Subgroups — C₂×#13067369 #14722753

Order 2816.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#4318	C₂	1	1	yes	maximal, characteristic
3	#13067369	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622117	C₂	1	2	yes	maximal, complemented
5	C₂×#10622118	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
8	C₂²×#4318	C₄	1	1	yes	characteristic
9	C₂²×#4318	C₄	1	1	yes	characteristic
10	C₂³×C₁₁⋊C₈	—	2	1	no	—
11	C₁₁×#431	—	2	2	no	—
12	C₁₁×#431	C₂²	1	4	yes	—
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₁₁	—	2	1	no	—
17	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
19	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
21	C₄×#4318	C₄	1	4	yes	—
22	C₄×#4318	C₂²	1	4	yes	—
23	C₂×#66867	C₄	1	2	yes	—
24	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
25	#10622118	C₂²	1	8	yes	complemented
26	#10622117	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	—	2	1	no	—
31	C₂²×C₈×C₁₁	—	2	1	no	—
32	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	D₄	1	2	yes	—
36	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
40	C₂×#4318	—	2	4	no	—
41	C₂×#4318	C₈	1	8	yes	complemented
42	C₂×#4318	C₂×C₄	1	4	yes	—
43	C₂×#4318	C₂×C₄	1	4	yes	—
44	C₂²×C₁₁⋊C₈	—	2	4	no	—
45	C₂²×C₁₁⋊C₈	—	2	2	no	—
46	C₂²×C₁₁⋊C₈	—	4	4	no	—
47	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₁×C₄⋊C₈	—	2	1	no	—
50	C₁₁×C₄⋊C₈	—	2	1	no	—
51	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
52	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₁×C₂²⋊C₈	—	2	4	no	—
54	C₄×C₈×C₁₁	—	2	4	no	—
55	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
56	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
57	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
58	#66867	C₂×C₄	1	8	yes	—
59	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
60	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
61	C₂×#1565	—	11	1	no	maximal, Sylow
62	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	—	2	2	no	—
64	C₂²×C₄×C₁₁	—	2	2	no	—
65	C₂²×C₄×C₁₁	—	4	2	no	—
66	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₁₁	M₁₆	1	4	yes	—
74	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	4	2	no	—
83	C₂×C₈×C₁₁	—	4	2	no	—
84	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
87	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₁	C₂×D₄	1	2	yes	—
90	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
92	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
94	#4318	—	2	8	no	—
95	#4318	C₂×C₈	1	16	yes	complemented
96	C₂×C₁₁⋊C₈	—	2	4	no	—
97	C₂×C₁₁⋊C₈	—	2	4	no	—
98	C₂×C₁₁⋊C₈	—	4	8	no	—
99	C₂×C₁₁⋊C₈	—	4	4	no	—
100	C₂×C₁₁⋊C₈	C₂×C₈	1	8	yes	complemented
101	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×#431	D₁₁	1	1	yes	characteristic
104	C₂²×C₄⋊C₈	—	11	1	no	—
105	C₂×C₄×M₁₆	—	11	1	no	—
106	#1565	—	11	8	no	—
107	C₂×#335	—	11	2	no	—
108	C₂×#334	—	11	2	no	—
109	C₂³×C₁₁	—	2	2	no	—
110	C₂³×C₁₁	—	2	2	no	—
111	C₂³×C₁₁	C₄⋊C₈	1	4	yes	—
112	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	2	2	no	—
119	C₂×C₄×C₁₁	—	2	4	no	—
120	C₂×C₄×C₁₁	—	4	2	no	—
121	C₂×C₄×C₁₁	—	4	2	no	—
122	C₂×C₄×C₁₁	C₄⋊C₈	1	8	yes	—
123	C₂×C₄×C₁₁	C₄⋊C₈	1	4	yes	—
124	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
132	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₁₁	#137	1	2	yes	—
136	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₁₁	—	4	2	no	—
140	C₈×C₁₁	—	4	2	no	—
141	C₁₁⋊C₈	—	2	4	no	—
142	C₁₁⋊C₈	—	4	4	no	—
143	C₁₁⋊C₈	C₂²×C₈	1	8	yes	complemented
144	C₂²×M₁₆	—	11	1	no	—
145	C₂²×M₁₆	—	11	1	no	—
146	C₂³×C₈	—	22	1	no	—
147	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
148	#431	—	2	2	no	—
149	#431	C₂×D₁₁	1	4	yes	—
150	C₂×C₄⋊C₈	—	11	2	no	—
151	C₂×C₄⋊C₈	—	11	4	no	—
152	C₂×C₄⋊C₈	—	22	2	no	—
153	C₂×C₄⋊C₈	—	22	1	no	—
154	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₄×M₁₆	—	11	4	no	—
158	C₄×M₁₆	—	11	4	no	—
159	C₂×#97	—	11	2	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	11	2	no	—
162	#335	—	11	8	no	—
163	#334	—	11	8	no	—
164	C₂²×C₁₁	—	2	8	no	—
165	C₂²×C₁₁	—	2	2	no	—
166	C₂²×C₁₁	—	2	2	no	—
167	C₂²×C₁₁	C₂×C₄⋊C₈	1	4	yes	—
168	C₂²×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
169	C₂²×C₁₁	#428	1	2	yes	—
170	C₂²×C₁₁	#430	1	2	yes	—
171	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₁₁	—	2	2	no	—
173	C₄×C₁₁	—	2	2	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
177	C₄×C₁₁	C₂×C₄⋊C₈	1	4	yes	—
178	C₄×C₁₁	#428	1	2	yes	—
179	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
180	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
183	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
186	C₂×M₁₆	—	11	8	no	—
187	C₂×M₁₆	—	11	4	no	—
188	C₂×M₁₆	—	11	4	no	—
189	C₂×M₁₆	—	22	4	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	11	2	no	—
193	C₂²×C₈	—	11	2	no	—
194	C₂²×C₈	—	22	4	no	—
195	C₂²×C₈	—	22	2	no	—
196	C₂²×C₈	—	44	4	no	—
197	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
198	C₂×C₄²	Dic₂₂	1	2	yes	—
199	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₁₁	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
202	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
204	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	11	4	no	—
208	C₄⋊C₈	—	22	4	no	—
209	C₄⋊C₈	—	22	2	no	—
210	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
211	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
212	C₂²⋊C₈	—	2	4	no	—
213	#97	—	11	8	no	—
214	C₄×C₈	—	2	4	no	—
215	C₄×C₈	—	11	4	no	—
216	C₄×C₈	—	11	4	no	—
217	C₂×C₁₁	—	2	4	no	—
218	C₂×C₁₁	#1565	1	4	yes	complemented
219	C₂×C₁₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
220	C₂×C₁₁	C₂×#428	1	1	yes	characteristic
221	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
222	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₈×D₁₁	1	4	yes	—
227	C₂²×C₄	#4313	1	4	yes	—
228	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
230	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
233	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
235	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
240	M₁₆	—	11	16	no	—
241	M₁₆	—	22	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	11	8	no	—
247	C₂×C₈	—	11	4	no	—
248	C₂×C₈	—	11	4	no	—
249	C₂×C₈	—	22	4	no	—
250	C₂×C₈	—	22	4	no	—
251	C₂×C₈	—	44	8	no	—
252	C₂×C₈	—	44	4	no	—
253	C₄²	Dic₁₁⋊C₄	1	2	yes	—
254	C₄²	Dic₁₁⋊C₄	1	2	yes	—
255	C₄²	Dic₁₁⋊C₄	1	2	yes	—
256	C₄²	C₂×Dic₂₂	1	2	yes	—
257	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₁₁	1	2	yes	—
260	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
262	C₄²	C₂³×D₁₁	1	1	yes	characteristic
263	C₁₁	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	Dic₁₁⋊C₈	1	4	yes	—
267	C₂³	C₂×Dic₁₁⋊C₄	1	2	yes	—
268	C₂³	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	Dic₁₁⋊C₈	1	8	yes	—
278	C₂×C₄	Dic₁₁⋊C₈	1	4	yes	—
279	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
280	C₂×C₄	C₂×C₈×D₁₁	1	4	yes	—
281	C₂×C₄	C₂×#4313	1	2	yes	—
282	C₂×C₄	C₂×#4313	1	4	yes	—
283	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
286	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
289	C₂×C₄	#67002	1	2	yes	—
290	C₂×C₄	#67004	1	2	yes	—
291	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	—	11	8	no	—
297	C₈	—	22	4	no	—
298	C₈	—	44	4	no	—
299	C₂²	—	2	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#10622608	1	2	yes	—
303	C₂²	C₂×Dic₁₁⋊C₈	1	4	yes	—
304	C₂²	C₂×Dic₁₁⋊C₈	1	2	yes	—
305	C₂²	#10622737	1	2	yes	—
306	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	#10622608	1	2	yes	—
312	C₄	C₂×Dic₁₁⋊C₈	1	2	yes	—
313	C₄	C₂×Dic₁₁⋊C₈	1	4	yes	—
314	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
315	C₄	C₂²×#4313	1	1	yes	characteristic
316	C₄	C₂×#67002	1	1	yes	characteristic
317	C₄	C₂×#67004	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	#13067369	1	4	yes	direct product factor
320	C₂	C₂²×Dic₁₁⋊C₈	1	1	yes	characteristic
321	C₂	C₂×#10622608	1	1	yes	characteristic
322	C₂	C₂×#10622737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#431	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	C₂×#1565
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13067369	C₂	direct (N × H)
4. C₂×#10622117	C₂	semidirect
5. C₂×#10622118	C₂	semidirect
25. #10622118	C₂²	semidirect
26. #10622117	C₂²	semidirect
41. C₂×#4318	C₈	semidirect
95. #4318	C₂×C₈	semidirect
100. C₂×C₁₁⋊C₈	C₂×C₈	semidirect
143. C₁₁⋊C₈	C₂²×C₈	semidirect
218. C₂×C₁₁	#1565	semidirect
263. C₁₁	C₂×#1565	semidirect
319. C₂	#13067369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂×C₄×#4318	2
#13067369	2
C₂×#10622117	2
C₂×#10622118	2
C₂×C₁₁×#431	2
C₂×#1565	11

Subgroups — C2×#13067369 #14722753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067369 #14722753