Subgroups — C₁₁×#23176 #14600369

Order 2816.

1124 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#436	C₂	1	4	yes	maximal, complemented
2	C₄×C₁₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1954	C₂	1	4	yes	maximal, complemented
4	C₁₁×#1870	C₂	1	4	yes	maximal, complemented
5	C₄×C₁₁×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₁×M₁₆	C₂²	1	4	yes	—
7	C₂×C₁₁×#127	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₁	C₂²	1	4	yes	—
9	C₂×C₄×D₄×C₁₁	C₂²	1	2	yes	—
10	C₁₁×#436	—	2	16	no	—
11	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₁×C₂²⋊C₈	C₂²	1	8	yes	—
14	C₂×C₁₁×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₁	C₂²	1	4	yes	—
16	C₁₁×#390	C₄	1	4	yes	—
17	C₁₁×#390	C₄	1	2	yes	—
18	C₁₁×#384	C₄	1	1	yes	characteristic
19	C₄³×C₁₁	C₂²	1	1	yes	characteristic
20	C₁₁×#336	C₂²	1	4	yes	—
21	C₂²×D₄×C₁₁	C₂×C₄	1	4	yes	—
22	C₂²×D₄×C₁₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
24	C₂×C₁₁×M₁₆	—	2	16	no	—
25	C₂×C₁₁×M₁₆	—	4	4	no	—
26	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
28	C₁₁×#127	—	2	4	no	—
29	C₄×D₄×C₁₁	—	2	8	no	—
30	C₄×D₄×C₁₁	—	2	8	no	—
31	C₄×D₄×C₁₁	—	4	2	no	—
32	C₄×D₄×C₁₁	—	4	2	no	—
33	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	8	yes	—
36	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
42	C₁₁×C₄⋊C₈	—	2	4	no	—
43	C₁₁×C₄⋊C₈	D₄	1	8	yes	complemented
44	C₁₁×C₂²⋊C₈	—	2	16	no	—
45	C₁₁×C₂²⋊C₈	—	4	4	no	—
46	C₄×C₈×C₁₁	—	2	8	no	—
47	#23176	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₁	C₂²×C₄	1	4	yes	—
49	C₂×D₄×C₁₁	—	2	8	no	—
50	C₂×D₄×C₁₁	—	2	4	no	—
51	C₂×D₄×C₁₁	—	2	8	no	—
52	C₂×D₄×C₁₁	—	2	8	no	—
53	C₂×D₄×C₁₁	—	4	2	no	—
54	C₂×D₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	—	2	8	no	—
56	C₂²×C₄×C₁₁	—	2	8	no	—
57	C₂²×C₄×C₁₁	—	2	4	no	—
58	C₂²×C₄×C₁₁	—	2	4	no	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
65	C₁₁×M₁₆	—	4	16	no	—
66	C₂×C₈×C₁₁	—	2	4	no	—
67	C₂×C₈×C₁₁	—	2	4	no	—
68	C₂×C₈×C₁₁	—	2	4	no	—
69	C₂×C₈×C₁₁	—	2	4	no	—
70	C₂×C₈×C₁₁	—	2	4	no	—
71	C₂×C₈×C₁₁	C₂×D₄	1	8	yes	—
72	C₁₁×C₄⋊C₄	—	2	8	no	—
73	C₁₁×C₄⋊C₄	—	4	2	no	—
74	C₁₁×C₂²⋊C₄	—	2	16	no	—
75	C₁₁×C₂²⋊C₄	—	4	4	no	—
76	C₄²×C₁₁	—	2	4	no	—
77	C₄²×C₁₁	—	2	1	no	—
78	C₄²×C₁₁	—	2	1	no	—
79	C₄²×C₁₁	—	4	1	no	—
80	C₄²×C₁₁	M₁₆	1	4	yes	complemented
81	C₄²×C₁₁	C₂×D₄	1	4	yes	—
82	C₄²×C₁₁	#41	1	4	yes	—
83	C₂×#436	C₂×C₁₁	1	4	yes	complemented
84	C₄×#127	C₂×C₁₁	1	1	yes	characteristic
85	#1954	C₂×C₁₁	1	4	yes	complemented
86	#1870	C₂×C₁₁	1	4	yes	complemented
87	C₄×C₄⋊C₈	C₂×C₁₁	1	2	yes	complemented
88	C₂³×C₁₁	—	2	8	no	—
89	C₂³×C₁₁	—	2	4	no	—
90	C₂³×C₁₁	—	2	4	no	—
91	C₂³×C₁₁	—	2	8	no	—
92	C₂³×C₁₁	—	2	4	no	—
93	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
94	D₄×C₁₁	—	4	8	no	—
95	D₄×C₁₁	—	4	16	no	—
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	2	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	2	8	no	—
101	C₂×C₄×C₁₁	—	2	1	no	—
102	C₂×C₄×C₁₁	—	2	4	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	4	4	no	—
105	C₂×C₄×C₁₁	—	4	8	no	—
106	C₂×C₄×C₁₁	—	4	4	no	—
107	C₂×C₄×C₁₁	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₁₁	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₁₁	#131	1	4	yes	—
112	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
116	C₈×C₁₁	—	2	8	no	—
117	C₈×C₁₁	—	4	4	no	—
118	C₂²×M₁₆	C₂²×C₁₁	1	4	yes	—
119	C₂×#127	C₄×C₁₁	1	1	yes	characteristic
120	C₂×C₄×D₄	C₂²×C₁₁	1	4	yes	—
121	C₂×C₄×D₄	C₂²×C₁₁	1	2	yes	—
122	#436	—	2	16	no	—
123	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
125	C₂×C₂²⋊C₈	C₂²×C₁₁	1	8	yes	—
126	C₂×C₂²⋊C₈	C₂²×C₁₁	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₁₁	1	4	yes	—
128	#390	C₄×C₁₁	1	4	yes	—
129	#390	C₄×C₁₁	1	2	yes	—
130	#384	C₄×C₁₁	1	1	yes	characteristic
131	C₄³	C₂²×C₁₁	1	1	yes	characteristic
132	#336	C₂²×C₁₁	1	4	yes	—
133	C₂²×C₁₁	—	4	4	no	—
134	C₂²×C₁₁	—	4	8	no	—
135	C₂²×C₁₁	—	4	8	no	—
136	C₂²×C₁₁	—	4	4	no	—
137	C₂²×C₁₁	—	4	4	no	—
138	C₂²×C₁₁	C₂×C₄×D₄	1	2	yes	—
139	C₂²×C₁₁	#518	1	1	yes	characteristic
140	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
141	C₂²×C₁₁	C₂×#131	1	2	yes	—
142	C₂²×C₁₁	#568	1	1	yes	characteristic
143	C₄×C₁₁	—	2	4	no	—
144	C₄×C₁₁	—	2	2	no	—
145	C₄×C₁₁	—	4	4	no	—
146	C₄×C₁₁	—	4	1	no	—
147	C₄×C₁₁	#436	1	8	yes	complemented
148	C₄×C₁₁	D₄²	1	1	yes	characteristic
149	C₄×C₁₁	#550	1	2	yes	—
150	C₄×C₁₁	#559	1	1	yes	characteristic
151	C₂²×D₄	C₂×C₄×C₁₁	1	4	yes	—
152	C₂²×D₄	C₂×C₄×C₁₁	1	2	yes	—
153	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
154	C₂×M₁₆	—	2	16	no	—
155	C₂×M₁₆	—	4	4	no	—
156	C₂²×C₈	C₂³×C₁₁	1	4	yes	—
157	C₂²×C₈	C₂³×C₁₁	1	4	yes	—
158	#127	—	2	4	no	—
159	C₄×D₄	—	2	8	no	—
160	C₄×D₄	—	2	8	no	—
161	C₄×D₄	—	4	2	no	—
162	C₄×D₄	—	4	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	8	yes	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	D₄×C₁₁	1	8	yes	complemented
174	C₂²⋊C₈	—	2	16	no	—
175	C₂²⋊C₈	—	4	4	no	—
176	C₄×C₈	—	2	8	no	—
177	C₂×C₁₁	—	8	4	no	—
178	C₂×C₁₁	#2366	1	1	yes	characteristic
179	C₂×C₁₁	C₂×#436	1	2	yes	—
180	C₂×C₁₁	#2935	1	2	yes	—
181	C₂×C₁₁	#2975	1	1	yes	characteristic
182	C₂×C₁₁	#2979	1	1	yes	characteristic
183	C₂⁴	C₂²×C₄×C₁₁	1	4	yes	—
184	C₂×D₄	—	2	8	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₁	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
200	M₁₆	—	4	16	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂×D₄×C₁₁	1	8	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	2	16	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₁₁×M₁₆	1	4	yes	complemented
216	C₄²	C₂×D₄×C₁₁	1	4	yes	—
217	C₄²	C₁₁×#41	1	4	yes	—
218	C₁₁	#23176	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	8	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
225	D₄	—	4	8	no	—
226	D₄	—	4	16	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×D₄×C₁₁	1	4	yes	—
239	C₂×C₄	C₄×D₄×C₁₁	1	4	yes	—
240	C₂×C₄	C₂×C₁₁×M₁₆	1	2	yes	—
241	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
242	C₂×C₄	C₁₁×#131	1	4	yes	—
243	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
244	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
245	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
247	C₈	—	2	8	no	—
248	C₈	—	4	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×C₄×D₄×C₁₁	1	2	yes	—
255	C₂²	C₁₁×#518	1	1	yes	characteristic
256	C₂²	C₂²×C₁₁×M₁₆	1	1	yes	characteristic
257	C₂²	C₂×C₁₁×#131	1	2	yes	—
258	C₂²	C₁₁×#568	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	4	no	—
263	C₄	C₁₁×#436	1	8	yes	complemented
264	C₄	D₄²×C₁₁	1	1	yes	characteristic
265	C₄	C₁₁×#550	1	2	yes	—
266	C₄	C₁₁×#559	1	1	yes	characteristic
267	C₂	—	8	4	no	—
268	C₂	C₁₁×#2366	1	1	yes	characteristic
269	C₂	C₂×C₁₁×#436	1	2	yes	—
270	C₂	C₁₁×#2935	1	2	yes	—
271	C₂	C₁₁×#2975	1	1	yes	characteristic
272	C₂	C₁₁×#2979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
socle	C₂³×C₁₁	C₂³×C₄
prime radical	C₂×C₁₁×#127	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
1. C₂×C₁₁×#436	C₂	semidirect
3. C₁₁×#1954	C₂	semidirect
4. C₁₁×#1870	C₂	semidirect
5. C₄×C₁₁×C₄⋊C₈	C₂	semidirect
43. C₁₁×C₄⋊C₈	D₄	semidirect
47. #23176	C₁₁	direct (N × H)
80. C₄²×C₁₁	M₁₆	semidirect
83. C₂×#436	C₂×C₁₁	semidirect
85. #1954	C₂×C₁₁	semidirect
86. #1870	C₂×C₁₁	semidirect
87. C₄×C₄⋊C₈	C₂×C₁₁	semidirect
147. C₄×C₁₁	#436	semidirect
173. C₄⋊C₈	D₄×C₁₁	semidirect
215. C₄²	C₁₁×M₁₆	semidirect
218. C₁₁	#23176	direct (N × H)
263. C₄	C₁₁×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#436	2
C₄×C₁₁×#127	2
C₁₁×#1954	2
C₁₁×#1870	2
C₄×C₁₁×C₄⋊C₈	2
#23176	11

Subgroups — C11×#23176 #14600369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23176 #14600369