Subgroups — C₂×#13067384 #14722792

Order 2816.

758 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622537	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#4318	C₂	1	1	yes	maximal, characteristic
3	#13067384	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×#432	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
8	C₂²×#4318	—	2	1	no	—
9	C₂²×#4318	C₄	1	2	yes	—
10	C₁₁×#432	—	2	2	no	—
11	C₁₁×#432	C₂²	1	4	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₁₁×#97	—	2	1	no	—
16	#10622537	—	2	2	no	—
17	#10622537	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
19	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
20	C₄×#4318	C₄	1	4	yes	—
21	C₄×#4318	C₂²	1	4	yes	—
22	C₂×#66867	C₄	1	1	yes	characteristic
23	C₂×#66867	C₄	1	1	yes	characteristic
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	—	2	8	no	—
42	C₂×#4318	—	4	4	no	—
43	C₂×#4318	C₂×C₄	1	8	yes	—
44	C₂²×C₁₁⋊C₈	—	2	2	no	—
45	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₁×C₄⋊C₈	—	2	1	no	—
48	C₁₁×C₄⋊C₈	—	2	1	no	—
49	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
50	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
51	C₁₁×C₂²⋊C₈	—	2	4	no	—
52	C₁₁×#97	—	2	4	no	—
53	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
54	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
56	#66867	C₂×C₄	1	4	yes	—
57	#66867	C₂×C₄	1	4	yes	—
58	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
59	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
60	C₂×#1569	—	11	1	no	maximal, Sylow
61	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	—	2	4	no	—
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	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₁	—	2	2	no	—
79	C₂×C₈×C₁₁	—	2	2	no	—
80	C₂×C₈×C₁₁	—	4	2	no	—
81	C₂×C₈×C₁₁	—	4	2	no	—
82	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₁	C₂×D₄	1	2	yes	—
88	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
92	#4318	—	2	16	no	—
93	#4318	—	4	8	no	—
94	C₂×C₁₁⋊C₈	—	2	4	no	—
95	C₂×C₁₁⋊C₈	—	2	4	no	—
96	C₂×C₁₁⋊C₈	—	4	4	no	—
97	C₂×C₁₁⋊C₈	M₁₆	1	8	yes	complemented
98	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×#432	D₁₁	1	1	yes	characteristic
101	C₂×#423	—	11	1	no	—
102	C₂×C₄×M₁₆	—	11	1	no	—
103	#1569	—	11	8	no	—
104	C₂×#335	—	11	2	no	—
105	C₂×#334	—	11	2	no	—
106	C₂³×C₁₁	—	2	2	no	—
107	C₂³×C₁₁	—	2	2	no	—
108	C₂³×C₁₁	—	2	2	no	—
109	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	4	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	2	4	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	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₁₁	#137	1	2	yes	—
131	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
134	C₈×C₁₁	—	4	2	no	—
135	C₈×C₁₁	—	4	2	no	—
136	C₁₁⋊C₈	—	2	4	no	—
137	C₁₁⋊C₈	—	4	4	no	—
138	C₁₁⋊C₈	C₂×M₁₆	1	8	yes	complemented
139	C₂²×M₁₆	—	11	2	no	—
140	C₂²×M₁₆	—	22	1	no	—
141	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
142	#432	—	2	2	no	—
143	#432	C₂×D₁₁	1	4	yes	—
144	#423	—	11	4	no	—
145	#423	—	22	2	no	—
146	C₂×C₄⋊C₈	—	11	2	no	—
147	C₂×C₄⋊C₈	—	22	1	no	—
148	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	—	2	1	no	—
151	C₄×M₁₆	—	11	4	no	—
152	C₄×M₁₆	—	11	4	no	—
153	C₂×#97	—	2	1	no	—
154	C₂×#97	—	11	1	no	—
155	C₂×#97	—	11	1	no	—
156	C₂×C₄×C₈	—	11	2	no	—
157	#335	—	11	8	no	—
158	#334	—	11	8	no	—
159	C₂²×C₁₁	—	2	2	no	—
160	C₂²×C₁₁	—	2	2	no	—
161	C₂²×C₁₁	—	2	2	no	—
162	C₂²×C₁₁	—	4	4	no	—
163	C₂²×C₁₁	#423	1	2	yes	—
164	C₂²×C₁₁	#428	1	2	yes	—
165	C₂²×C₁₁	#430	1	2	yes	—
166	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₁₁	—	2	2	no	—
168	C₄×C₁₁	—	2	2	no	—
169	C₄×C₁₁	—	2	2	no	—
170	C₄×C₁₁	—	2	1	no	—
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	#423	1	2	yes	—
173	C₄×C₁₁	#428	1	2	yes	—
174	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
175	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
177	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
178	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
179	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
180	C₂×M₁₆	—	11	8	no	—
181	C₂×M₁₆	—	22	8	no	—
182	C₂×M₁₆	—	22	4	no	—
183	C₂×M₁₆	—	44	4	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	11	2	no	—
187	C₂²×C₈	—	11	2	no	—
188	C₂²×C₈	—	22	2	no	—
189	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
190	C₂×C₄²	Dic₂₂	1	2	yes	—
191	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
192	C₂×C₄²	C₄×D₁₁	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
194	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
196	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	11	4	no	—
200	C₄⋊C₈	—	22	2	no	—
201	C₄⋊C₈	—	22	4	no	—
202	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
203	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
204	C₂²⋊C₈	—	2	4	no	—
205	#97	—	2	4	no	—
206	#97	—	11	4	no	—
207	#97	—	11	4	no	—
208	C₄×C₈	—	11	4	no	—
209	C₄×C₈	—	11	4	no	—
210	C₂×C₁₁	—	4	2	no	—
211	C₂×C₁₁	#1569	1	4	yes	complemented
212	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
213	C₂×C₁₁	C₂×#428	1	1	yes	characteristic
214	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
215	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
222	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
224	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
232	M₁₆	—	22	16	no	—
233	M₁₆	—	44	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	11	8	no	—
239	C₂×C₈	—	11	4	no	—
240	C₂×C₈	—	11	4	no	—
241	C₂×C₈	—	22	4	no	—
242	C₂×C₈	—	22	4	no	—
243	C₂×C₈	—	44	4	no	—
244	C₄²	Dic₁₁⋊C₄	1	2	yes	—
245	C₄²	Dic₁₁⋊C₄	1	2	yes	—
246	C₄²	Dic₁₁⋊C₄	1	2	yes	—
247	C₄²	C₂×Dic₂₂	1	2	yes	—
248	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₁₁	1	2	yes	—
251	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
252	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
253	C₄²	C₂³×D₁₁	1	1	yes	characteristic
254	C₁₁	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×Dic₁₁⋊C₄	1	2	yes	—
259	C₂³	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	M₁₆×D₁₁	1	4	yes	—
272	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
273	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
274	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
276	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
278	C₂×C₄	#67002	1	2	yes	—
279	C₂×C₄	#67004	1	2	yes	—
280	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	11	8	no	—
286	C₈	—	22	4	no	—
287	C₈	—	44	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#10622608	1	2	yes	—
293	C₂²	#10622731	1	2	yes	—
294	C₂²	#10622737	1	2	yes	—
295	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	#10622608	1	2	yes	—
302	C₄	#10622731	1	2	yes	—
303	C₄	C₂×M₁₆×D₁₁	1	2	yes	—
304	C₄	C₂×#67002	1	1	yes	characteristic
305	C₄	C₂×#67004	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	#13067384	1	4	yes	direct product factor
308	C₂	C₂×#10622731	1	1	yes	characteristic
309	C₂	C₂×#10622608	1	1	yes	characteristic
310	C₂	C₂×#10622737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#432	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#432	D₁₁
hypocenter	C₁₁	C₂×#1569
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #13067384	C₂	direct (N × H)
4. C₂×#10622117	C₂	semidirect
5. C₂×#10622118	C₂	semidirect
25. #10622118	C₂²	semidirect
26. #10622117	C₂²	semidirect
97. C₂×C₁₁⋊C₈	M₁₆	semidirect
138. C₁₁⋊C₈	C₂×M₁₆	semidirect
211. C₂×C₁₁	#1569	semidirect
254. C₁₁	C₂×#1569	semidirect
307. C₂	#13067384	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×#10622537	2
C₂×C₄×#4318	2
#13067384	2
C₂×#10622117	2
C₂×#10622118	2
C₂×C₁₁×#432	2
C₂×#1569	11

Subgroups — C2×#13067384 #14722792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067384 #14722792