Subgroups — D₁₁×#344 #13068356

Order 1408.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₁	C₂	1	1	yes	maximal, characteristic
3	D₁₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#344	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622197	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622194	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622171	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
11	C₁₁×#117	C₂²	1	1	yes	characteristic
12	#66979	C₂²	1	1	yes	characteristic
13	C₂×#4318	C₂²	1	1	yes	characteristic
14	C₂×#4318	C₂²	1	1	yes	characteristic
15	M₁₆×D₁₁	—	2	1	no	—
16	M₁₆×D₁₁	—	2	1	no	—
17	M₁₆×D₁₁	—	2	1	no	—
18	M₁₆×D₁₁	—	2	1	no	—
19	M₁₆×D₁₁	—	2	1	no	—
20	M₁₆×D₁₁	—	2	1	no	—
21	M₁₆×D₁₁	C₄	1	2	yes	complemented
22	M₁₆×D₁₁	C₄	1	2	yes	complemented
23	C₂×#4313	—	2	1	no	—
24	C₂×#4313	C₄	1	2	yes	complemented
25	C₂×C₈×D₁₁	—	2	1	no	—
26	C₂×C₈×D₁₁	C₄	1	2	yes	complemented
27	#66945	—	2	1	no	—
28	C₄⋊C₄×D₁₁	—	2	1	no	—
29	C₂²⋊C₄×D₁₁	C₄	1	2	yes	—
30	#66930	C₄	1	2	yes	—
31	#66925	—	2	1	no	—
32	C₄²×D₁₁	—	2	1	no	—
33	C₂³×D₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₁	—	2	2	no	—
37	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
42	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
43	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
44	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
45	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₁	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₁	Q₈	1	1	yes	characteristic
48	C₂×C₄×D₁₁	Q₈	1	1	yes	characteristic
49	C₁₁×M₁₆	—	2	1	no	—
50	C₁₁×M₁₆	—	2	1	no	—
51	C₁₁×M₁₆	—	2	1	no	—
52	C₁₁×M₁₆	C₂×C₄	1	2	yes	complemented
53	C₂×C₈×C₁₁	—	2	1	no	—
54	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	complemented
55	C₁₁×C₄⋊C₄	—	2	1	no	—
56	C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₄²×C₁₁	—	2	1	no	—
58	C₂²⋊Dic₁₁	C₂×C₄	1	2	yes	—
59	(C₂×D₁₁)⋊C₄	—	2	2	no	—
60	C₄⋊Dic₁₁	—	2	1	no	—
61	Dic₁₁⋊C₄	—	2	2	no	—
62	C₄×Dic₁₁	—	2	2	no	—
63	C₄×Dic₁₁	—	2	1	no	—
64	#4318	—	2	1	no	—
65	#4318	—	2	1	no	—
66	#4318	—	2	1	no	—
67	#4318	C₂×C₄	1	2	yes	complemented
68	C₂×C₁₁⋊C₈	—	2	1	no	—
69	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	complemented
70	#4313	—	2	2	no	—
71	#4313	—	2	2	no	—
72	#4313	—	2	2	no	—
73	#4313	—	2	2	no	—
74	C₈×D₁₁	—	2	2	no	—
75	C₈×D₁₁	—	2	2	no	—
76	C₈×D₁₁	—	2	2	no	—
77	C₈×D₁₁	—	2	2	no	—
78	C₂×#344	—	11	1	no	maximal, Sylow
79	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₁	—	4	2	no	—
81	C₂²×D₁₁	C₄²	1	1	yes	characteristic
82	C₂²×D₁₁	C₄²	1	1	yes	characteristic
83	C₂²×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×D₁₁	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×D₁₁	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁	—	2	2	no	—
88	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
94	C₂×Dic₁₁	—	2	2	no	—
95	C₂×Dic₁₁	C₄²	1	1	yes	characteristic
96	C₂×Dic₁₁	C₄²	1	1	yes	characteristic
97	C₂×Dic₁₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×Dic₁₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₁₁	—	4	4	no	—
102	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×D₁₁	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
111	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
112	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
113	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
114	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
115	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
116	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
117	C₄×D₁₁	C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₁₁	—	2	2	no	—
119	C₈×C₁₁	—	2	2	no	—
120	C₁₁⋊C₈	—	2	2	no	—
121	C₁₁⋊C₈	—	2	2	no	—
122	C₂²×M₁₆	—	11	1	no	—
123	C₂²×M₁₆	—	11	1	no	—
124	C₂×#117	—	11	1	no	—
125	#344	—	11	1	no	—
126	#344	—	11	1	no	—
127	#344	—	11	1	no	—
128	#344	D₁₁	1	1	yes	characteristic, direct product factor
129	C₂²×C₁₁	—	4	1	no	—
130	C₂²×C₁₁	C₂×C₄²	1	1	yes	characteristic
131	C₂²×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×D₁₁	—	2	2	no	—
134	C₂×D₁₁	—	2	2	no	—
135	C₂×D₁₁	—	2	2	no	—
136	C₂×D₁₁	—	4	1	no	—
137	C₂×D₁₁	—	4	1	no	—
138	C₂×D₁₁	—	4	1	no	—
139	C₂×D₁₁	#95	1	1	yes	characteristic
140	C₂×D₁₁	#95	1	1	yes	characteristic
141	C₂×D₁₁	#95	1	1	yes	characteristic
142	C₂×D₁₁	#95	1	1	yes	characteristic
143	C₄×C₁₁	—	4	2	no	—
144	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
148	Dic₁₁	—	4	2	no	—
149	Dic₁₁	#95	1	1	yes	characteristic
150	Dic₁₁	#95	1	1	yes	characteristic
151	Dic₁₁	#95	1	1	yes	characteristic
152	Dic₁₁	#95	1	1	yes	characteristic
153	C₂³×C₄	—	11	1	no	—
154	C₂×M₁₆	—	11	1	no	—
155	C₂×M₁₆	—	11	2	no	—
156	C₂×M₁₆	—	11	1	no	—
157	C₂×M₁₆	—	11	2	no	—
158	C₂×M₁₆	—	11	2	no	—
159	C₂×M₁₆	—	22	1	no	—
160	C₂×M₁₆	—	22	1	no	—
161	C₂×M₁₆	—	22	1	no	—
162	C₂×M₁₆	—	22	1	no	—
163	C₂×M₁₆	—	22	1	no	—
164	C₂×M₁₆	—	22	1	no	—
165	C₂×M₁₆	—	22	1	no	—
166	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
167	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
168	C₂²×C₈	—	11	2	no	—
169	C₂²×C₈	—	22	1	no	—
170	#117	—	11	1	no	—
171	#117	—	11	2	no	—
172	#117	—	22	1	no	—
173	#117	—	22	1	no	—
174	#117	C₂×D₁₁	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	22	1	no	—
176	C₂×C₂²⋊C₄	—	11	2	no	—
177	C₂×C₄²	—	22	1	no	—
178	C₂×C₁₁	—	2	1	no	—
179	C₂×C₁₁	—	2	1	no	—
180	C₂×C₁₁	—	2	1	no	—
181	C₂×C₁₁	C₂×#95	1	1	yes	characteristic
182	D₁₁	—	2	1	no	—
183	D₁₁	—	2	1	no	—
184	D₁₁	—	2	1	no	—
185	D₁₁	#344	1	2	yes	direct product factor
186	C₂⁴	—	11	1	no	—
187	C₂²×C₄	—	11	1	no	—
188	C₂²×C₄	—	11	1	no	—
189	C₂²×C₄	—	11	1	no	—
190	C₂²×C₄	—	11	1	no	—
191	C₂²×C₄	—	11	1	no	—
192	C₂²×C₄	—	11	1	no	—
193	C₂²×C₄	—	11	1	no	—
194	C₂²×C₄	—	11	1	no	—
195	C₂²×C₄	—	11	1	no	—
196	C₂²×C₄	—	11	1	no	—
197	C₂²×C₄	—	11	1	no	—
198	C₂²×C₄	—	11	1	no	—
199	C₂²×C₄	—	11	1	no	—
200	C₂²×C₄	—	22	2	no	—
201	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
202	M₁₆	—	2	1	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	11	2	no	—
206	M₁₆	—	22	2	no	—
207	M₁₆	—	22	2	no	—
208	M₁₆	—	22	2	no	—
209	M₁₆	—	22	2	no	—
210	M₁₆	—	22	1	no	—
211	M₁₆	—	22	1	no	—
212	M₁₆	—	22	1	no	—
213	M₁₆	C₄×D₁₁	1	2	yes	complemented
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	11	2	no	—
216	C₂×C₈	—	22	2	no	—
217	C₂×C₈	—	22	2	no	—
218	C₂×C₈	—	22	2	no	—
219	C₂×C₈	—	22	2	no	—
220	C₂×C₈	—	22	1	no	—
221	C₂×C₈	C₄×D₁₁	1	2	yes	complemented
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	22	2	no	—
224	C₄⋊C₄	—	22	1	no	—
225	C₂²⋊C₄	—	11	2	no	—
226	C₂²⋊C₄	—	22	2	no	—
227	C₂²⋊C₄	C₄×D₁₁	1	2	yes	—
228	C₄²	—	2	1	no	—
229	C₄²	—	22	2	no	—
230	C₄²	—	22	1	no	—
231	C₁₁	C₂×#344	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	11	1	no	—
233	C₂³	—	11	1	no	—
234	C₂³	—	11	1	no	—
235	C₂³	—	11	1	no	—
236	C₂³	—	11	1	no	—
237	C₂³	—	11	1	no	—
238	C₂³	—	44	2	no	—
239	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	11	1	no	—
242	C₂×C₄	—	11	1	no	—
243	C₂×C₄	—	11	1	no	—
244	C₂×C₄	—	11	1	no	—
245	C₂×C₄	—	11	1	no	—
246	C₂×C₄	—	11	1	no	—
247	C₂×C₄	—	11	1	no	—
248	C₂×C₄	—	11	1	no	—
249	C₂×C₄	—	11	1	no	—
250	C₂×C₄	—	11	1	no	—
251	C₂×C₄	—	11	1	no	—
252	C₂×C₄	—	11	1	no	—
253	C₂×C₄	—	11	1	no	—
254	C₂×C₄	—	11	1	no	—
255	C₂×C₄	—	11	1	no	—
256	C₂×C₄	—	11	1	no	—
257	C₂×C₄	—	11	1	no	—
258	C₂×C₄	—	11	1	no	—
259	C₂×C₄	—	11	1	no	—
260	C₂×C₄	—	11	1	no	—
261	C₂×C₄	—	11	1	no	—
262	C₂×C₄	—	11	1	no	—
263	C₂×C₄	—	22	2	no	—
264	C₂×C₄	—	44	4	no	—
265	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
268	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
269	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
270	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	22	2	no	—
274	C₈	—	22	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	11	1	no	—
277	C₂²	—	11	1	no	—
278	C₂²	—	11	1	no	—
279	C₂²	—	11	1	no	—
280	C₂²	—	22	2	no	—
281	C₂²	—	22	2	no	—
282	C₂²	—	22	2	no	—
283	C₂²	—	44	1	no	—
284	C₂²	—	44	1	no	—
285	C₂²	—	44	1	no	—
286	C₂²	C₄²×D₁₁	1	1	yes	characteristic
287	C₂²	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
288	C₂²	C₄⋊C₄×D₁₁	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	11	1	no	—
291	C₄	—	11	1	no	—
292	C₄	—	11	1	no	—
293	C₄	—	11	1	no	—
294	C₄	—	44	2	no	—
295	C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
296	C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
297	C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
298	C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	11	2	no	—
303	C₂	—	22	1	no	—
304	C₂	—	22	1	no	—
305	C₂	—	22	1	no	—
306	C₂	D₁₁×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₁₁
commutator	C₂²×C₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#344	C₂
socle	C₂×C₁₁	C₂×#95
prime radical	C₂³×D₁₁	C₂×C₄
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	#344	D₁₁
hypocenter	C₁₁	C₂×#344
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. C₁₁×#344	C₂	semidirect
5. #10622197	C₂	semidirect
6. #10622194	C₂	semidirect
7. #10622171	C₂	semidirect
21. M₁₆×D₁₁	C₄	semidirect
22. M₁₆×D₁₁	C₄	semidirect
24. C₂×#4313	C₄	semidirect
26. C₂×C₈×D₁₁	C₄	semidirect
52. C₁₁×M₁₆	C₂×C₄	semidirect
54. C₂×C₈×C₁₁	C₂×C₄	semidirect
67. #4318	C₂×C₄	semidirect
69. C₂×C₁₁⋊C₈	C₂×C₄	semidirect
128. #344	D₁₁	direct (N × H)
185. D₁₁	#344	direct (N × H)
213. M₁₆	C₄×D₁₁	semidirect
221. C₂×C₈	C₄×D₁₁	semidirect
231. C₁₁	C₂×#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₁	2
D₁₁×#117	2
C₁₁×#344	2
#10622197	2
#10622194	2
#10622171	2
C₂×#344	11

Subgroups — D11×#344 #13068356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₁×#344 #13068356