Subgroups — C₂×#13066706 #14668643

Order 2816.

606 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622714	C₂	1	1	yes	maximal, characteristic
3	#13066706	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×#66979	C₄	1	1	yes	characteristic
7	C₂²×#4318	C₄	1	1	yes	characteristic
8	C₂×C₁₁×M₃₂	—	2	2	no	—
9	C₂×C₁₁×M₃₂	—	2	1	no	—
10	C₂×C₁₁×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₁×C₁₆	—	2	1	no	—
13	#10622714	—	2	1	no	—
14	#10622714	—	2	1	no	—
15	#10622714	C₄	1	2	yes	—
16	#10622714	C₂²	1	2	yes	—
17	C₂×#66879	—	2	1	no	—
18	C₂×C₈×Dic₁₁	—	2	1	no	—
19	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
20	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
21	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
22	C₂²×C₁₁⋊C₁₆	C₄	1	4	yes	complemented
23	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
24	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	D₄	1	2	yes	—
31	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₁	—	2	1	no	—
36	#66979	—	2	1	no	—
37	#66979	—	2	1	no	—
38	#66979	C₈	1	2	yes	—
39	#66979	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₁₁	C₈	1	2	yes	—
41	C₂×C₄×Dic₁₁	—	2	1	no	—
42	C₂×#4318	—	2	1	no	—
43	C₂×#4318	—	2	2	no	—
44	C₂×#4318	C₈	1	4	yes	—
45	C₂×#4318	C₂×C₄	1	2	yes	—
46	C₂²×C₁₁⋊C₈	—	2	1	no	—
47	C₁₁×M₃₂	—	2	4	no	—
48	C₁₁×M₃₂	—	2	2	no	—
49	C₁₁×M₃₂	C₂×C₄	1	4	yes	complemented
50	C₂×C₁₁×C₁₆	—	2	4	no	—
51	C₂×C₁₁×C₁₆	—	2	2	no	—
52	#66879	—	2	2	no	—
53	#66879	—	2	2	no	—
54	C₈×Dic₁₁	—	2	2	no	—
55	C₈×Dic₁₁	—	2	2	no	—
56	C₂×C₁₁⋊C₁₆	—	2	4	no	—
57	C₂×C₁₁⋊C₁₆	—	2	4	no	—
58	C₂×C₁₁⋊C₁₆	—	2	2	no	—
59	C₂×C₁₁⋊C₁₆	—	2	2	no	—
60	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
61	C₂×#1379	—	11	1	no	maximal, Sylow
62	C₂⁴×C₁₁	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₁₁	—	2	2	no	—
74	C₂×C₈×C₁₁	C₄²	1	2	yes	—
75	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
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	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
92	C₂²⋊Dic₁₁	—	2	2	no	—
93	C₂²⋊Dic₁₁	—	2	2	no	—
94	C₄⋊Dic₁₁	—	2	2	no	—
95	C₄⋊Dic₁₁	C₂×C₈	1	4	yes	—
96	C₄×Dic₁₁	—	2	2	no	—
97	C₄×Dic₁₁	—	2	2	no	—
98	#4318	—	2	4	no	—
99	#4318	—	2	2	no	—
100	#4318	C₂×C₈	1	4	yes	—
101	C₂×C₁₁⋊C₈	—	2	4	no	—
102	C₂×C₁₁⋊C₈	—	2	2	no	—
103	C₁₁×C₁₆	—	2	4	no	—
104	C₁₁⋊C₁₆	—	2	4	no	—
105	C₂²×M₃₂	D₁₁	1	1	yes	characteristic
106	C₂³×C₁₆	—	11	1	no	—
107	C₂×#405	—	11	1	no	—
108	#1379	—	11	4	no	—
109	C₂³×C₁₁	—	2	4	no	—
110	C₂³×C₁₁	#97	1	2	yes	—
111	C₂³×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₂³×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
113	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
114	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#95	1	2	yes	—
116	C₂×C₄×C₁₁	#95	1	2	yes	—
117	C₂×C₄×C₁₁	C₄×C₈	1	2	yes	—
118	C₂×C₄×C₁₁	C₄×C₈	1	2	yes	—
119	C₂×C₄×C₁₁	#97	1	2	yes	—
120	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
123	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
125	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
127	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
133	C₂×Dic₁₁	—	2	4	no	—
134	C₂×Dic₁₁	—	4	4	no	—
135	C₈×C₁₁	#95	1	2	yes	—
136	C₈×C₁₁	#95	1	2	yes	—
137	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₁₁⋊C₈	—	2	4	no	—
142	C₂²×M₁₆	—	11	1	no	—
143	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
144	C₂×#117	—	11	1	no	—
145	C₂×M₃₂	—	2	2	no	—
146	C₂×M₃₂	—	2	1	no	—
147	C₂×M₃₂	Dic₁₁	1	4	yes	complemented
148	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
149	C₂²×C₁₆	—	2	1	no	—
150	C₂²×C₁₆	—	11	2	no	—
151	C₂²×C₁₆	—	11	4	no	—
152	C₂²×C₁₆	—	22	1	no	—
153	C₂²×C₁₆	—	22	1	no	—
154	C₂²×C₁₆	—	22	2	no	—
155	#405	—	11	2	no	—
156	#405	—	11	2	no	—
157	#405	—	22	1	no	—
158	#405	—	22	1	no	—
159	C₂×#97	—	22	1	no	—
160	C₂×C₄×C₈	—	22	1	no	—
161	C₂²×C₁₁	—	2	2	no	—
162	C₂²×C₁₁	—	2	2	no	—
163	C₂²×C₁₁	—	2	4	no	—
164	C₂²×C₁₁	—	2	4	no	—
165	C₂²×C₁₁	#336	1	2	yes	—
166	C₂²×C₁₁	#336	1	2	yes	—
167	C₂²×C₁₁	#350	1	4	yes	—
168	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
169	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₄×C₁₁	#336	1	2	yes	—
172	C₄×C₁₁	#336	1	2	yes	—
173	C₄×C₁₁	C₂×#95	1	1	yes	characteristic
174	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
175	C₄×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
177	Dic₁₁	—	4	4	no	—
178	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
179	C₂×M₁₆	—	11	4	no	—
180	C₂×M₁₆	—	11	2	no	—
181	C₂×M₁₆	—	22	2	no	—
182	C₂×M₁₆	—	22	1	no	—
183	C₂²×C₈	—	22	1	no	—
184	C₂²×C₈	Dic₂₂	1	2	yes	—
185	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₁₁	1	2	yes	—
187	C₂²×C₈	D₄₄	1	1	yes	characteristic
188	C₂²×C₈	D₄₄	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
194	#117	—	11	2	no	—
195	#117	—	11	2	no	—
196	#117	—	22	1	no	—
197	#117	—	22	1	no	—
198	C₂×C₄⋊C₄	—	11	2	no	—
199	C₂×C₂²⋊C₄	—	22	1	no	—
200	C₂×C₄²	—	22	1	no	—
201	M₃₂	—	2	4	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	C₂×Dic₁₁	1	4	yes	complemented
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	11	4	no	—
207	C₂×C₁₆	—	22	4	no	—
208	C₂×C₁₆	—	22	4	no	—
209	C₂×C₁₆	—	22	2	no	—
210	C₂×C₁₆	—	22	2	no	—
211	#97	—	22	2	no	—
212	#97	—	22	2	no	—
213	C₄×C₈	—	22	2	no	—
214	C₄×C₈	—	22	2	no	—
215	C₂×C₁₁	—	2	2	no	—
216	C₂×C₁₁	—	2	1	no	—
217	C₂×C₁₁	—	2	1	no	—
218	C₂×C₁₁	#1379	1	4	yes	complemented
219	C₂×C₁₁	C₂×#336	1	1	yes	characteristic
220	C₂×C₁₁	C₂×#350	1	2	yes	—
221	C₂⁴	#4313	1	1	yes	characteristic
222	C₂²×C₄	—	22	2	no	—
223	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
224	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
225	C₂²×C₄	C₈×D₁₁	1	2	yes	—
226	C₂²×C₄	#4313	1	1	yes	characteristic
227	C₂²×C₄	#4313	1	2	yes	—
228	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
229	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
233	M₁₆	—	11	4	no	—
234	M₁₆	—	22	4	no	—
235	M₁₆	—	22	2	no	—
236	C₂×C₈	—	22	4	no	—
237	C₂×C₈	—	22	2	no	—
238	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
239	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
240	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
241	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
242	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
243	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
244	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
245	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
246	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
247	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
249	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
250	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
254	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
256	C₄⋊C₄	—	11	4	no	—
257	C₄⋊C₄	—	22	2	no	—
258	C₂²⋊C₄	—	22	2	no	—
259	C₂²⋊C₄	—	22	2	no	—
260	C₄²	—	22	2	no	—
261	C₄²	—	22	2	no	—
262	C₁₆	—	2	4	no	—
263	C₁₆	—	22	4	no	—
264	C₁₁	C₂×#1379	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	2	4	no	—
266	C₂³	Dic₁₁⋊C₈	1	2	yes	—
267	C₂³	#66879	1	2	yes	—
268	C₂³	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
269	C₂³	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
270	C₂³	C₂×#4313	1	1	yes	characteristic
271	C₂×C₄	—	22	4	no	—
272	C₂×C₄	—	44	4	no	—
273	C₂×C₄	C₈×Dic₁₁	1	2	yes	—
274	C₂×C₄	C₈×Dic₁₁	1	2	yes	—
275	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
276	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
277	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
278	C₂×C₄	#66879	1	2	yes	—
279	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
280	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
281	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
282	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
283	C₂×C₄	C₁₁⋊#95	1	2	yes	—
284	C₂×C₄	C₁₁⋊#95	1	2	yes	—
285	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
287	C₂×C₄	C₂×#4313	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
289	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
291	C₈	—	22	4	no	—
292	C₈	C₁₁⋊#95	1	2	yes	—
293	C₈	C₁₁⋊#95	1	2	yes	—
294	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
295	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
296	C₈	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
297	C₈	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#10622153	1	4	yes	—
303	C₂²	#10622188	1	2	yes	—
304	C₂²	#10622188	1	2	yes	—
305	C₂²	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
306	C₂²	C₂×#66879	1	1	yes	characteristic
307	C₂²	C₂×(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
308	C₄	—	44	4	no	—
309	C₄	#10622188	1	2	yes	—
310	C₄	#10622188	1	2	yes	—
311	C₄	C₂×C₈×Dic₁₁	1	1	yes	characteristic
312	C₄	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
313	C₄	C₂×(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
314	C₄	C₂×C₁₁⋊#95	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	#13066706	1	4	yes	direct product factor
319	C₂	C₂×#10622188	1	1	yes	characteristic
320	C₂	C₂×#10622153	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂²×C₁₁×M₃₂	C₂
socle	C₂³×C₁₁	C₂²⋊C₈
prime radical	C₂⁴×C₁₁	M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂²×M₃₂	D₁₁
hypocenter	C₁₁	C₂×#1379
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13066706	C₂	direct (N × H)
10. C₂×C₁₁×M₃₂	C₄	semidirect
22. C₂²×C₁₁⋊C₁₆	C₄	semidirect
49. C₁₁×M₃₂	C₂×C₄	semidirect
60. C₂×C₁₁⋊C₁₆	C₂×C₄	semidirect
147. C₂×M₃₂	Dic₁₁	semidirect
203. M₃₂	C₂×Dic₁₁	semidirect
218. C₂×C₁₁	#1379	semidirect
264. C₁₁	C₂×#1379	semidirect
318. C₂	#13066706	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₁⋊C₁₆	2
C₂×#10622714	2
#13066706	2
C₂²×C₁₁×M₃₂	2
C₂×#1379	11

Subgroups — C2×#13066706 #14668643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13066706 #14668643