Subgroups — C₁₁×#22024 #14599217

Order 2816.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#1985	C₂	1	2	yes	maximal
3	C₁₁×#1940	C₂	1	2	yes	maximal
4	C₁₁×#1917	C₂	1	4	yes	maximal
5	C₁₁×#1839	C₂	1	2	yes	maximal
6	C₁₁×#1831	C₂	1	2	yes	maximal
7	C₁₁×#1767	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₁	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₁₁×#97	C₂²	1	2	yes	—
14	C₂×C₁₁×#97	C₂²	1	2	yes	—
15	C₄×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
16	C₄×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
17	C₄×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄³×C₁₁	C₂²	1	1	yes	characteristic
21	C₄³×C₁₁	C₂²	1	2	yes	—
22	C₁₁×#336	C₂²	1	2	yes	—
23	C₁₁×#336	C₂²	1	4	yes	—
24	C₁₁×#336	C₂²	1	4	yes	—
25	C₁₁×#336	C₂²	1	4	yes	—
26	C₁₁×#336	C₂²	1	1	yes	characteristic
27	C₁₁×#336	C₂²	1	1	yes	characteristic
28	C₁₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
33	C₄×Q₈×C₁₁	—	2	8	no	—
34	C₄×Q₈×C₁₁	—	2	4	no	—
35	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	C₁₁×C₄⋊C₈	—	2	4	no	—
52	C₁₁×C₄⋊C₈	—	4	4	no	—
53	C₁₁×#97	—	2	4	no	—
54	C₁₁×#97	—	4	2	no	—
55	C₁₁×#95	C₂×C₄	1	2	yes	—
56	C₁₁×#95	C₂×C₄	1	1	yes	characteristic
57	C₁₁×#95	C₂×C₄	1	2	yes	—
58	C₁₁×#95	C₂×C₄	1	1	yes	characteristic
59	#22024	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₁	—	2	2	no	—
61	C₂×Q₈×C₁₁	—	2	4	no	—
62	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₁	—	2	2	no	—
72	C₂×C₈×C₁₁	—	2	2	no	—
73	C₂×C₈×C₁₁	—	2	4	no	—
74	C₂×C₈×C₁₁	—	2	4	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	2	no	—
79	C₂×C₈×C₁₁	—	2	4	no	—
80	C₁₁×C₄⋊C₄	—	2	4	no	—
81	C₁₁×C₄⋊C₄	—	2	2	no	—
82	C₁₁×C₄⋊C₄	—	2	4	no	—
83	C₁₁×C₄⋊C₄	—	2	4	no	—
84	C₁₁×C₄⋊C₄	—	2	4	no	—
85	C₄²×C₁₁	—	2	4	no	—
86	C₄²×C₁₁	—	2	4	no	—
87	C₄²×C₁₁	—	2	4	no	—
88	C₄²×C₁₁	—	2	1	no	—
89	C₄²×C₁₁	—	2	2	no	—
90	C₄²×C₁₁	—	2	2	no	—
91	C₄²×C₁₁	—	2	2	no	—
92	C₄²×C₁₁	—	2	1	no	—
93	C₄²×C₁₁	—	4	2	no	—
94	C₄²×C₁₁	—	4	2	no	—
95	C₄²×C₁₁	—	4	2	no	—
96	C₄²×C₁₁	M₁₆	1	8	yes	—
97	C₄²×C₁₁	#41	1	4	yes	—
98	C₄²×Q₈	C₂×C₁₁	1	1	yes	characteristic
99	#1985	C₂×C₁₁	1	2	yes	—
100	#1940	C₂×C₁₁	1	2	yes	—
101	#1917	C₂×C₁₁	1	4	yes	—
102	#1839	C₂×C₁₁	1	2	yes	—
103	#1831	C₂×C₁₁	1	2	yes	—
104	#1767	C₂×C₁₁	1	1	yes	characteristic
105	#1745	C₂×C₁₁	1	1	yes	characteristic
106	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
107	Q₈×C₁₁	—	2	8	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
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	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	2	no	—
120	C₂×C₄×C₁₁	—	2	2	no	—
121	C₂×C₄×C₁₁	—	2	2	no	—
122	C₂×C₄×C₁₁	—	2	2	no	—
123	C₂×C₄×C₁₁	—	2	2	no	—
124	C₂×C₄×C₁₁	—	2	2	no	—
125	C₂×C₄×C₁₁	#117	1	2	yes	—
126	C₂×C₄×C₁₁	#117	1	2	yes	—
127	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
134	C₈×C₁₁	—	4	2	no	—
135	C₈×C₁₁	—	4	2	no	—
136	C₈×C₁₁	—	4	4	no	—
137	C₂×C₄×Q₈	C₄×C₁₁	1	2	yes	—
138	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₁₁	1	4	yes	—
141	C₂×#97	C₂²×C₁₁	1	2	yes	—
142	C₂×#97	C₂²×C₁₁	1	2	yes	—
143	C₄×C₄⋊C₄	C₄×C₁₁	1	2	yes	—
144	C₄×C₄⋊C₄	C₄×C₁₁	1	2	yes	—
145	C₄×C₄⋊C₄	C₄×C₁₁	1	2	yes	—
146	C₄×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
148	C₄³	C₂²×C₁₁	1	2	yes	—
149	C₄³	C₂²×C₁₁	1	1	yes	characteristic
150	#336	C₂²×C₁₁	1	1	yes	characteristic
151	#336	C₂²×C₁₁	1	2	yes	—
152	#336	C₂²×C₁₁	1	4	yes	—
153	#336	C₂²×C₁₁	1	1	yes	characteristic
154	#336	C₂²×C₁₁	1	4	yes	—
155	#336	C₂²×C₁₁	1	4	yes	—
156	#336	C₂²×C₁₁	1	2	yes	—
157	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
158	C₂²×C₁₁	#519	1	1	yes	characteristic
159	C₂²×C₁₁	#520	1	1	yes	characteristic
160	C₂²×C₁₁	C₂²×M₁₆	1	2	yes	—
161	C₂²×C₁₁	#568	1	2	yes	—
162	C₄×C₁₁	—	2	2	no	—
163	C₄×C₁₁	—	2	4	no	—
164	C₄×C₁₁	—	2	2	no	—
165	C₄×C₁₁	—	2	4	no	—
166	C₄×C₁₁	—	2	4	no	—
167	C₄×C₁₁	—	4	2	no	—
168	C₄×C₁₁	—	4	2	no	—
169	C₄×C₁₁	—	4	2	no	—
170	C₄×C₁₁	#541	1	2	yes	—
171	C₄×C₁₁	#542	1	2	yes	—
172	C₂²×Q₈	C₂×C₄×C₁₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₁	1	4	yes	—
176	C₄×Q₈	—	2	8	no	—
177	C₄×Q₈	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
185	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
188	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
193	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	4	4	no	—
196	#97	—	2	4	no	—
197	#97	—	4	2	no	—
198	#95	C₂×C₄×C₁₁	1	2	yes	—
199	#95	C₂×C₄×C₁₁	1	1	yes	characteristic
200	#95	C₂×C₄×C₁₁	1	2	yes	—
201	#95	C₂×C₄×C₁₁	1	1	yes	characteristic
202	C₂×C₁₁	#2320	1	1	yes	characteristic
203	C₂×C₁₁	#2920	1	1	yes	characteristic
204	C₂×C₁₁	#2923	1	1	yes	characteristic
205	C₂×C₁₁	#2983	1	2	yes	—
206	C₂×C₁₁	#2989	1	2	yes	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₁₁×M₁₆	1	8	yes	—
244	C₄²	C₁₁×#41	1	4	yes	—
245	C₁₁	#22024	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
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	2	no	—
265	C₂×C₄	C₁₁×#117	1	2	yes	—
266	C₂×C₄	C₁₁×#117	1	2	yes	—
267	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
268	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
269	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
270	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
271	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
273	C₂×C₄	C₁₁×2₋¹⁺⁴	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	C₂×C₁₁×#117	1	1	yes	characteristic
278	C₂²	C₁₁×#519	1	1	yes	characteristic
279	C₂²	C₁₁×#520	1	1	yes	characteristic
280	C₂²	C₂²×C₁₁×M₁₆	1	2	yes	—
281	C₂²	C₁₁×#568	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	C₁₁×#541	1	2	yes	—
291	C₄	C₁₁×#542	1	2	yes	—
292	C₂	C₁₁×#2320	1	1	yes	characteristic
293	C₂	C₁₁×#2920	1	1	yes	characteristic
294	C₂	C₁₁×#2923	1	1	yes	characteristic
295	C₂	C₁₁×#2983	1	2	yes	—
296	C₂	C₁₁×#2989	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
socle, prime radical	C₂³×C₁₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
59. #22024	C₁₁	direct (N × H)
245. C₁₁	#22024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₁₁	2
C₁₁×#1985	2
C₁₁×#1940	2
C₁₁×#1917	2
C₁₁×#1839	2
C₁₁×#1831	2
C₁₁×#1767	2
C₁₁×#1745	2
#22024	11

Subgroups — C11×#22024 #14599217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#22024 #14599217