Subgroups — C₁₁×#23196 #14600389

Order 2816.

884 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#2957	C₂	1	4	yes	maximal, complemented
3	C₁₁×#1976	C₂	1	4	yes	maximal, complemented
4	C₁₁×#1884	C₂	1	4	yes	maximal, complemented
5	C₁₁×#1772	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#135	C₂²	1	4	yes	—
7	C₁₁×#568	C₂²	1	4	yes	—
8	C₁₁×#532	C₂²	1	2	yes	—
9	C₁₁×#495	C₄	1	4	yes	complemented
10	C₁₁×#494	C₄	1	1	yes	characteristic, complemented
11	C₁₁×#493	C₄	1	1	yes	characteristic, complemented
12	C₁₁×#492	C₄	1	2	yes	complemented
13	C₁₁×#444	—	2	8	no	—
14	C₁₁×#443	—	2	8	no	—
15	C₂×C₁₁×#108	C₂²	1	2	yes	—
16	C₁₁×#427	C₂²	1	2	yes	—
17	C₁₁×#421	C₂²	1	8	yes	—
18	C₁₁×#410	C₂²	1	4	yes	—
19	C₁₁×#405	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
21	C₁₁×#339	C₂²	1	4	yes	—
22	C₂×C₁₁×#41	C₂³	1	4	yes	—
23	C₁₁×#135	—	2	4	no	—
24	C₁₁×#135	—	2	4	no	—
25	C₁₁×#135	—	2	4	no	—
26	C₁₁×#135	—	2	4	no	—
27	C₂×C₁₁×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₁×SD₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₁₁×D₈	C₂×C₄	1	4	yes	—
30	C₁₁×#131	—	2	16	no	—
31	C₂×C₁₁×M₁₆	—	4	4	no	—
32	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
34	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₁×#127	C₂×C₄	1	2	yes	—
36	C₁₁×#124	C₂×C₄	1	4	yes	—
37	C₁₁×#121	—	4	2	no	—
38	C₄×D₄×C₁₁	—	4	2	no	—
39	C₁₁×#117	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
41	C₁₁×#108	D₄	1	4	yes	complemented
42	C₁₁×#108	D₄	1	4	yes	complemented
43	C₁₁×#107	—	2	2	no	—
44	C₁₁×#106	—	2	2	no	—
45	C₁₁×#104	—	2	16	no	—
46	C₁₁×#99	—	4	4	no	—
47	C₁₁×#97	—	2	4	no	—
48	C₄×C₈×C₁₁	—	2	4	no	—
49	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
52	#23196	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₁×#41	—	2	8	no	—
54	C₁₁×#41	—	2	8	no	—
55	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
56	C₂×D₄×C₁₁	—	2	4	no	—
57	C₂×D₄×C₁₁	—	4	2	no	—
58	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₁	—	2	1	no	—
60	C₂²×C₄×C₁₁	—	2	4	no	—
61	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
62	C₁₁×Q₁₆	—	2	4	no	—
63	C₁₁×Q₁₆	—	2	4	no	—
64	C₁₁×SD₁₆	—	2	8	no	—
65	C₁₁×SD₁₆	—	2	8	no	—
66	C₁₁×D₈	—	2	4	no	—
67	C₁₁×D₈	—	2	4	no	—
68	C₁₁×M₁₆	—	2	4	no	—
69	C₁₁×M₁₆	—	4	8	no	—
70	C₁₁×M₁₆	C₂×D₄	1	8	yes	—
71	C₂×C₈×C₁₁	—	2	4	no	—
72	C₂×C₈×C₁₁	—	4	8	no	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₁	#41	1	2	yes	—
78	C₂×C₈×C₁₁	#41	1	2	yes	—
79	C₁₁×C₄⋊C₄	—	2	4	no	—
80	C₁₁×C₄⋊C₄	—	4	2	no	—
81	C₁₁×C₂²⋊C₄	—	2	4	no	—
82	C₁₁×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₁₁	—	2	1	no	—
84	C₄²×C₁₁	—	2	4	no	—
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	#3147	C₂×C₁₁	1	1	yes	characteristic
88	#2957	C₂×C₁₁	1	4	yes	complemented
89	#1976	C₂×C₁₁	1	4	yes	complemented
90	#1884	C₂×C₁₁	1	4	yes	complemented
91	#1772	C₂×C₁₁	1	2	yes	complemented
92	C₂³×C₁₁	—	2	4	no	—
93	C₂³×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
94	Q₈×C₁₁	—	2	8	no	—
95	D₄×C₁₁	—	2	8	no	—
96	D₄×C₁₁	—	4	8	no	—
97	D₄×C₁₁	—	8	4	no	—
98	C₂×C₄×C₁₁	—	2	1	no	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	2	4	no	—
101	C₂×C₄×C₁₁	—	4	2	no	—
102	C₂×C₄×C₁₁	—	4	4	no	—
103	C₂×C₄×C₁₁	—	4	4	no	—
104	C₂×C₄×C₁₁	—	4	4	no	—
105	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
108	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
109	C₈×C₁₁	—	2	8	no	—
110	C₈×C₁₁	C₄×D₄	1	2	yes	—
111	C₈×C₁₁	C₄×D₄	1	2	yes	—
112	C₈×C₁₁	C₄×D₄	1	2	yes	—
113	C₈×C₁₁	C₄×D₄	1	2	yes	—
114	C₂×#135	C₂²×C₁₁	1	4	yes	—
115	#568	C₂²×C₁₁	1	4	yes	—
116	#532	C₂²×C₁₁	1	2	yes	—
117	#495	C₄×C₁₁	1	4	yes	complemented
118	#494	C₄×C₁₁	1	1	yes	characteristic, complemented
119	#493	C₄×C₁₁	1	1	yes	characteristic, complemented
120	#492	C₄×C₁₁	1	2	yes	complemented
121	#444	—	2	8	no	—
122	#443	—	2	8	no	—
123	C₂×#108	C₂²×C₁₁	1	2	yes	—
124	#427	C₂²×C₁₁	1	2	yes	—
125	#421	C₂²×C₁₁	1	8	yes	—
126	#410	C₂²×C₁₁	1	4	yes	—
127	#405	C₂²×C₁₁	1	4	yes	—
128	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
129	#339	C₂²×C₁₁	1	4	yes	—
130	C₂²×C₁₁	—	4	1	no	—
131	C₂²×C₁₁	—	4	4	no	—
132	C₂²×C₁₁	—	8	4	no	—
133	C₂²×C₁₁	#518	1	1	yes	characteristic
134	C₂²×C₁₁	#550	1	2	yes	—
135	C₄×C₁₁	—	2	2	no	—
136	C₄×C₁₁	—	4	1	no	—
137	C₄×C₁₁	—	4	4	no	—
138	C₄×C₁₁	—	8	4	no	—
139	C₄×C₁₁	C₂×C₄×D₄	1	2	yes	—
140	C₄×C₁₁	D₄²	1	1	yes	characteristic
141	C₄×C₁₁	#559	1	1	yes	characteristic
142	C₂×#41	C₂³×C₁₁	1	4	yes	—
143	#135	—	2	4	no	—
144	#135	—	2	4	no	—
145	#135	—	2	4	no	—
146	#135	—	2	4	no	—
147	C₂×Q₁₆	C₂×C₄×C₁₁	1	4	yes	—
148	C₂×SD₁₆	C₂×C₄×C₁₁	1	8	yes	—
149	C₂×D₈	C₂×C₄×C₁₁	1	4	yes	—
150	#131	—	2	16	no	—
151	C₂×M₁₆	—	4	4	no	—
152	C₂×M₁₆	C₂³×C₁₁	1	4	yes	—
153	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
154	C₄⋊Q₈	C₂×C₄×C₁₁	1	2	yes	—
155	#127	C₂×C₄×C₁₁	1	2	yes	—
156	#124	C₂×C₄×C₁₁	1	4	yes	—
157	#121	—	4	2	no	—
158	C₄×D₄	—	4	2	no	—
159	#117	C₂³×C₁₁	1	4	yes	—
160	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
161	#108	D₄×C₁₁	1	4	yes	complemented
162	#108	D₄×C₁₁	1	4	yes	complemented
163	#107	—	2	2	no	—
164	#106	—	2	2	no	—
165	#104	—	2	16	no	—
166	#99	—	4	4	no	—
167	#97	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
170	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₁₁	1	2	yes	—
172	C₂×C₁₁	—	2	1	no	—
173	C₂×C₁₁	—	2	2	no	—
174	C₂×C₁₁	—	8	4	no	—
175	C₂×C₁₁	#2366	1	1	yes	characteristic
176	#41	—	2	8	no	—
177	#41	—	2	8	no	—
178	C₂×Q₈	C₂²×C₄×C₁₁	1	4	yes	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	C₂²×C₄×C₁₁	1	4	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	2	4	no	—
187	SD₁₆	—	2	8	no	—
188	SD₁₆	—	2	8	no	—
189	D₈	—	2	4	no	—
190	D₈	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	8	no	—
193	M₁₆	C₂×D₄×C₁₁	1	8	yes	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	8	no	—
196	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
200	C₂×C₈	C₁₁×#41	1	2	yes	—
201	C₂×C₈	C₁₁×#41	1	2	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	4	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	4	no	—
208	C₄²	C₂²×C₄×C₁₁	1	1	yes	characteristic
209	C₄²	C₂²×C₄×C₁₁	1	1	yes	characteristic
210	C₁₁	#23196	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	—	2	4	no	—
212	C₂³	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	D₄	—	2	8	no	—
215	D₄	—	4	8	no	—
216	D₄	—	8	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
226	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
227	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
228	C₈	—	2	8	no	—
229	C₈	C₄×D₄×C₁₁	1	2	yes	—
230	C₈	C₄×D₄×C₁₁	1	2	yes	—
231	C₈	C₄×D₄×C₁₁	1	2	yes	—
232	C₈	C₄×D₄×C₁₁	1	2	yes	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₁₁×#518	1	1	yes	characteristic
237	C₂²	C₁₁×#550	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₂×C₄×D₄×C₁₁	1	2	yes	—
243	C₄	D₄²×C₁₁	1	1	yes	characteristic
244	C₄	C₁₁×#559	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	2	no	—
247	C₂	—	8	4	no	—
248	C₂	C₁₁×#2366	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
socle	C₂×C₁₁	#2366
prime radical	C₁₁×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁×#2957	C₂	semidirect
3. C₁₁×#1976	C₂	semidirect
4. C₁₁×#1884	C₂	semidirect
5. C₁₁×#1772	C₂	semidirect
9. C₁₁×#495	C₄	semidirect
10. C₁₁×#494	C₄	semidirect
11. C₁₁×#493	C₄	semidirect
12. C₁₁×#492	C₄	semidirect
41. C₁₁×#108	D₄	semidirect
42. C₁₁×#108	D₄	semidirect
52. #23196	C₁₁	direct (N × H)
88. #2957	C₂×C₁₁	semidirect
89. #1976	C₂×C₁₁	semidirect
90. #1884	C₂×C₁₁	semidirect
91. #1772	C₂×C₁₁	semidirect
117. #495	C₄×C₁₁	semidirect
118. #494	C₄×C₁₁	semidirect
119. #493	C₄×C₁₁	semidirect
120. #492	C₄×C₁₁	semidirect
161. #108	D₄×C₁₁	semidirect
162. #108	D₄×C₁₁	semidirect
210. C₁₁	#23196	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁×#3147	2
C₁₁×#2957	2
C₁₁×#1976	2
C₁₁×#1884	2
C₁₁×#1772	2
#23196	11

Subgroups — C11×#23196 #14600389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23196 #14600389