Subgroups — C₂×#13067370 #14722769

Order 2816.

854 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622450	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₁	C₂	1	1	yes	maximal, characteristic
3	#13067370	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622118	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#4313	—	2	2	no	—
8	C₂²×C₈×D₁₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₁	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₁⋊C₈	—	2	2	no	—
14	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
18	#10622450	—	2	4	no	—
19	#10622450	C₂²	1	8	yes	—
20	C₄×C₈×D₁₁	C₄	1	4	yes	—
21	C₄×C₈×D₁₁	C₂²	1	4	yes	—
22	C₁₁×#334	C₂²	1	4	yes	complemented
23	#10622118	C₂²	1	8	yes	complemented
24	#10622095	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	—	2	2	no	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₁	Q₈	1	1	yes	characteristic
35	C₂²×C₁₁⋊C₈	—	2	2	no	—
36	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×#4313	—	2	8	no	—
39	C₂×#4313	—	4	8	no	—
40	C₂×C₈×D₁₁	—	2	4	no	—
41	C₂×C₈×D₁₁	—	2	4	no	—
42	C₂×C₈×D₁₁	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₁₁	C₂×C₄	1	4	yes	—
44	C₄²×D₁₁	C₂×C₄	1	2	yes	—
45	C₄²×D₁₁	D₄	1	2	yes	—
46	C₄²×D₁₁	Q₈	1	2	yes	—
47	C₄²×D₁₁	C₂³	1	2	yes	—
48	C₁₁×C₄⋊C₈	—	2	2	no	—
49	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
50	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₁	C₂³	1	2	yes	—
52	Dic₁₁⋊C₈	—	2	8	no	—
53	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
55	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
57	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
59	C₂×#1566	—	11	1	no	maximal, Sylow
60	C₂³×D₁₁	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₁	—	2	4	no	—
68	C₂×C₄×D₁₁	—	2	2	no	—
69	C₂×C₄×D₁₁	—	2	2	no	—
70	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₁	SD₁₆	1	8	yes	—
74	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×D₁₁	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	4	no	—
79	C₂×C₈×C₁₁	—	4	4	no	—
80	C₂×C₈×C₁₁	M₁₆	1	4	yes	complemented
81	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
87	C₄×Dic₁₁	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₁₁	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₁₁	C₄⋊C₄	1	2	yes	—
90	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
91	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
92	C₄×Dic₁₁	C₂×Q₈	1	2	yes	—
93	C₂×C₁₁⋊C₈	—	2	2	no	—
94	C₂×C₁₁⋊C₈	—	2	4	no	—
95	C₂×C₁₁⋊C₈	—	4	4	no	—
96	C₂×C₁₁⋊C₈	M₁₆	1	4	yes	complemented
97	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
99	#4313	—	4	16	no	—
100	C₈×D₁₁	—	2	8	no	—
101	C₈×D₁₁	—	2	8	no	—
102	C₂×#423	—	11	2	no	—
103	C₂²×C₄×C₈	—	11	1	no	—
104	#1566	—	11	8	no	—
105	C₂×#334	—	11	2	no	—
106	C₂×#334	—	11	1	no	—
107	C₂×#334	D₁₁	1	1	yes	characteristic, complemented
108	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×D₁₁	—	2	2	no	—
110	C₂²×D₁₁	—	2	2	no	—
111	C₂²×D₁₁	#106	1	4	yes	—
112	C₂²×D₁₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×SD₁₆	1	4	yes	—
120	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
123	C₂×Dic₁₁	—	2	2	no	—
124	C₂×Dic₁₁	#106	1	4	yes	—
125	C₂×Dic₁₁	#108	1	4	yes	—
126	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₄×D₁₁	—	2	2	no	—
130	C₄×D₁₁	—	2	2	no	—
131	C₄×D₁₁	—	2	4	no	—
132	C₄×D₁₁	—	2	4	no	—
133	C₄×D₁₁	—	2	4	no	—
134	C₄×D₁₁	#106	1	8	yes	—
135	C₄×D₁₁	C₂×SD₁₆	1	8	yes	—
136	C₈×C₁₁	—	2	2	no	—
137	C₈×C₁₁	—	4	4	no	—
138	C₈×C₁₁	C₂×M₁₆	1	4	yes	complemented
139	C₁₁⋊C₈	—	2	2	no	—
140	C₁₁⋊C₈	—	4	4	no	—
141	C₁₁⋊C₈	C₂×M₁₆	1	4	yes	complemented
142	C₂²×M₁₆	—	22	2	no	—
143	C₂³×C₈	—	11	1	no	—
144	C₂³×C₈	—	11	1	no	—
145	C₂²×C₄²	—	11	1	no	—
146	#423	—	11	8	no	—
147	#423	—	22	4	no	—
148	C₂×C₄⋊C₈	—	11	2	no	—
149	C₂×C₄⋊C₈	—	22	2	no	—
150	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
151	C₂×C₄×C₈	—	11	1	no	—
152	C₂×C₄×C₈	—	11	1	no	—
153	C₂×C₄×C₈	—	11	1	no	—
154	C₂×C₄×C₈	—	11	4	no	—
155	C₂×C₄×C₈	—	11	4	no	—
156	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
157	#334	—	11	8	no	—
158	#334	—	11	4	no	—
159	#334	C₂×D₁₁	1	4	yes	complemented
160	C₂²×C₁₁	#423	1	2	yes	—
161	C₂²×C₁₁	C₂×#106	1	2	yes	—
162	C₂²×C₁₁	C₂×#108	1	2	yes	—
163	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂×D₁₁	—	2	8	no	—
165	C₂×D₁₁	—	2	2	no	—
166	C₂×D₁₁	—	2	2	no	—
167	C₂×D₁₁	C₂×#106	1	4	yes	—
168	C₄×C₁₁	—	2	1	no	—
169	C₄×C₁₁	—	2	1	no	—
170	C₄×C₁₁	#423	1	2	yes	—
171	C₄×C₁₁	C₂×#106	1	2	yes	—
172	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
173	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
174	C₄×C₁₁	C₂²×SD₁₆	1	2	yes	—
175	Dic₁₁	—	2	2	no	—
176	Dic₁₁	C₂×#106	1	4	yes	—
177	Dic₁₁	C₂×#108	1	4	yes	—
178	C₂³×C₄	—	11	1	no	—
179	C₂³×C₄	—	11	1	no	—
180	C₂³×C₄	—	11	1	no	—
181	C₂×M₁₆	—	22	8	no	—
182	C₂×M₁₆	—	44	8	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	11	1	no	—
185	C₂²×C₈	—	11	1	no	—
186	C₂²×C₈	—	11	4	no	—
187	C₂²×C₈	—	11	4	no	—
188	C₂²×C₈	—	22	4	no	—
189	C₂²×C₈	—	22	4	no	—
190	C₂²×C₈	—	22	2	no	—
191	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
193	C₂×C₄²	—	11	1	no	—
194	C₂×C₄²	—	11	1	no	—
195	C₂×C₄²	—	11	1	no	—
196	C₂×C₄²	—	11	2	no	—
197	C₂×C₄²	—	11	2	no	—
198	C₂×C₄²	—	11	2	no	—
199	C₂×C₄²	—	11	2	no	—
200	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	11	4	no	—
203	C₄⋊C₈	—	22	8	no	—
204	C₄⋊C₈	—	22	2	no	—
205	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
206	C₄×C₈	—	11	4	no	—
207	C₄×C₈	—	11	4	no	—
208	C₄×C₈	—	11	2	no	—
209	C₄×C₈	—	11	2	no	—
210	C₄×C₈	C₄×D₁₁	1	2	yes	—
211	C₄×C₈	C₂²×D₁₁	1	2	yes	—
212	C₂×C₁₁	#1566	1	4	yes	complemented
213	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
214	C₂×C₁₁	C₂²×#106	1	1	yes	characteristic
215	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
216	D₁₁	—	2	4	no	—
217	C₂⁴	—	11	1	no	—
218	C₂²×C₄	—	11	1	no	—
219	C₂²×C₄	—	11	1	no	—
220	C₂²×C₄	—	11	2	no	—
221	C₂²×C₄	—	11	8	no	—
222	C₂²×C₄	—	11	2	no	—
223	C₂²×C₄	—	11	1	no	—
224	C₂²×C₄	—	11	2	no	—
225	C₂²×C₄	—	11	2	no	—
226	C₂²×C₄	—	11	2	no	—
227	C₂²×C₄	—	11	2	no	—
228	C₂²×C₄	—	22	4	no	—
229	C₂²×C₄	—	22	2	no	—
230	C₂²×C₄	—	22	2	no	—
231	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
233	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
234	M₁₆	—	44	16	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	11	4	no	—
239	C₂×C₈	—	11	2	no	—
240	C₂×C₈	—	11	2	no	—
241	C₂×C₈	—	22	8	no	—
242	C₂×C₈	—	22	2	no	—
243	C₂×C₈	—	22	8	no	—
244	C₂×C₈	—	22	4	no	—
245	C₂×C₈	—	44	4	no	—
246	C₂×C₈	#4313	1	4	yes	complemented
247	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
249	C₄²	—	11	2	no	—
250	C₄²	—	11	2	no	—
251	C₄²	—	11	2	no	—
252	C₄²	—	11	2	no	—
253	C₄²	—	11	2	no	—
254	C₄²	—	11	2	no	—
255	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
256	C₄²	D₄×D₁₁	1	1	yes	characteristic
257	C₄²	Q₈×D₁₁	1	1	yes	characteristic
258	C₄²	C₂³×D₁₁	1	1	yes	characteristic
259	C₁₁	C₂×#1566	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	11	4	no	—
261	C₂³	—	11	2	no	—
262	C₂³	—	22	2	no	—
263	C₂³	—	22	2	no	—
264	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	11	4	no	—
267	C₂×C₄	—	11	4	no	—
268	C₂×C₄	—	11	2	no	—
269	C₂×C₄	—	11	2	no	—
270	C₂×C₄	—	11	8	no	—
271	C₂×C₄	—	11	8	no	—
272	C₂×C₄	—	11	2	no	—
273	C₂×C₄	—	22	4	no	—
274	C₂×C₄	—	22	4	no	—
275	C₂×C₄	—	22	2	no	—
276	C₂×C₄	—	22	2	no	—
277	C₂×C₄	—	22	2	no	—
278	C₂×C₄	—	22	4	no	—
279	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
280	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
281	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
282	C₂×C₄	C₂×#4313	1	2	yes	—
283	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
284	C₂×C₄	SD₁₆×D₁₁	1	4	yes	—
285	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	4	4	no	—
290	C₈	—	11	4	no	—
291	C₈	—	22	2	no	—
292	C₈	—	44	4	no	—
293	C₈	C₂×#4313	1	4	yes	complemented
294	C₂²	—	11	4	no	—
295	C₂²	—	22	8	no	—
296	C₂²	—	22	2	no	—
297	C₂²	—	22	2	no	—
298	C₂²	#10622450	1	2	yes	—
299	C₂²	D₁₁×#106	1	2	yes	—
300	C₂²	D₁₁×#108	1	2	yes	—
301	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	11	4	no	—
305	C₄	—	11	4	no	—
306	C₄	—	22	2	no	—
307	C₄	#10622450	1	2	yes	—
308	C₄	D₁₁×#106	1	2	yes	—
309	C₄	C₂²×#4313	1	1	yes	characteristic
310	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
311	C₄	C₂×SD₁₆×D₁₁	1	2	yes	—
312	C₂	—	22	4	no	—
313	C₂	#13067370	1	4	yes	direct product factor
314	C₂	C₂×#10622450	1	1	yes	characteristic
315	C₂	C₂×D₁₁×#106	1	1	yes	characteristic
316	C₂	C₂×D₁₁×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#334	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂³×D₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#334	D₁₁
hypocenter	C₁₁	C₂×#1566
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #13067370	C₂	direct (N × H)
4. C₂×#10622095	C₂	semidirect
5. C₂×#10622118	C₂	semidirect
6. C₂×C₁₁×#334	C₂	semidirect
22. C₁₁×#334	C₂²	semidirect
23. #10622118	C₂²	semidirect
24. #10622095	C₂²	semidirect
80. C₂×C₈×C₁₁	M₁₆	semidirect
96. C₂×C₁₁⋊C₈	M₁₆	semidirect
107. C₂×#334	D₁₁	semidirect
138. C₈×C₁₁	C₂×M₁₆	semidirect
141. C₁₁⋊C₈	C₂×M₁₆	semidirect
159. #334	C₂×D₁₁	semidirect
212. C₂×C₁₁	#1566	semidirect
246. C₂×C₈	#4313	semidirect
259. C₁₁	C₂×#1566	semidirect
293. C₈	C₂×#4313	semidirect
313. C₂	#13067370	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×#10622450	2
C₂×C₄×C₈×D₁₁	2
#13067370	2
C₂×#10622095	2
C₂×#10622118	2
C₂×C₁₁×#334	2
C₂×#1566	11

Subgroups — C2×#13067370 #14722769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067370 #14722769