Subgroups — C₁₁×#23361 #14600554

Order 2816.

756 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	C₁₁×#3151	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#1904	C₂	1	4	yes	maximal, complemented
3	C₁₁×#1903	C₂	1	4	yes	maximal, complemented
4	C₁₁×#1887	C₂	1	2	yes	maximal, complemented
5	C₁₁×#1884	C₂	1	2	yes	maximal, complemented
6	C₁₁×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
8	C₁₁×#568	C₂²	1	2	yes	—
9	C₁₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#127	C₂²	1	1	yes	characteristic
11	C₁₁×#518	C₂²	1	2	yes	—
12	C₁₁×#496	C₄	1	4	yes	complemented
13	C₁₁×#493	C₄	1	2	yes	complemented
14	C₁₁×#492	C₄	1	2	yes	complemented
15	C₁₁×#421	C₂²	1	4	yes	—
16	C₁₁×#418	C₂²	1	4	yes	—
17	C₂×C₁₁×#100	C₂²	1	2	yes	—
18	C₁₁×#410	C₂²	1	2	yes	—
19	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
20	C₁₁×#344	C₂²	1	4	yes	—
21	C₁₁×#339	C₂²	1	4	yes	—
22	C₁₁×#338	C₂²	1	2	yes	—
23	C₁₁×H(3,ℤ/4)	C₂²	1	2	yes	—
24	C₂×C₁₁×#41	C₂³	1	2	yes	—
25	C₂²×D₄×C₁₁	—	4	1	no	—
26	C₂²×D₄×C₁₁	C₂³	1	2	yes	—
27	C₁₁×AGL(1,ℤ/8)	—	4	4	no	—
28	C₁₁×AGL(1,ℤ/8)	—	4	4	no	—
29	C₂×C₁₁×SD₁₆	C₂×C₄	1	8	yes	complemented
30	C₂×C₁₁×D₈	C₂×C₄	1	8	yes	complemented
31	C₁₁×#131	—	4	4	no	—
32	C₂×C₁₁×M₁₆	—	4	2	no	—
33	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
34	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
35	C₁₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
36	C₁₁×#127	—	4	1	no	—
37	C₁₁×#127	C₂×C₄	1	2	yes	—
38	C₁₁×#127	C₂×C₄	1	2	yes	—
39	C₁₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
40	C₁₁×#124	C₂×C₄	1	2	yes	complemented
41	C₁₁×#121	—	4	1	no	—
42	C₄×D₄×C₁₁	—	4	4	no	—
43	C₄×D₄×C₁₁	—	4	1	no	—
44	C₁₁×#117	C₂³	1	4	yes	—
45	C₂×C₁₁×C₂²⋊C₄	—	4	2	no	—
46	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
47	C₁₁×#104	—	4	4	no	—
48	C₁₁×#102	—	4	4	no	—
49	C₁₁×#100	—	4	2	no	—
50	C₁₁×#99	—	4	2	no	—
51	C₁₁×#97	C₂×C₄	1	2	yes	complemented
52	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	complemented
53	#23361	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₁	—	2	2	no	—
55	C₁₁×#41	—	2	4	no	—
56	C₁₁×#41	—	2	4	no	—
57	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
58	C₂×D₄×C₁₁	—	2	2	no	—
59	C₂×D₄×C₁₁	—	2	4	no	—
60	C₂×D₄×C₁₁	—	2	4	no	—
61	C₂×D₄×C₁₁	—	2	2	no	—
62	C₂×D₄×C₁₁	—	4	1	no	—
63	C₂×D₄×C₁₁	—	4	2	no	—
64	C₂×D₄×C₁₁	—	4	2	no	—
65	C₂×D₄×C₁₁	—	8	4	no	—
66	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₁	—	2	1	no	—
69	C₂²×C₄×C₁₁	—	2	2	no	—
70	C₂²×C₄×C₁₁	—	4	4	no	—
71	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
72	C₁₁×SD₁₆	—	4	8	no	—
73	C₁₁×D₈	—	4	8	no	—
74	C₁₁×M₁₆	—	2	4	no	—
75	C₁₁×M₁₆	—	2	4	no	—
76	C₁₁×M₁₆	—	2	4	no	—
77	C₁₁×M₁₆	—	4	4	no	—
78	C₂×C₈×C₁₁	—	2	4	no	—
79	C₂×C₈×C₁₁	—	4	4	no	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
81	C₁₁×C₄⋊C₄	—	2	4	no	—
82	C₁₁×C₄⋊C₄	—	4	1	no	—
83	C₁₁×C₂²⋊C₄	—	2	4	no	—
84	C₁₁×C₂²⋊C₄	—	4	2	no	—
85	C₁₁×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₁₁	—	2	1	no	—
87	C₄²×C₁₁	—	2	4	no	—
88	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	#3151	C₂×C₁₁	1	1	yes	characteristic
91	#1904	C₂×C₁₁	1	4	yes	complemented
92	#1903	C₂×C₁₁	1	4	yes	complemented
93	#1887	C₂×C₁₁	1	2	yes	complemented
94	#1884	C₂×C₁₁	1	2	yes	complemented
95	#1841	C₂×C₁₁	1	2	yes	complemented
96	C₂³×C₁₁	—	2	2	no	—
97	C₂³×C₁₁	—	2	4	no	—
98	C₂³×C₁₁	—	4	2	no	—
99	C₂³×C₁₁	—	4	2	no	—
100	C₂³×C₁₁	—	8	4	no	—
101	C₂³×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
102	Q₈×C₁₁	—	2	4	no	—
103	D₄×C₁₁	—	2	4	no	—
104	D₄×C₁₁	—	2	8	no	—
105	D₄×C₁₁	—	4	4	no	—
106	D₄×C₁₁	—	4	4	no	—
107	D₄×C₁₁	—	8	2	no	—
108	D₄×C₁₁	—	8	4	no	—
109	D₄×C₁₁	—	8	4	no	—
110	C₂×C₄×C₁₁	—	2	1	no	—
111	C₂×C₄×C₁₁	—	2	1	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	4	2	no	—
114	C₂×C₄×C₁₁	—	4	2	no	—
115	C₂×C₄×C₁₁	—	4	2	no	—
116	C₂×C₄×C₁₁	—	4	4	no	—
117	C₂×C₄×C₁₁	—	16	4	no	—
118	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	H(5,𝔽₂)	1	4	yes	—
120	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
121	C₈×C₁₁	—	2	4	no	—
122	C₈×C₁₁	—	4	4	no	—
123	C₂×AGL(1,ℤ/8)	C₂²×C₁₁	1	4	yes	—
124	#568	C₂²×C₁₁	1	2	yes	—
125	#532	C₂²×C₁₁	1	1	yes	characteristic
126	C₂×#127	C₂²×C₁₁	1	1	yes	characteristic
127	#518	C₂²×C₁₁	1	2	yes	—
128	#496	C₄×C₁₁	1	4	yes	complemented
129	#493	C₄×C₁₁	1	2	yes	complemented
130	#492	C₄×C₁₁	1	2	yes	complemented
131	#421	C₂²×C₁₁	1	4	yes	—
132	#418	C₂²×C₁₁	1	4	yes	—
133	C₂×#100	C₂²×C₁₁	1	2	yes	—
134	#410	C₂²×C₁₁	1	2	yes	—
135	C₄×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
136	#344	C₂²×C₁₁	1	4	yes	—
137	#339	C₂²×C₁₁	1	4	yes	—
138	#338	C₂²×C₁₁	1	2	yes	—
139	H(3,ℤ/4)	C₂²×C₁₁	1	2	yes	—
140	C₂²×C₁₁	—	4	1	no	—
141	C₂²×C₁₁	—	4	2	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	—	8	4	no	—
144	C₂²×C₁₁	—	8	2	no	—
145	C₂²×C₁₁	—	8	4	no	—
146	C₂²×C₁₁	—	16	2	no	—
147	C₂²×C₁₁	#518	1	1	yes	characteristic
148	C₂²×C₁₁	#560	1	2	yes	—
149	C₄×C₁₁	—	2	2	no	—
150	C₄×C₁₁	—	4	1	no	—
151	C₄×C₁₁	—	4	2	no	—
152	C₄×C₁₁	—	8	4	no	—
153	C₄×C₁₁	#518	1	2	yes	—
154	C₄×C₁₁	#560	1	2	yes	—
155	C₂×#41	C₂³×C₁₁	1	2	yes	—
156	C₂²×D₄	—	4	1	no	—
157	C₂²×D₄	C₂³×C₁₁	1	2	yes	—
158	AGL(1,ℤ/8)	—	4	4	no	—
159	AGL(1,ℤ/8)	—	4	4	no	—
160	C₂×SD₁₆	C₂×C₄×C₁₁	1	8	yes	complemented
161	C₂×D₈	C₂×C₄×C₁₁	1	8	yes	complemented
162	#131	—	4	4	no	—
163	C₂×M₁₆	—	4	2	no	—
164	C₂×M₁₆	C₂³×C₁₁	1	4	yes	—
165	C₂×M₁₆	C₂³×C₁₁	1	2	yes	—
166	C₄⋊Q₈	C₂×C₄×C₁₁	1	1	yes	characteristic, complemented
167	#127	—	4	1	no	—
168	#127	C₂×C₄×C₁₁	1	2	yes	—
169	#127	C₂×C₄×C₁₁	1	2	yes	—
170	#127	C₂×C₄×C₁₁	1	1	yes	characteristic, complemented
171	#124	C₂×C₄×C₁₁	1	2	yes	complemented
172	#121	—	4	1	no	—
173	C₄×D₄	—	4	4	no	—
174	C₄×D₄	—	4	1	no	—
175	#117	C₂³×C₁₁	1	4	yes	—
176	C₂×C₂²⋊C₄	—	4	2	no	—
177	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
178	#104	—	4	4	no	—
179	#102	—	4	4	no	—
180	#100	—	4	2	no	—
181	#99	—	4	2	no	—
182	#97	C₂×C₄×C₁₁	1	2	yes	complemented
183	C₄×C₈	C₂×C₄×C₁₁	1	2	yes	complemented
184	C₂×C₁₁	—	2	1	no	—
185	C₂×C₁₁	—	2	2	no	—
186	C₂×C₁₁	—	8	2	no	—
187	C₂×C₁₁	—	8	4	no	—
188	C₂×C₁₁	#2380	1	1	yes	characteristic
189	C₂⁴	—	2	2	no	—
190	#41	—	2	4	no	—
191	#41	—	2	4	no	—
192	C₂×Q₈	C₂²×C₄×C₁₁	1	2	yes	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	8	4	no	—
201	C₂×D₄	C₂²×C₄×C₁₁	1	2	yes	—
202	C₂×D₄	C₂²×C₄×C₁₁	1	4	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
207	SD₁₆	—	4	8	no	—
208	D₈	—	4	8	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	4	no	—
223	C₄²	C₂²×C₄×C₁₁	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₁₁	1	1	yes	characteristic
225	C₁₁	#23361	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	8	4	no	—
231	C₂³	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	8	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	16	4	no	—
248	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
249	C₂×C₄	C₁₁×H(5,𝔽₂)	1	4	yes	—
250	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
251	C₈	—	2	4	no	—
252	C₈	—	4	4	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	16	2	no	—
260	C₂²	C₁₁×#518	1	1	yes	characteristic
261	C₂²	C₁₁×#560	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₁₁×#518	1	2	yes	—
267	C₄	C₁₁×#560	1	2	yes	—
268	C₂	—	2	1	no	—
269	C₂	—	2	2	no	—
270	C₂	—	8	4	no	—
271	C₂	—	8	2	no	—
272	C₂	C₁₁×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₁	#2380
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₁₁×#3151	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁×#1904	C₂	semidirect
3. C₁₁×#1903	C₂	semidirect
4. C₁₁×#1887	C₂	semidirect
5. C₁₁×#1884	C₂	semidirect
6. C₁₁×#1841	C₂	semidirect
12. C₁₁×#496	C₄	semidirect
13. C₁₁×#493	C₄	semidirect
14. C₁₁×#492	C₄	semidirect
29. C₂×C₁₁×SD₁₆	C₂×C₄	semidirect
30. C₂×C₁₁×D₈	C₂×C₄	semidirect
35. C₁₁×C₄⋊Q₈	C₂×C₄	semidirect
39. C₁₁×#127	C₂×C₄	semidirect
40. C₁₁×#124	C₂×C₄	semidirect
51. C₁₁×#97	C₂×C₄	semidirect
52. C₄×C₈×C₁₁	C₂×C₄	semidirect
53. #23361	C₁₁	direct (N × H)
91. #1904	C₂×C₁₁	semidirect
92. #1903	C₂×C₁₁	semidirect
93. #1887	C₂×C₁₁	semidirect
94. #1884	C₂×C₁₁	semidirect
95. #1841	C₂×C₁₁	semidirect
128. #496	C₄×C₁₁	semidirect
129. #493	C₄×C₁₁	semidirect
130. #492	C₄×C₁₁	semidirect
160. C₂×SD₁₆	C₂×C₄×C₁₁	semidirect
161. C₂×D₈	C₂×C₄×C₁₁	semidirect
166. C₄⋊Q₈	C₂×C₄×C₁₁	semidirect
170. #127	C₂×C₄×C₁₁	semidirect
171. #124	C₂×C₄×C₁₁	semidirect
182. #97	C₂×C₄×C₁₁	semidirect
183. C₄×C₈	C₂×C₄×C₁₁	semidirect
225. C₁₁	#23361	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×#3151	2
C₁₁×#1904	2
C₁₁×#1903	2
C₁₁×#1887	2
C₁₁×#1884	2
C₁₁×#1841	2
#23361	11

Subgroups — C11×#23361 #14600554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23361 #14600554