Subgroups — C₂×#13066916 #14719397

Order 2816.

802 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622538	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622538	C₂	1	1	yes	maximal, characteristic
3	#13066916	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622092	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10622093	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	C₁₁×#431	C₄	1	4	yes	—
11	C₁₁×#431	C₂²	1	4	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₁₁×C₂²⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
17	C₂×C₂²⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
18	#10622538	C₄	1	4	yes	—
19	#10622538	C₄	1	4	yes	—
20	#10622538	C₂²	1	4	yes	—
21	#10622538	C₂²	1	4	yes	—
22	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
23	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
24	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
28	#10622093	C₂²	1	4	yes	complemented
29	#10622092	C₂²	1	4	yes	complemented
30	#10622092	C₂²	1	4	yes	complemented
31	C₈×C₁₁⋊C₈	C₂²	1	4	yes	complemented
32	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
45	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₁₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₁	C₂³	1	2	yes	—
54	C₄×C₈×C₁₁	C₂³	1	2	yes	—
55	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
56	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
57	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
58	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
59	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
60	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
64	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
65	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
66	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
67	C₂×#1454	—	11	1	no	maximal, Sylow
68	C₂⁴×C₁₁	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	—	2	4	no	—
70	C₂²×C₄×C₁₁	—	2	4	no	—
71	C₂²×C₄×C₁₁	—	2	4	no	—
72	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
76	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
77	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
78	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₁	—	2	4	no	—
85	C₂×C₈×C₁₁	—	2	4	no	—
86	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₁	C₄²	1	2	yes	—
89	C₄²×C₁₁	C₄²	1	2	yes	—
90	C₄²×C₁₁	C₄²	1	2	yes	—
91	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
98	C₂×C₁₁⋊C₈	—	2	4	no	—
99	C₂×C₁₁⋊C₈	—	2	4	no	—
100	C₂×C₁₁⋊C₈	—	2	4	no	—
101	C₂×C₁₁⋊C₈	—	2	4	no	—
102	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×#431	—	11	1	no	—
107	C₂×#431	—	11	1	no	—
108	C₂×#431	D₁₁	1	1	yes	characteristic
109	#1454	—	11	8	no	—
110	C₂×#322	—	11	1	no	—
111	C₂×#322	—	11	1	no	—
112	C₂×#322	—	11	1	no	—
113	C₂×C₈²	—	11	1	no	—
114	C₂³×C₁₁	—	2	2	no	—
115	C₂³×C₁₁	—	2	2	no	—
116	C₂³×C₁₁	—	2	2	no	—
117	C₂³×C₁₁	C₂×C₄²	1	2	yes	—
118	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	—	2	2	no	—
121	C₂×C₄×C₁₁	—	2	2	no	—
122	C₂×C₄×C₁₁	—	2	4	no	—
123	C₂×C₄×C₁₁	—	2	4	no	—
124	C₂×C₄×C₁₁	—	2	4	no	—
125	C₂×C₄×C₁₁	—	2	4	no	—
126	C₂×C₄×C₁₁	—	2	4	no	—
127	C₂×C₄×C₁₁	—	2	4	no	—
128	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₁₁	#131	1	4	yes	—
135	C₂×C₄×C₁₁	#131	1	4	yes	—
136	C₂×C₄×C₁₁	#131	1	4	yes	—
137	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₁₁	—	2	4	no	—
141	C₈×C₁₁	—	2	4	no	—
142	C₁₁⋊C₈	—	2	4	no	—
143	C₁₁⋊C₈	—	2	4	no	—
144	C₁₁⋊C₈	—	2	4	no	—
145	C₁₁⋊C₈	—	2	4	no	—
146	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
147	#431	—	11	4	no	—
148	#431	—	11	4	no	—
149	#431	—	11	4	no	—
150	#431	—	11	4	no	—
151	#431	Dic₁₁	1	4	yes	—
152	#431	C₂×D₁₁	1	4	yes	—
153	C₂×C₄⋊C₈	—	11	2	no	—
154	C₂×C₄⋊C₈	—	11	2	no	—
155	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
156	C₂×C₂²⋊C₈	—	11	2	no	—
157	C₂×C₂²⋊C₈	—	11	2	no	—
158	C₂×C₂²⋊C₈	Dic₁₁	1	2	yes	—
159	C₂×C₄×C₈	—	11	1	no	—
160	C₂×C₄×C₈	—	11	1	no	—
161	C₂×C₄×C₈	—	11	1	no	—
162	C₂×C₄×C₈	—	11	1	no	—
163	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
165	#322	—	11	4	no	—
166	#322	—	11	4	no	—
167	#322	—	11	4	no	—
168	C₈²	—	11	4	no	—
169	C₂²×C₁₁	—	2	2	no	—
170	C₂²×C₁₁	—	2	2	no	—
171	C₂²×C₁₁	—	2	2	no	—
172	C₂²×C₁₁	—	4	4	no	—
173	C₂²×C₁₁	#405	1	2	yes	—
174	C₂²×C₁₁	#405	1	2	yes	—
175	C₂²×C₁₁	#405	1	2	yes	—
176	C₂²×C₁₁	C₂²×C₄²	1	1	yes	characteristic
177	C₄×C₁₁	—	2	2	no	—
178	C₄×C₁₁	—	2	2	no	—
179	C₄×C₁₁	—	2	2	no	—
180	C₄×C₁₁	#405	1	2	yes	—
181	C₄×C₁₁	#405	1	2	yes	—
182	C₄×C₁₁	#405	1	2	yes	—
183	C₄×C₁₁	C₂×#131	1	2	yes	—
184	C₄×C₁₁	C₂×#131	1	2	yes	—
185	C₄×C₁₁	C₂×#131	1	2	yes	—
186	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
187	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
188	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
189	C₂²×C₈	—	11	2	no	—
190	C₂²×C₈	—	11	2	no	—
191	C₂²×C₈	—	11	2	no	—
192	C₂²×C₈	—	11	2	no	—
193	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
194	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
195	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₁₁	1	2	yes	—
198	C₂×C₄²	C₄×D₁₁	1	2	yes	—
199	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
201	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
202	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
203	C₄⋊C₈	—	11	8	no	—
204	C₄⋊C₈	—	11	8	no	—
205	C₄⋊C₈	C₂×Dic₁₁	1	8	yes	—
206	C₂²⋊C₈	—	11	8	no	—
207	C₂²⋊C₈	—	11	8	no	—
208	C₂²⋊C₈	C₂×Dic₁₁	1	8	yes	—
209	C₄×C₈	—	11	2	no	—
210	C₄×C₈	—	11	2	no	—
211	C₄×C₈	—	11	2	no	—
212	C₄×C₈	—	11	2	no	—
213	C₄×C₈	—	11	2	no	—
214	C₄×C₈	—	11	2	no	—
215	C₄×C₈	—	11	2	no	—
216	C₄×C₈	—	11	2	no	—
217	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
218	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
219	C₄×C₈	C₂²×D₁₁	1	2	yes	—
220	C₄×C₈	C₂²×D₁₁	1	2	yes	—
221	C₂×C₁₁	—	4	2	no	—
222	C₂×C₁₁	#1454	1	4	yes	complemented
223	C₂×C₁₁	C₂×#405	1	1	yes	characteristic
224	C₂×C₁₁	C₂×#405	1	1	yes	characteristic
225	C₂×C₁₁	C₂×#405	1	1	yes	characteristic
226	C₂⁴	C₄×Dic₁₁	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
231	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
232	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
233	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
234	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
235	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	11	4	no	—
245	C₂×C₈	—	11	4	no	—
246	C₂×C₈	—	11	4	no	—
247	C₂×C₈	—	11	4	no	—
248	C₂×C₈	—	22	4	no	—
249	C₂×C₈	—	22	4	no	—
250	C₂×C₈	—	22	4	no	—
251	C₂×C₈	—	22	4	no	—
252	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
253	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
254	C₄²	C₄×Dic₁₁	1	2	yes	—
255	C₄²	C₄×Dic₁₁	1	2	yes	—
256	C₄²	C₄×Dic₁₁	1	2	yes	—
257	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₁₁	1	2	yes	—
260	C₄²	C₂×C₄×D₁₁	1	2	yes	—
261	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
262	C₄²	C₂²×Dic₁₁	1	2	yes	—
263	C₄²	C₂³×D₁₁	1	1	yes	characteristic
264	C₁₁	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₄×Dic₁₁	1	2	yes	—
269	C₂³	C₂×C₄×Dic₁₁	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	#66954	1	4	yes	—
280	C₂×C₄	#66960	1	4	yes	—
281	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
282	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
284	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
287	C₂×C₄	#67001	1	4	yes	—
288	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
290	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	22	4	no	—
294	C₈	—	22	4	no	—
295	C₈	—	22	4	no	—
296	C₈	—	22	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	#10622610	1	2	yes	—
302	C₂²	#10622714	1	2	yes	—
303	C₂²	#10622735	1	2	yes	—
304	C₂²	C₂²×C₄×Dic₁₁	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	#10622610	1	2	yes	—
309	C₄	#10622714	1	2	yes	—
310	C₄	#10622735	1	2	yes	—
311	C₄	C₂×#66954	1	2	yes	—
312	C₄	C₂×#66960	1	2	yes	—
313	C₄	C₂×#67001	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	#13066916	1	4	yes	direct product factor
316	C₂	C₂×#10622714	1	1	yes	characteristic
317	C₂	C₂×#10622610	1	1	yes	characteristic
318	C₂	C₂×#10622735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#431	C₂
socle	C₂³×C₁₁	C₂×C₄²
prime radical	C₂⁴×C₁₁	C₄²
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	C₂×#1454
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13066916	C₂	direct (N × H)
4. C₂×#10622092	C₂	semidirect
5. C₂×#10622092	C₂	semidirect
6. C₂×#10622093	C₂	semidirect
7. C₂×C₈×C₁₁⋊C₈	C₂	semidirect
28. #10622093	C₂²	semidirect
29. #10622092	C₂²	semidirect
30. #10622092	C₂²	semidirect
31. C₈×C₁₁⋊C₈	C₂²	semidirect
222. C₂×C₁₁	#1454	semidirect
264. C₁₁	C₂×#1454	semidirect
315. C₂	#13066916	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10622538	2
#13066916	2
C₂×#10622092	2
C₂×#10622093	2
C₂×C₈×C₁₁⋊C₈	2
C₂×C₁₁×#431	2
C₂×#1454	11

Subgroups — C2×#13066916 #14719397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13066916 #14719397