Subgroups — C₁₁×#22270 #14599463

Order 2816.

886 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₁×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₁₁×#1956	C₂	1	2	yes	maximal
4	C₁₁×#1878	C₂	1	8	yes	maximal
5	C₁₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×Q₁₆	C₂²	1	4	yes	—
8	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₁	C₂²	1	4	yes	—
10	C₁₁×#494	C₄	1	8	yes	complemented
11	C₂×C₁₁×#103	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×#391	C₂²	1	4	yes	—
15	C₁₁×#389	—	2	2	no	—
16	C₁₁×#389	—	2	2	no	—
17	C₁₁×#384	C₂²	1	2	yes	—
18	C₄×C₁₁×C₄⋊C₄	—	2	2	no	—
19	C₄×C₁₁×C₄⋊C₄	—	2	2	no	—
20	C₄³×C₁₁	C₂²	1	1	yes	characteristic
21	C₁₁×#340	C₂²	1	4	yes	—
22	C₁₁×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₁	C₂³	1	4	yes	—
24	C₂×C₁₁×Q₁₆	—	4	8	no	—
25	C₂×C₁₁×Q₁₆	C₂×C₄	1	16	yes	—
26	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
28	C₁₁×C₄⋊Q₈	—	2	4	no	—
29	C₁₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₁₁	—	4	8	no	—
31	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
36	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₁	—	2	4	no	—
38	C₂×C₄²×C₁₁	D₄	1	2	yes	—
39	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
43	C₁₁×#103	—	4	8	no	—
44	C₄×C₈×C₁₁	—	2	2	no	—
45	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
46	C₁₁×#95	—	2	4	no	—
47	C₁₁×#95	—	2	4	no	—
48	#22270	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×Q₈×C₁₁	—	2	4	no	—
50	C₂×Q₈×C₁₁	—	2	8	no	—
51	C₂×Q₈×C₁₁	—	2	8	no	—
52	C₂×Q₈×C₁₁	C₂²×C₄	1	8	yes	—
53	C₂²×C₄×C₁₁	—	2	4	no	—
54	C₂²×C₄×C₁₁	—	2	4	no	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
60	C₁₁×Q₁₆	—	4	16	no	—
61	C₂×C₈×C₁₁	—	2	2	no	—
62	C₂×C₈×C₁₁	—	2	4	no	—
63	C₂×C₈×C₁₁	—	2	4	no	—
64	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
65	C₁₁×C₄⋊C₄	—	2	8	no	—
66	C₁₁×C₄⋊C₄	—	2	8	no	—
67	C₁₁×C₄⋊C₄	—	2	8	no	—
68	C₁₁×C₄⋊C₄	—	4	8	no	—
69	C₁₁×C₄⋊C₄	—	4	4	no	—
70	C₁₁×C₄⋊C₄	—	4	4	no	—
71	C₄²×C₁₁	—	2	2	no	—
72	C₄²×C₁₁	—	2	4	no	—
73	C₄²×C₁₁	—	2	4	no	—
74	C₄²×C₁₁	—	4	8	no	—
75	C₄²×C₁₁	Q₁₆	1	4	yes	complemented
76	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₁	#41	1	2	yes	—
78	C₂×#494	C₂×C₁₁	1	1	yes	characteristic
79	C₄×C₄⋊Q₈	C₂×C₁₁	1	2	yes	—
80	#1956	C₂×C₁₁	1	2	yes	—
81	#1878	C₂×C₁₁	1	8	yes	—
82	#1834	C₂×C₁₁	1	1	yes	characteristic
83	#1744	C₂×C₁₁	1	1	yes	characteristic
84	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₁	—	2	16	no	—
86	Q₈×C₁₁	—	4	8	no	—
87	C₂×C₄×C₁₁	—	2	2	no	—
88	C₂×C₄×C₁₁	—	2	2	no	—
89	C₂×C₄×C₁₁	—	2	8	no	—
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	—	2	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	4	4	no	—
94	C₂×C₄×C₁₁	—	4	4	no	—
95	C₂×C₄×C₁₁	—	4	4	no	—
96	C₂×C₄×C₁₁	—	4	4	no	—
97	C₂×C₄×C₁₁	—	4	4	no	—
98	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₁	C₂×Q₁₆	1	4	yes	—
101	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₁₁	#135	1	2	yes	—
103	C₂×C₄×C₁₁	#137	1	4	yes	—
104	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₁₁	H(5,𝔽₂)	1	4	yes	—
107	C₈×C₁₁	—	2	4	no	—
108	C₈×C₁₁	—	4	2	no	—
109	C₂²×Q₁₆	C₂²×C₁₁	1	4	yes	—
110	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
111	C₂×C₄×Q₈	C₂²×C₁₁	1	4	yes	—
112	#494	C₄×C₁₁	1	8	yes	complemented
113	C₂×#103	C₂²×C₁₁	1	8	yes	—
114	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
116	#391	C₂²×C₁₁	1	4	yes	—
117	#389	—	2	2	no	—
118	#389	—	2	2	no	—
119	#384	C₂²×C₁₁	1	2	yes	—
120	C₄×C₄⋊C₄	—	2	2	no	—
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄³	C₂²×C₁₁	1	1	yes	characteristic
123	#340	C₂²×C₁₁	1	4	yes	—
124	#336	C₂²×C₁₁	1	4	yes	—
125	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₁	#535	1	2	yes	—
127	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
129	C₂²×C₁₁	C₂×#137	1	2	yes	—
130	C₄×C₁₁	—	2	4	no	—
131	C₄×C₁₁	—	4	8	no	—
132	C₄×C₁₁	—	4	2	no	—
133	C₄×C₁₁	—	8	4	no	—
134	C₄×C₁₁	C₄×Q₁₆	1	4	yes	—
135	C₄×C₁₁	#441	1	4	yes	—
136	C₄×C₁₁	#518	1	2	yes	—
137	C₄×C₁₁	#540	1	2	yes	—
138	C₂²×Q₈	C₂³×C₁₁	1	4	yes	—
139	C₂×Q₁₆	—	4	8	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₁	1	16	yes	—
141	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
142	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	C₂×C₄×C₁₁	1	8	yes	—
145	C₄×Q₈	—	4	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
152	C₂×C₄²	—	2	4	no	—
153	C₂×C₄²	D₄×C₁₁	1	2	yes	—
154	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
155	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
157	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
158	#103	—	4	8	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	C₂×C₄×C₁₁	1	4	yes	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂×C₁₁	#2328	1	1	yes	characteristic
164	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
165	C₂×C₁₁	C₂×#441	1	1	yes	characteristic
166	C₂×C₁₁	#3209	1	2	yes	—
167	C₂×C₁₁	#3210	1	2	yes	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	2	8	no	—
170	C₂×Q₈	—	2	8	no	—
171	C₂×Q₈	C₂²×C₄×C₁₁	1	8	yes	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
178	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
179	Q₁₆	—	4	16	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	4	8	no	—
194	C₄²	C₁₁×Q₁₆	1	4	yes	complemented
195	C₄²	C₂²×C₄×C₁₁	1	2	yes	—
196	C₄²	C₁₁×#41	1	2	yes	—
197	C₁₁	#22270	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
199	Q₈	—	2	16	no	—
200	Q₈	—	4	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
213	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
214	C₂×C₄	C₂×C₁₁×Q₁₆	1	4	yes	—
215	C₂×C₄	C₂×C₁₁×Q₁₆	1	2	yes	—
216	C₂×C₄	C₁₁×#135	1	2	yes	—
217	C₂×C₄	C₁₁×#137	1	4	yes	—
218	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
219	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
220	C₂×C₄	C₁₁×H(5,𝔽₂)	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
224	C₂²	C₁₁×#535	1	2	yes	—
225	C₂²	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
226	C₂²	C₂×C₁₁×#135	1	1	yes	characteristic
227	C₂²	C₂×C₁₁×#137	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	8	no	—
231	C₄	—	8	4	no	—
232	C₄	C₄×C₁₁×Q₁₆	1	4	yes	—
233	C₄	C₁₁×#441	1	4	yes	—
234	C₄	C₁₁×#518	1	2	yes	—
235	C₄	C₁₁×#540	1	2	yes	—
236	C₂	C₁₁×#2328	1	1	yes	characteristic
237	C₂	C₂×C₄×C₁₁×Q₁₆	1	1	yes	characteristic
238	C₂	C₂×C₁₁×#441	1	1	yes	characteristic
239	C₂	C₁₁×#3209	1	2	yes	—
240	C₂	C₁₁×#3210	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₄³×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₁₁×#494	C₄	semidirect
48. #22270	C₁₁	direct (N × H)
75. C₄²×C₁₁	Q₁₆	semidirect
112. #494	C₄×C₁₁	semidirect
194. C₄²	C₁₁×Q₁₆	semidirect
197. C₁₁	#22270	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×#494	2
C₄×C₁₁×C₄⋊Q₈	2
C₁₁×#1956	2
C₁₁×#1878	2
C₁₁×#1834	2
C₁₁×#1744	2
#22270	11

Subgroups — C11×#22270 #14599463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#22270 #14599463