Subgroups — #14725270

Order 2816.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077567	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₄₄	C₂	1	1	yes	maximal, characteristic
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	#13067044	C₂	1	2	yes	maximal
7	#13066905	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#1566	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₄₄	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₂	C₂²	1	2	yes	—
11	C₁₁×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622731	—	2	2	no	—
14	M₁₆×Dic₁₁	—	2	2	no	—
15	C₂×#66882	C₄	1	1	yes	characteristic
16	C₂×#66880	C₄	1	2	yes	—
17	#10622538	C₂²	1	1	yes	characteristic
18	C₄×#4318	C₂²	1	1	yes	characteristic
19	#10622444	C₂²	1	4	yes	—
20	#10622443	C₂²	1	4	yes	—
21	#10622441	—	2	4	no	—
22	C₄×Dic₄₄	C₄	1	2	yes	—
23	C₄×Dic₄₄	C₄	1	1	yes	characteristic
24	C₄×Dic₄₄	C₄	1	1	yes	characteristic
25	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
27	C₄×Dic₄₄	C₂²	1	2	yes	—
28	C₁₁×#334	C₂²	1	2	yes	—
29	C₁₁×#334	C₂²	1	2	yes	—
30	#10622122	C₂²	1	4	yes	—
31	#10622122	C₂²	1	4	yes	—
32	#10622093	C₂²	1	2	yes	—
33	#10622092	C₂²	1	2	yes	—
34	C₂²×Dic₂₂	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	—	2	2	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
40	C₂×Dic₁₁⋊C₄	—	2	2	no	—
41	C₂×C₄×Dic₁₁	—	2	2	no	—
42	C₂×#4318	D₄	1	2	yes	—
43	C₂×Dic₄₄	—	2	4	no	—
44	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₂₂	—	2	2	no	—
48	C₄×Dic₂₂	—	2	2	no	—
49	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
50	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₂	C₂³	1	2	yes	—
52	C₄×Dic₂₂	C₂³	1	2	yes	—
53	C₁₁×C₄⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
60	#66882	C₂×C₄	1	2	yes	—
61	#66882	C₂×C₄	1	1	yes	characteristic
62	#66882	C₂×C₄	1	1	yes	characteristic
63	#66880	C₂×C₄	1	2	yes	—
64	#66880	C₂×C₄	1	2	yes	—
65	#66880	C₂×C₄	1	2	yes	—
66	#66880	C₂×C₄	1	2	yes	—
67	#66879	—	2	4	no	—
68	Dic₁₁⋊C₈	—	2	4	no	—
69	Dic₁₁⋊C₈	—	2	4	no	—
70	C₈×Dic₁₁	—	2	4	no	—
71	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
72	#66867	D₄	1	2	yes	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	#18042	—	11	1	no	maximal, Sylow
76	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
79	C₂²×Dic₁₁	—	2	2	no	—
80	C₂²×Dic₁₁	—	2	2	no	—
81	C₂×Dic₂₂	—	2	4	no	—
82	C₂×Dic₂₂	—	2	2	no	—
83	C₂×Dic₂₂	—	4	2	no	—
84	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
86	C₁₁×M₁₆	—	4	4	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	4	no	—
89	C₂×C₈×C₁₁	—	4	1	no	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₁	#41	1	1	yes	characteristic
97	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊Dic₁₁	—	2	2	no	—
99	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
101	Dic₁₁⋊C₄	—	2	2	no	—
102	Dic₁₁⋊C₄	—	2	2	no	—
103	Dic₁₁⋊C₄	—	2	2	no	—
104	Dic₁₁⋊C₄	—	2	2	no	—
105	C₄×Dic₁₁	—	2	2	no	—
106	C₄×Dic₁₁	—	2	2	no	—
107	C₄×Dic₁₁	—	2	2	no	—
108	C₄×Dic₁₁	—	2	2	no	—
109	#4318	—	2	4	no	—
110	C₂×C₁₁⋊C₈	—	2	2	no	—
111	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
112	Dic₄₄	—	2	4	no	—
113	Dic₄₄	—	2	2	no	—
114	Dic₄₄	M₁₆	1	4	yes	complemented
115	Q₈×M₁₆	—	11	2	no	—
116	C₂×C₄×Q₁₆	—	11	1	no	—
117	#1593	—	11	2	no	—
118	#1587	—	11	4	no	—
119	#1579	—	11	2	no	—
120	#1566	D₁₁	1	1	yes	characteristic
121	#1487	—	11	2	no	—
122	#1451	—	11	1	no	—
123	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	—	2	1	no	—
125	C₂×C₄×C₁₁	—	2	1	no	—
126	C₂×C₄×C₁₁	—	4	1	no	—
127	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	#137	1	4	yes	—
131	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₁₁	—	2	2	no	—
135	C₂×Dic₁₁	—	2	2	no	—
136	C₂×Dic₁₁	—	2	2	no	—
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	4	4	no	—
139	C₂×Dic₁₁	—	8	2	no	—
140	Dic₂₂	—	2	2	no	—
141	Dic₂₂	—	2	2	no	—
142	Dic₂₂	—	2	2	no	—
143	Dic₂₂	—	4	4	no	—
144	Dic₂₂	C₂×M₁₆	1	4	yes	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	4	4	no	—
149	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
150	C₁₁⋊C₈	—	2	4	no	—
151	C₁₁⋊C₈	—	4	2	no	—
152	C₂²×Q₁₆	—	11	1	no	—
153	C₂×C₄×Q₈	—	11	2	no	—
154	#446	—	11	4	no	—
155	#446	—	22	4	no	—
156	C₈×Q₈	—	11	4	no	—
157	C₄×Q₁₆	—	11	1	no	—
158	C₄×Q₁₆	—	11	2	no	—
159	C₄×Q₁₆	—	11	1	no	—
160	C₄×Q₁₆	—	11	1	no	—
161	C₄×Q₁₆	—	11	2	no	—
162	C₄×Q₁₆	—	11	1	no	—
163	#431	—	11	1	no	—
164	C₂×#107	—	11	1	no	—
165	#423	—	22	2	no	—
166	#423	C₂×D₁₁	1	2	yes	—
167	C₂×#103	—	11	2	no	—
168	C₄×M₁₆	—	11	1	no	—
169	C₄×M₁₆	—	22	2	no	—
170	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
171	#334	C₂×D₁₁	1	2	yes	—
172	#334	C₂×D₁₁	1	2	yes	—
173	Q₈⋊C₈	—	11	4	no	—
174	Q₈⋊C₈	—	11	4	no	—
175	#322	—	11	2	no	—
176	#322	—	11	2	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	#441	1	2	yes	—
180	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₁₁	—	2	1	no	—
182	C₄×C₁₁	—	2	1	no	—
183	C₄×C₁₁	—	2	1	no	—
184	C₄×C₁₁	#441	1	2	yes	—
185	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
186	C₄×C₁₁	#568	1	1	yes	characteristic
187	C₄×C₁₁	C₂×#137	1	2	yes	—
188	Dic₁₁	—	2	4	no	—
189	Dic₁₁	—	4	2	no	—
190	Dic₁₁	—	4	2	no	—
191	Dic₁₁	—	4	2	no	—
192	C₂²×Q₈	—	11	2	no	—
193	C₂×Q₁₆	—	11	1	no	—
194	C₂×Q₁₆	—	11	2	no	—
195	C₂×Q₁₆	—	11	1	no	—
196	C₂×Q₁₆	—	22	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	11	2	no	—
199	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
200	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₄×Q₈	—	11	2	no	—
202	C₄×Q₈	—	11	2	no	—
203	C₄×Q₈	—	11	2	no	—
204	C₄×Q₈	—	11	2	no	—
205	C₄×Q₈	—	22	2	no	—
206	C₄×Q₈	—	22	2	no	—
207	C₂×C₄⋊C₄	—	11	2	no	—
208	C₂×C₄⋊C₄	—	22	2	no	—
209	C₂×C₄²	—	22	2	no	—
210	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
211	#107	—	11	2	no	—
212	#107	—	11	1	no	—
213	#107	—	11	1	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	22	4	no	—
216	C₄⋊C₈	—	22	4	no	—
217	C₄⋊C₈	—	22	1	no	—
218	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
219	#103	—	11	2	no	—
220	#103	—	11	2	no	—
221	#103	—	11	2	no	—
222	#103	—	11	2	no	—
223	C₂²⋊C₈	—	22	1	no	—
224	#97	—	11	2	no	—
225	#97	—	22	4	no	—
226	C₄×C₈	—	11	2	no	—
227	C₄×C₈	—	11	2	no	—
228	C₄×C₈	—	22	4	no	—
229	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
230	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
233	C₂×C₁₁	—	2	2	no	—
234	C₂×C₁₁	D₄×M₁₆	1	1	yes	characteristic
235	C₂×C₁₁	C₂×#441	1	1	yes	characteristic
236	C₂×C₁₁	#2957	1	1	yes	characteristic
237	C₂×Q₈	—	11	2	no	—
238	C₂×Q₈	—	11	2	no	—
239	C₂×Q₈	—	22	4	no	—
240	C₂×Q₈	—	22	2	no	—
241	C₂×Q₈	—	44	2	no	—
242	C₂²×C₄	—	22	2	no	—
243	C₂²×C₄	—	22	2	no	—
244	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
246	C₂²×C₄	#4343	1	1	yes	characteristic
247	Q₁₆	—	11	4	no	—
248	Q₁₆	—	22	4	no	—
249	Q₁₆	—	22	2	no	—
250	M₁₆	—	4	4	no	—
251	M₁₆	—	22	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	11	4	no	—
256	C₂×C₈	—	22	2	no	—
257	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
261	C₄⋊C₄	—	11	2	no	—
262	C₄⋊C₄	—	11	2	no	—
263	C₄⋊C₄	—	22	2	no	—
264	C₄⋊C₄	—	22	2	no	—
265	C₄⋊C₄	—	22	2	no	—
266	C₄⋊C₄	—	22	2	no	—
267	C₄⋊C₄	—	22	2	no	—
268	C₄²	—	22	2	no	—
269	C₄²	—	22	2	no	—
270	C₄²	—	22	2	no	—
271	C₄²	—	22	2	no	—
272	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
273	C₄²	D₄×D₁₁	1	1	yes	characteristic
274	C₄²	#4343	1	1	yes	characteristic
275	C₄²	C₂³×D₁₁	1	1	yes	characteristic
276	C₁₁	#18042	1	1	yes	characteristic, complemented, Sylow
277	C₂³	#66946	1	1	yes	characteristic
278	Q₈	—	11	4	no	—
279	Q₈	—	22	2	no	—
280	Q₈	—	22	2	no	—
281	Q₈	—	22	2	no	—
282	Q₈	—	44	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	22	2	no	—
287	C₂×C₄	—	22	2	no	—
288	C₂×C₄	—	22	2	no	—
289	C₂×C₄	—	22	2	no	—
290	C₂×C₄	—	44	4	no	—
291	C₂×C₄	—	88	2	no	—
292	C₂×C₄	#66946	1	1	yes	characteristic
293	C₂×C₄	#66946	1	1	yes	characteristic
294	C₂×C₄	#66946	1	1	yes	characteristic
295	C₂×C₄	#66968	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
298	C₂×C₄	C₂×#4343	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	4	no	—
303	C₈	—	22	4	no	—
304	C₈	—	44	2	no	—
305	C₈	M₁₆×D₁₁	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#10622466	1	2	yes	—
309	C₂²	C₂×#66946	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	22	4	no	—
314	C₄	—	44	2	no	—
315	C₄	—	44	2	no	—
316	C₄	—	44	2	no	—
317	C₄	#10622466	1	2	yes	—
318	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
319	C₄	#10623276	1	1	yes	characteristic
320	C₄	C₂×#66968	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	#13077351	1	1	yes	characteristic
323	C₂	C₂×#10622466	1	1	yes	characteristic
324	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	C₁₁×#1566	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1566	D₁₁
hypocenter	C₁₁	#18042
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

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
114. Dic₄₄	M₁₆	semidirect
276. C₁₁	#18042	semidirect

Maximal subgroups

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

subgroup	index
#13077567	2
C₂×C₄×Dic₄₄	2
#13067524	2
#13067486	2
#13067434	2
#13067044	2
#13066905	2
C₁₁×#1566	2
#18042	11