Subgroups — #14646608

Order 2816.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068596	C₂	1	2	yes	maximal
2	C₂×#10622095	C₂	1	2	yes	maximal, complemented
3	#13066451	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	—	2	4	no	—
9	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
15	#10622162	—	2	2	no	—
16	#10622095	—	2	4	no	—
17	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	—	2	2	no	—
23	C₂²×C₈×C₁₁	—	2	2	no	—
24	C₂²×C₈×C₁₁	—	2	2	no	—
25	C₂²×C₈×C₁₁	C₈	1	4	yes	complemented
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	—	2	2	no	—
31	C₂×C₄²×C₁₁	—	2	2	no	—
32	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
36	C₂²×C₁₁⋊C₈	—	2	4	no	—
37	C₄×C₈×C₁₁	—	2	2	no	—
38	C₄×C₈×C₁₁	—	2	2	no	—
39	C₄×C₈×C₁₁	—	2	2	no	—
40	C₄×C₈×C₁₁	D₄	1	4	yes	—
41	C₂²⋊(C₁₁⋊C₈)	—	4	4	no	—
42	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
44	#8595	—	11	1	no	maximal, Sylow
45	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	—	2	2	no	—
47	C₂²×C₄×C₁₁	—	2	2	no	—
48	C₂²×C₄×C₁₁	—	2	1	no	—
49	C₂²×C₄×C₁₁	—	2	1	no	—
50	C₂²×C₄×C₁₁	—	2	2	no	—
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	—	2	1	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	—	2	2	no	—
55	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
60	C₂²×C₄×C₁₁	SD₁₆	1	4	yes	—
61	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₁	—	2	8	no	—
65	C₂×C₈×C₁₁	—	2	8	no	—
66	C₂×C₈×C₁₁	—	2	1	no	—
67	C₂×C₈×C₁₁	—	2	1	no	—
68	C₂×C₈×C₁₁	—	2	1	no	—
69	C₂×C₈×C₁₁	—	2	1	no	—
70	C₂×C₈×C₁₁	—	2	1	no	—
71	C₂×C₈×C₁₁	—	2	1	no	—
72	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
77	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
78	C₄²×C₁₁	—	2	1	no	—
79	C₄²×C₁₁	—	2	1	no	—
80	C₄²×C₁₁	—	2	2	no	—
81	C₄²×C₁₁	—	2	2	no	—
82	C₄²×C₁₁	C₂×D₄	1	2	yes	—
83	C₄²×C₁₁	#41	1	2	yes	—
84	C₂×C₁₁⋊C₈	—	4	4	no	—
85	C₂×C₁₁⋊C₈	—	4	4	no	—
86	C₂×C₁₁⋊C₈	—	4	4	no	—
87	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
88	#1797	—	11	2	no	—
89	C₂×#334	—	11	2	no	—
90	#1277	—	11	2	no	—
91	C₂³×C₁₁	—	2	1	no	—
92	C₂³×C₁₁	—	2	1	no	—
93	C₂³×C₁₁	—	2	1	no	—
94	C₂³×C₁₁	—	2	1	no	—
95	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
96	C₂³×C₁₁	#106	1	2	yes	—
97	C₂³×C₁₁	#108	1	2	yes	—
98	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₁	—	2	8	no	—
100	C₂×C₄×C₁₁	—	2	1	no	—
101	C₂×C₄×C₁₁	—	2	1	no	—
102	C₂×C₄×C₁₁	—	2	1	no	—
103	C₂×C₄×C₁₁	—	2	1	no	—
104	C₂×C₄×C₁₁	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	2	8	no	—
113	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₁₁	#106	1	2	yes	—
115	C₂×C₄×C₁₁	#108	1	2	yes	—
116	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#121	1	2	yes	—
119	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₁₁	#135	1	2	yes	—
126	C₈×C₁₁	—	2	2	no	—
127	C₈×C₁₁	—	2	2	no	—
128	C₈×C₁₁	—	2	2	no	—
129	C₈×C₁₁	C₂²⋊C₈	1	4	yes	complemented
130	C₁₁⋊C₈	—	8	4	no	—
131	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
132	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
133	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	11	4	no	—
135	C₂×C₂²⋊C₈	—	22	2	no	—
136	C₂×C₄×C₈	—	2	4	no	—
137	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
138	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
141	#336	—	22	2	no	—
142	#334	—	22	4	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	—	2	2	no	—
145	C₂²×C₁₁	—	2	2	no	—
146	C₂²×C₁₁	—	2	2	no	—
147	C₂²×C₁₁	—	2	2	no	—
148	C₂²×C₁₁	—	2	2	no	—
149	C₂²×C₁₁	#334	1	4	yes	complemented
150	C₂²×C₁₁	#380	1	1	yes	characteristic
151	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂²×C₁₁	#423	1	1	yes	characteristic
153	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
154	C₂²×C₁₁	#427	1	1	yes	characteristic
155	C₂²×C₁₁	C₂×#108	1	1	yes	characteristic
156	C₂²×C₁₁	C₂×#108	1	1	yes	characteristic
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	2	2	no	—
159	C₄×C₁₁	—	2	2	no	—
160	C₄×C₁₁	—	2	2	no	—
161	C₄×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₁₁	#407	1	1	yes	characteristic
163	C₄×C₁₁	#431	1	1	yes	characteristic
164	C₄×C₁₁	#432	1	1	yes	characteristic
165	C₄×C₁₁	#465	1	4	yes	—
166	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
167	C₂³×C₄	D₄₄	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	22	4	no	—
175	C₂²×C₈	C₁₁⋊C₈	1	4	yes	complemented
176	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
183	C₂×C₄²	D₄₄	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
186	C₄⋊C₈	—	22	4	no	—
187	C₄⋊C₈	—	22	4	no	—
188	C₂²⋊C₈	—	44	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	C₂²⋊D₁₁	1	4	yes	—
193	C₂×C₁₁	—	2	4	no	—
194	C₂×C₁₁	C₂×#334	1	1	yes	characteristic
195	C₂×C₁₁	#1564	1	1	yes	characteristic
196	C₂×C₁₁	#1566	1	1	yes	characteristic
197	C₂×C₁₁	#1568	1	1	yes	characteristic
198	C₂×C₁₁	#1797	1	1	yes	characteristic
199	C₂×C₁₁	#1807	1	1	yes	characteristic
200	C₂×C₁₁	#1810	1	1	yes	characteristic
201	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	#4314	1	4	yes	—
212	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
213	C₂²×C₄	#4318	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	44	4	no	—
229	C₂×C₈	—	44	4	no	—
230	C₂×C₈	—	44	4	no	—
231	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
232	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
233	C₂×C₈	#4318	1	2	yes	—
234	C₂×C₈	#4318	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
236	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	#4339	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
243	C₁₁	#8595	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
249	C₂³	#66881	1	2	yes	—
250	C₂³	#66883	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
267	C₂×C₄	#66881	1	2	yes	—
268	C₂×C₄	#66883	1	2	yes	—
269	C₂×C₄	C₂×#4314	1	2	yes	—
270	C₂×C₄	#66957	1	2	yes	—
271	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#4318	1	1	yes	characteristic
273	C₂×C₄	C₂×#4318	1	1	yes	characteristic
274	C₂×C₄	C₂×#4318	1	1	yes	characteristic
275	C₂×C₄	#66977	1	2	yes	—
276	C₂×C₄	#66979	1	1	yes	characteristic
277	C₂×C₄	#66983	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	88	4	no	—
283	C₈	C₂²⋊(C₁₁⋊C₈)	1	4	yes	complemented
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	#10622095	1	4	yes	complemented
291	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
292	C₂²	#10622537	1	1	yes	characteristic
293	C₂²	C₂×#66881	1	1	yes	characteristic
294	C₂²	#10622719	1	1	yes	characteristic
295	C₂²	C₂×#66883	1	1	yes	characteristic
296	C₂²	C₂×#66883	1	1	yes	characteristic
297	C₂²	#10622825	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	#10622538	1	1	yes	characteristic
303	C₄	#10622539	1	1	yes	characteristic
304	C₄	#10622727	1	4	yes	—
305	C₄	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
306	C₄	#10622820	1	1	yes	characteristic
307	C₂	—	2	4	no	—
308	C₂	C₂×#10622095	1	1	yes	characteristic
309	C₂	#13067364	1	1	yes	characteristic
310	C₂	#13067372	1	1	yes	characteristic
311	C₂	#13067379	1	1	yes	characteristic
312	C₂	#13068596	1	1	yes	characteristic
313	C₂	#13068641	1	1	yes	characteristic
314	C₂	#13068654	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	#8595
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10622095	C₂	semidirect
3. #13066451	C₂	semidirect
25. C₂²×C₈×C₁₁	C₈	semidirect
129. C₈×C₁₁	C₂²⋊C₈	semidirect
149. C₂²×C₁₁	#334	semidirect
175. C₂²×C₈	C₁₁⋊C₈	semidirect
243. C₁₁	#8595	semidirect
283. C₈	C₂²⋊(C₁₁⋊C₈)	semidirect
290. C₂²	#10622095	semidirect

Maximal subgroups

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

subgroup	index
#13068596	2
C₂×#10622095	2
#13066451	2
C₂²×C₄×C₈×C₁₁	2
#8595	11