁	#131	1	2	yes	—
130	C₁₁⋊C₈	—	2	4	no	—
131	C₁₁⋊C₈	—	4	2	no	—
132	#533	—	11	2	no	—
133	#500	—	11	1	no	—
134	#494	—	11	1	no	—
135	#487	—	11	2	no	—
136	#467	—	11	2	no	—
137	#467	—	11	2	no	—
138	#446	—	11	4	no	—
139	#446	—	22	4	no	—
140	C₈×Q₈	—	11	4	no	—
141	C₄×Q₁₆	—	11	2	no	—
142	C₄×Q₁₆	—	11	2	no	—
143	#431	—	11	1	no	—
144	#424	—	22	2	no	—
145	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
146	#405	—	22	2	no	—
147	C₄×M₁₆	—	11	1	no	—
148	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
149	#334	C₂×D₁₁	1	1	yes	characteristic
150	#334	C₂×D₁₁	1	2	yes	—
151	#334	C₂×D₁₁	1	1	yes	characteristic
152	Q₈⋊C₈	—	11	4	no	—
153	Q₈⋊C₈	—	11	4	no	—
154	#322	—	11	2	no	—
155	#322	—	11	2	no	—
156	C₂²×C₁₁	—	2	1	no	—
157	C₂²×C₁₁	—	2	1	no	—
158	C₂²×C₁₁	#441	1	2	yes	—
159	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₁₁	—	2	1	no	—
161	C₄×C₁₁	—	2	1	no	—
162	C₄×C₁₁	—	2	1	no	—
163	C₄×C₁₁	#441	1	2	yes	—
164	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
165	C₄×C₁₁	#568	1	1	yes	characteristic
166	C₄×C₁₁	C₂×#137	1	2	yes	—
167	Dic₁₁	—	4	4	no	—
168	Dic₁₁	—	4	4	no	—
169	C₂×Q₁₆	—	11	2	no	—
170	C₂×Q₁₆	—	11	2	no	—
171	C₂×M₁₆	—	11	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
174	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
175	C₄⋊Q₈	—	11	2	no	—
176	C₄⋊Q₈	—	11	2	no	—
177	#125	—	22	2	no	—
178	C₂²⋊Q₈	—	11	4	no	—
179	C₂²⋊Q₈	—	22	2	no	—
180	C₄×Q₈	—	11	4	no	—
181	C₄×Q₈	—	22	2	no	—
182	#117	—	22	2	no	—
183	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
184	#107	—	11	2	no	—
185	#107	—	11	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	22	4	no	—
188	C₄⋊C₈	—	22	4	no	—
189	C₄⋊C₈	—	22	1	no	—
190	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
191	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
192	#103	—	11	4	no	—
193	#103	—	11	4	no	—
194	C₂²⋊C₈	—	22	1	no	—
195	#97	—	11	2	no	—
196	#97	—	22	4	no	—
197	C₄×C₈	—	11	2	no	—
198	C₄×C₈	—	11	2	no	—
199	C₄×C₈	—	22	4	no	—
200	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
204	C₂×C₁₁	—	2	2	no	—
205	C₂×C₁₁	#2935	1	1	yes	characteristic
206	C₂×C₁₁	C₂×#441	1	1	yes	characteristic
207	C₂×C₁₁	#2957	1	1	yes	characteristic
208	C₂×Q₈	—	11	4	no	—
209	C₂×Q₈	—	22	2	no	—
210	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
212	C₂²×C₄	#4343	1	1	yes	characteristic
213	Q₁₆	—	22	4	no	—
214	Q₁₆	—	44	2	no	—
215	M₁₆	—	22	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	11	4	no	—
222	C₂×C₈	—	22	2	no	—
223	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
227	C₄⋊C₄	—	11	4	no	—
228	C₄⋊C₄	—	22	2	no	—
229	C₄⋊C₄	—	22	2	no	—
230	C₄⋊C₄	—	22	4	no	—
231	C₄⋊C₄	—	22	2	no	—
232	C₄⋊C₄	—	22	4	no	—
233	C₂²⋊C₄	—	22	4	no	—
234	C₄²	—	22	4	no	—
235	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
236	C₄²	D₄×D₁₁	1	1	yes	characteristic
237	C₄²	#4343	1	1	yes	characteristic
238	C₄²	C₂³×D₁₁	1	1	yes	characteristic
239	C₁₁	#18072	1	1	yes	characteristic, complemented, Sylow
240	C₂³	#66946	1	1	yes	characteristic
241	Q₈	—	22	4	no	—
242	Q₈	—	22	4	no	—
243	Q₈	—	88	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	22	4	no	—
248	C₂×C₄	—	22	4	no	—
249	C₂×C₄	#66946	1	1	yes	characteristic
250	C₂×C₄	#66946	1	1	yes	characteristic
251	C₂×C₄	#66946	1	1	yes	characteristic
252	C₂×C₄	#66968	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
255	C₂×C₄	C₂×#4343	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	22	4	no	—
262	C₈	—	44	2	no	—
263	C₈	#66954	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#10622466	1	2	yes	—
267	C₂²	C₂×#66946	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	44	4	no	—
272	C₄	—	44	4	no	—
273	C₄	#10622466	1	2	yes	—
274	C₄	C₂×#66954	1	1	yes	characteristic
275	C₄	#10623276	1	1	yes	characteristic
276	C₄	C₂×#66968	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#13077359	1	1	yes	characteristic
279	C₂	C₂×#10622466	1	1	yes	characteristic
280	C₂	#13077532	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#334	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#334	D₁₁
hypocenter	C₁₁	#18072
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #13067524	C₂	semidirect
4. #13067486	C₂	semidirect
5. #13067434	C₂	semidirect
239. C₁₁	#18072	semidirect

Maximal subgroups

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

subgroup	index
#13079202	2
#13077559	2
#13067524	2
#13067486	2
#13067434	2
#13067018	2
#13066905	2
C₂×C₁₁×#334	2
#18072	11