Subgroups — C₂×C₁₁×#2110 #14610521

Order 2816.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#2110	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₁×#347	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#117	C₄	1	1	yes	characteristic
8	C₂×C₁₁×M₃₂	C₄	1	4	yes	complemented
9	C₂×C₁₁×M₃₂	C₄	1	8	yes	complemented
10	C₂×C₁₁×M₃₂	C₂²	1	4	yes	—
11	C₂×C₁₁×M₃₂	C₂²	1	8	yes	—
12	C₂²×C₁₁×C₁₆	C₄	1	4	yes	complemented
13	C₁₁×#405	C₄	1	4	yes	—
14	C₁₁×#405	C₂²	1	4	yes	—
15	C₂×C₁₁×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
17	C₁₁×#347	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₁×M₁₆	—	2	2	no	—
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
30	C₁₁×#117	—	2	2	no	—
31	C₁₁×#117	C₂×C₄	1	4	yes	—
32	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
34	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
35	C₁₁×M₃₂	C₂×C₄	1	8	yes	—
36	C₁₁×M₃₂	C₂×C₄	1	16	yes	complemented
37	C₁₁×M₃₂	C₂³	1	8	yes	—
38	C₂×C₁₁×C₁₆	C₂×C₄	1	8	yes	—
39	C₂×C₁₁×C₁₆	C₂×C₄	1	16	yes	complemented
40	C₁₁×#97	C₂×C₄	1	4	yes	—
41	C₁₁×#97	C₂³	1	4	yes	—
42	C₄×C₈×C₁₁	C₂×C₄	1	8	yes	—
43	C₂×#2110	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₁	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	—	2	2	no	—
46	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
51	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
54	C₁₁×M₁₆	—	2	8	no	—
55	C₂×C₈×C₁₁	—	2	2	no	—
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	C₄²	1	4	yes	—
58	C₂×C₈×C₁₁	C₄²	1	4	yes	—
59	C₂×C₈×C₁₁	C₄²	1	2	yes	—
60	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
70	C₁₁×C₄⋊C₄	—	2	4	no	—
71	C₁₁×C₂²⋊C₄	—	2	4	no	—
72	C₄²×C₁₁	—	2	2	no	—
73	C₄²×C₁₁	C₂²×C₄	1	4	yes	—
74	C₁₁×C₁₆	C₂²×C₄	1	16	yes	—
75	C₂²×M₃₂	C₂×C₁₁	1	2	yes	—
76	C₂×#405	C₂×C₁₁	1	1	yes	characteristic
77	#2110	C₂×C₁₁	1	8	yes	direct product factor
78	C₂×#347	C₂×C₁₁	1	4	yes	complemented
79	C₂³×C₁₁	—	4	2	no	—
80	C₂³×C₁₁	#97	1	2	yes	—
81	C₂³×C₁₁	#97	1	2	yes	—
82	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂³×C₁₁	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₁	—	2	4	no	—
85	C₂×C₄×C₁₁	—	4	4	no	—
86	C₂×C₄×C₁₁	#97	1	2	yes	—
87	C₂×C₄×C₁₁	#97	1	4	yes	—
88	C₂×C₄×C₁₁	#97	1	4	yes	—
89	C₂×C₄×C₁₁	#97	1	2	yes	—
90	C₂×C₄×C₁₁	C₂×C₄²	1	4	yes	—
91	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
96	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
99	C₈×C₁₁	—	2	4	no	—
100	C₈×C₁₁	C₂×C₄²	1	2	yes	—
101	C₈×C₁₁	C₂×C₄²	1	2	yes	—
102	C₈×C₁₁	C₂×C₄²	1	2	yes	—
103	C₈×C₁₁	C₂³×C₄	1	2	yes	—
104	C₂²×M₁₆	C₄×C₁₁	1	1	yes	characteristic
105	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
106	C₂×#117	C₄×C₁₁	1	1	yes	characteristic
107	C₂×M₃₂	C₄×C₁₁	1	4	yes	complemented
108	C₂×M₃₂	C₄×C₁₁	1	8	yes	complemented
109	C₂×M₃₂	C₂²×C₁₁	1	4	yes	—
110	C₂×M₃₂	C₂²×C₁₁	1	8	yes	—
111	C₂²×C₁₆	C₄×C₁₁	1	4	yes	complemented
112	#405	C₄×C₁₁	1	4	yes	—
113	#405	C₂²×C₁₁	1	4	yes	—
114	C₂×#97	C₂²×C₁₁	1	2	yes	—
115	C₂×C₄×C₈	C₄×C₁₁	1	2	yes	—
116	#347	C₂²×C₁₁	1	16	yes	complemented
117	C₂²×C₁₁	—	2	2	no	—
118	C₂²×C₁₁	—	2	4	no	—
119	C₂²×C₁₁	—	4	2	no	—
120	C₂²×C₁₁	—	4	2	no	—
121	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
122	C₂²×C₁₁	C₂×#97	1	4	yes	—
123	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
124	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
125	C₄×C₁₁	—	4	4	no	—
126	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
127	C₄×C₁₁	C₂×#97	1	4	yes	—
128	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
129	C₄×C₁₁	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
131	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂×C₄×C₁₁	1	4	yes	—
135	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
136	C₂²×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₁₁	1	4	yes	—
141	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
143	#117	—	2	2	no	—
144	#117	C₂×C₄×C₁₁	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
148	M₃₂	C₂×C₄×C₁₁	1	8	yes	—
149	M₃₂	C₂×C₄×C₁₁	1	16	yes	complemented
150	M₃₂	C₂³×C₁₁	1	8	yes	—
151	C₂×C₁₆	C₂×C₄×C₁₁	1	8	yes	—
152	C₂×C₁₆	C₂×C₄×C₁₁	1	16	yes	complemented
153	#97	C₂×C₄×C₁₁	1	4	yes	—
154	#97	C₂³×C₁₁	1	4	yes	—
155	C₄×C₈	C₂×C₄×C₁₁	1	8	yes	—
156	C₂×C₁₁	—	2	1	no	—
157	C₂×C₁₁	—	2	1	no	—
158	C₂×C₁₁	—	2	4	no	—
159	C₂×C₁₁	#2110	1	2	yes	direct product factor
160	C₂×C₁₁	C₂²×#97	1	1	yes	characteristic
161	C₂⁴	C₁₁×M₁₆	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₄²×C₁₁	1	2	yes	—
164	C₂²×C₄	C₄²×C₁₁	1	2	yes	—
165	C₂²×C₄	C₁₁×M₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₁₁×M₁₆	1	2	yes	—
167	C₂²×C₄	C₁₁×M₁₆	1	2	yes	—
168	C₂²×C₄	C₁₁×M₁₆	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
171	M₁₆	—	2	8	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄²×C₁₁	1	1	yes	characteristic
175	C₂×C₈	C₄²×C₁₁	1	2	yes	—
176	C₂×C₈	C₄²×C₁₁	1	4	yes	—
177	C₂×C₈	C₄²×C₁₁	1	4	yes	—
178	C₂×C₈	C₄²×C₁₁	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
180	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
184	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
186	C₂×C₈	C₂⁴×C₁₁	1	1	yes	characteristic
187	C₄⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂²×C₄×C₁₁	1	4	yes	—
191	C₁₆	C₂²×C₄×C₁₁	1	16	yes	—
192	C₁₁	C₂×#2110	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	—	4	2	no	—
194	C₂³	C₁₁×#97	1	2	yes	—
195	C₂³	C₁₁×#97	1	2	yes	—
196	C₂³	C₂×C₁₁×M₁₆	1	1	yes	characteristic
197	C₂³	C₂×C₁₁×M₁₆	1	2	yes	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₁₁×#97	1	2	yes	—
201	C₂×C₄	C₁₁×#97	1	4	yes	—
202	C₂×C₄	C₁₁×#97	1	4	yes	—
203	C₂×C₄	C₁₁×#97	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₁₁	1	4	yes	—
205	C₂×C₄	C₂×C₄²×C₁₁	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄²×C₁₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₁₁×M₁₆	1	1	yes	characteristic
208	C₂×C₄	C₂×C₁₁×M₁₆	1	2	yes	—
209	C₂×C₄	C₂×C₁₁×M₁₆	1	1	yes	characteristic
210	C₂×C₄	C₂×C₁₁×M₁₆	1	1	yes	characteristic
211	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
212	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄²×C₁₁	1	2	yes	—
215	C₈	C₂×C₄²×C₁₁	1	2	yes	—
216	C₈	C₂×C₄²×C₁₁	1	2	yes	—
217	C₈	C₂³×C₄×C₁₁	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×C₁₁×#97	1	1	yes	characteristic
223	C₂²	C₂×C₁₁×#97	1	4	yes	—
224	C₂²	C₂×C₁₁×#97	1	1	yes	characteristic
225	C₂²	C₂²×C₁₁×M₁₆	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	C₂×C₁₁×#97	1	1	yes	characteristic
228	C₄	C₂×C₁₁×#97	1	1	yes	characteristic
229	C₄	C₂×C₁₁×#97	1	4	yes	—
230	C₄	C₂²×C₄²×C₁₁	1	1	yes	characteristic
231	C₄	C₂²×C₁₁×M₁₆	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	C₁₁×#2110	1	2	yes	direct product factor
236	C₂	C₂²×C₁₁×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₁	C₄²
commutator	C₄	C₂²×C₄²×C₁₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₁
socle	C₂²×C₁₁	C₂×#97
prime radical	C₂⁴×C₁₁	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁×#2110	C₂	direct (N × H)
4. C₂×C₁₁×#347	C₂	semidirect
8. C₂×C₁₁×M₃₂	C₄	semidirect
9. C₂×C₁₁×M₃₂	C₄	semidirect
12. C₂²×C₁₁×C₁₆	C₄	semidirect
17. C₁₁×#347	C₂²	semidirect
36. C₁₁×M₃₂	C₂×C₄	semidirect
39. C₂×C₁₁×C₁₆	C₂×C₄	semidirect
43. C₂×#2110	C₁₁	direct (N × H)
77. #2110	C₂×C₁₁	direct (N × H)
78. C₂×#347	C₂×C₁₁	semidirect
107. C₂×M₃₂	C₄×C₁₁	semidirect
108. C₂×M₃₂	C₄×C₁₁	semidirect
111. C₂²×C₁₆	C₄×C₁₁	semidirect
116. #347	C₂²×C₁₁	semidirect
149. M₃₂	C₂×C₄×C₁₁	semidirect
152. C₂×C₁₆	C₂×C₄×C₁₁	semidirect
159. C₂×C₁₁	#2110	direct (N × H)
192. C₁₁	C₂×#2110	direct (N × H)
235. C₂	C₁₁×#2110	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×M₃₂	2
C₂×C₁₁×#405	2
C₁₁×#2110	2
C₂×C₁₁×#347	2
C₂×#2110	11

Subgroups — C2×C11×#2110 #14610521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#2110 #14610521