Subgroups — C₁₁×#3046 #13065884

Order 1408.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₁×#135	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#464	C₂	1	2	yes	maximal, complemented
6	C₁₁×#463	C₂	1	2	yes	maximal, complemented
7	C₁₁×#462	C₂	1	2	yes	maximal, complemented
8	C₁₁×#461	C₂	1	2	yes	maximal, complemented
9	C₁₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₁₁×#137	—	2	4	no	—
16	C₁₁×#135	—	2	4	no	—
17	C₂×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₁×Q₁₆	C₂²	1	2	yes	—
19	C₂×C₁₁×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₁×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
22	C₂×C₁₁×D₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
25	C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₁₁×#124	C₂²	1	2	yes	—
27	C₁₁×#123	—	2	2	no	—
28	C₁₁×C₂²⋊Q₈	—	2	1	no	—
29	C₁₁×C₂²⋊Q₈	—	2	1	no	—
30	C₄×Q₈×C₁₁	C₂²	1	2	yes	—
31	C₄×D₄×C₁₁	—	2	2	no	—
32	C₄×D₄×C₁₁	C₂²	1	2	yes	—
33	C₁₁×#117	—	2	1	no	—
34	C₁₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
38	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
39	C₁₁×#103	C₂²	1	2	yes	—
40	C₁₁×#103	C₂²	1	2	yes	—
41	C₁₁×#103	C₂²	1	2	yes	—
42	C₁₁×#102	C₂²	1	2	yes	—
43	C₁₁×#41	—	2	2	no	—
44	C₁₁×#41	—	2	2	no	—
45	C₁₁×#41	—	2	2	no	—
46	C₁₁×#41	D₄	1	4	yes	—
47	C₂×Q₈×C₁₁	—	2	1	no	—
48	C₂×Q₈×C₁₁	—	2	2	no	—
49	C₂×Q₈×C₁₁	—	2	2	no	—
50	C₂×Q₈×C₁₁	C₂³	1	2	yes	—
51	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₁	—	2	1	no	—
54	C₂×D₄×C₁₁	—	2	1	no	—
55	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	—	2	1	no	—
58	C₂²×C₄×C₁₁	—	2	1	no	—
59	C₂²×C₄×C₁₁	—	2	1	no	—
60	C₂²×C₄×C₁₁	—	2	1	no	—
61	C₂²×C₄×C₁₁	—	2	2	no	—
62	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
63	C₁₁×Q₁₆	—	2	2	no	—
64	C₁₁×Q₁₆	—	2	4	no	—
65	C₁₁×SD₁₆	—	2	2	no	—
66	C₁₁×SD₁₆	—	2	2	no	—
67	C₁₁×SD₁₆	—	2	4	no	—
68	C₁₁×D₈	—	2	2	no	—
69	C₁₁×M₁₆	—	2	2	no	—
70	C₂×C₈×C₁₁	—	2	2	no	—
71	C₂×C₈×C₁₁	C₂³	1	2	yes	—
72	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	—	2	1	no	—
75	C₁₁×C₄⋊C₄	—	2	2	no	—
76	C₁₁×C₄⋊C₄	—	2	2	no	—
77	C₁₁×C₄⋊C₄	—	2	2	no	—
78	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
80	C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
81	C₁₁×C₂²⋊C₄	—	2	2	no	—
82	C₁₁×C₂²⋊C₄	—	2	2	no	—
83	C₁₁×C₂²⋊C₄	—	2	2	no	—
84	C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
85	C₄²×C₁₁	—	2	2	no	—
86	C₄²×C₁₁	C₂³	1	2	yes	—
87	#3046	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₁	—	2	1	no	—
89	C₂³×C₁₁	—	2	1	no	—
90	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₁₁	—	2	1	no	—
92	Q₈×C₁₁	—	2	2	no	—
93	Q₈×C₁₁	—	2	1	no	—
94	Q₈×C₁₁	—	2	2	no	—
95	Q₈×C₁₁	—	2	1	no	—
96	Q₈×C₁₁	—	4	2	no	—
97	Q₈×C₁₁	C₂×D₄	1	2	yes	—
98	D₄×C₁₁	—	2	4	no	—
99	D₄×C₁₁	—	2	1	no	—
100	D₄×C₁₁	—	2	1	no	—
101	D₄×C₁₁	—	2	1	no	—
102	D₄×C₁₁	—	4	2	no	—
103	D₄×C₁₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₁	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	4	no	—
106	C₂×C₄×C₁₁	—	2	2	no	—
107	C₂×C₄×C₁₁	—	2	2	no	—
108	C₂×C₄×C₁₁	—	2	1	no	—
109	C₂×C₄×C₁₁	—	4	2	no	—
110	C₂×C₄×C₁₁	—	4	2	no	—
111	C₂×C₄×C₁₁	—	8	2	no	—
112	C₂×C₄×C₁₁	—	8	1	no	—
113	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₁	#41	1	2	yes	—
118	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
119	C₈×C₁₁	—	2	1	no	—
120	C₈×C₁₁	—	2	1	no	—
121	C₈×C₁₁	—	2	2	no	—
122	C₂×#137	C₂×C₁₁	1	1	yes	characteristic, complemented
123	C₂×#135	C₂×C₁₁	1	1	yes	characteristic
124	#536	C₂×C₁₁	1	1	yes	characteristic, complemented
125	#520	C₂×C₁₁	1	1	yes	characteristic, complemented
126	#464	C₂×C₁₁	1	2	yes	complemented
127	#463	C₂×C₁₁	1	2	yes	complemented
128	#462	C₂×C₁₁	1	2	yes	complemented
129	#461	C₂×C₁₁	1	2	yes	complemented
130	#424	C₂×C₁₁	1	1	yes	characteristic, complemented
131	#417	C₂×C₁₁	1	1	yes	characteristic, complemented
132	C₂×#103	C₂×C₁₁	1	1	yes	characteristic, complemented
133	C₂²×C₁₁	—	2	1	no	—
134	C₂²×C₁₁	—	2	1	no	—
135	C₂²×C₁₁	—	2	2	no	—
136	C₂²×C₁₁	—	4	1	no	—
137	C₂²×C₁₁	—	4	1	no	—
138	C₂²×C₁₁	—	4	1	no	—
139	C₂²×C₁₁	—	4	1	no	—
140	C₂²×C₁₁	—	4	1	no	—
141	C₂²×C₁₁	#121	1	2	yes	—
142	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₁	—	2	2	no	—
144	C₄×C₁₁	—	2	2	no	—
145	C₄×C₁₁	—	2	2	no	—
146	C₄×C₁₁	—	4	1	no	—
147	C₄×C₁₁	—	4	2	no	—
148	C₄×C₁₁	—	4	2	no	—
149	C₄×C₁₁	#121	1	2	yes	—
150	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
152	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
153	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
154	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
155	#137	—	2	4	no	—
156	#135	—	2	4	no	—
157	C₂×Q₁₆	C₂²×C₁₁	1	1	yes	characteristic
158	C₂×Q₁₆	C₂²×C₁₁	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₁₁	1	1	yes	characteristic
160	C₂×SD₁₆	C₂²×C₁₁	1	1	yes	characteristic
161	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
162	C₂×D₈	C₂²×C₁₁	1	1	yes	characteristic
163	C₂×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²×C₁₁	1	1	yes	characteristic
165	C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
166	#124	C₂²×C₁₁	1	2	yes	—
167	#123	—	2	2	no	—
168	C₂²⋊Q₈	—	2	1	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₄×Q₈	C₂²×C₁₁	1	2	yes	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	C₂²×C₁₁	1	2	yes	—
173	#117	—	2	1	no	—
174	#117	C₂²×C₁₁	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
177	C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
178	C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
179	#103	C₂²×C₁₁	1	2	yes	—
180	#103	C₂²×C₁₁	1	2	yes	—
181	#103	C₂²×C₁₁	1	2	yes	—
182	#102	C₂²×C₁₁	1	2	yes	—
183	C₂×C₁₁	—	2	2	no	—
184	C₂×C₁₁	—	4	2	no	—
185	C₂×C₁₁	—	8	1	no	—
186	C₂×C₁₁	C₂×#121	1	1	yes	characteristic
187	C₂×C₁₁	#577	1	1	yes	characteristic
188	C₂×C₁₁	#578	1	1	yes	characteristic
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	D₄×C₁₁	1	4	yes	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂³×C₁₁	1	2	yes	—
197	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
198	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
202	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
209	Q₁₆	—	2	2	no	—
210	Q₁₆	—	2	4	no	—
211	SD₁₆	—	2	2	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	4	no	—
214	D₈	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂³×C₁₁	1	2	yes	—
218	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
219	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
225	C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	D₄×C₁₁	1	2	yes	—
231	C₄²	—	2	2	no	—
232	C₄²	C₂³×C₁₁	1	2	yes	—
233	C₁₁	#3046	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	2	2	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	1	no	—
242	Q₈	—	4	2	no	—
243	Q₈	C₂×D₄×C₁₁	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	1	no	—
248	D₄	—	4	2	no	—
249	D₄	C₂×D₄×C₁₁	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
263	C₂×C₄	C₁₁×#41	1	2	yes	—
264	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₁₁×#121	1	2	yes	—
277	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₁₁×#121	1	2	yes	—
285	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
286	C₄	C₂×C₁₁×#41	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×C₁₁×#121	1	1	yes	characteristic
291	C₂	C₁₁×#577	1	1	yes	characteristic
292	C₂	C₁₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₁₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#137	C₂	semidirect
3. C₁₁×#536	C₂	semidirect
4. C₁₁×#520	C₂	semidirect
5. C₁₁×#464	C₂	semidirect
6. C₁₁×#463	C₂	semidirect
7. C₁₁×#462	C₂	semidirect
8. C₁₁×#461	C₂	semidirect
9. C₁₁×#424	C₂	semidirect
10. C₁₁×#417	C₂	semidirect
11. C₂×C₁₁×#103	C₂	semidirect
87. #3046	C₁₁	direct (N × H)
122. C₂×#137	C₂×C₁₁	semidirect
124. #536	C₂×C₁₁	semidirect
125. #520	C₂×C₁₁	semidirect
126. #464	C₂×C₁₁	semidirect
127. #463	C₂×C₁₁	semidirect
128. #462	C₂×C₁₁	semidirect
129. #461	C₂×C₁₁	semidirect
130. #424	C₂×C₁₁	semidirect
131. #417	C₂×C₁₁	semidirect
132. C₂×#103	C₂×C₁₁	semidirect
233. C₁₁	#3046	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×C₁₁×#137	2
C₂×C₁₁×#135	2
C₁₁×#536	2
C₁₁×#520	2
C₁₁×#464	2
C₁₁×#463	2
C₁₁×#462	2
C₁₁×#461	2
C₁₁×#424	2
C₁₁×#417	2
C₂×C₁₁×#103	2
#3046	11

Subgroups — C11×#3046 #13065884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3046 #13065884