Subgroups — C₁₁×#3252 #13066090

Order 1408.

348 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₁×#529	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#479	C₂	1	2	yes	maximal, complemented
4	C₁₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#470	C₂	1	2	yes	maximal, complemented
7	C₁₁×#468	C₂	1	2	yes	maximal
8	C₁₁×#464	C₂	1	2	yes	maximal, complemented
9	C₁₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₁×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
15	C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₁×#126	—	2	1	no	—
18	C₁₁×#125	—	2	1	no	—
19	C₁₁×#125	C₂²	1	1	yes	characteristic
20	C₁₁×#124	C₂²	1	1	yes	characteristic
21	C₁₁×#123	—	2	1	no	—
22	C₁₁×C₂²⋊Q₈	—	2	1	no	—
23	C₁₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₁₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₁₁×#121	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₁	—	2	1	no	—
27	C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₁	C₂²	1	2	yes	—
29	C₄×D₄×C₁₁	—	2	1	no	—
30	C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
31	C₁₁×#117	—	2	1	no	—
32	C₁₁×#117	—	2	1	no	—
33	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
34	C₁₁×#107	C₂²	1	2	yes	—
35	C₁₁×#106	C₂²	1	2	yes	—
36	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
39	C₁₁×#103	C₂²	1	2	yes	—
40	C₁₁×#103	C₂²	1	2	yes	—
41	C₁₁×#103	C₂²	1	2	yes	—
42	C₁₁×#102	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₁	—	2	1	no	—
44	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₁	C₂³	1	2	yes	—
46	C₂×D₄×C₁₁	—	2	1	no	—
47	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	—	2	1	no	—
49	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
52	C₁₁×Q₁₆	—	4	2	no	—
53	C₁₁×SD₁₆	—	4	2	no	—
54	C₁₁×M₁₆	—	4	2	no	—
55	C₂×C₈×C₁₁	C₂³	1	2	yes	—
56	C₂×C₈×C₁₁	C₂³	1	2	yes	—
57	C₁₁×C₄⋊C₄	—	2	1	no	—
58	C₁₁×C₄⋊C₄	—	2	1	no	—
59	C₁₁×C₄⋊C₄	—	2	1	no	—
60	C₁₁×C₄⋊C₄	—	2	1	no	—
61	C₁₁×C₄⋊C₄	—	2	1	no	—
62	C₁₁×C₄⋊C₄	—	2	1	no	—
63	C₁₁×C₄⋊C₄	—	2	2	no	—
64	C₁₁×C₄⋊C₄	—	2	2	no	—
65	C₁₁×C₄⋊C₄	—	2	1	no	—
66	C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₁×C₂²⋊C₄	—	2	2	no	—
70	C₁₁×C₂²⋊C₄	—	2	1	no	—
71	C₁₁×C₂²⋊C₄	—	2	1	no	—
72	C₁₁×C₂²⋊C₄	—	2	1	no	—
73	C₁₁×C₂²⋊C₄	—	2	1	no	—
74	C₁₁×C₂²⋊C₄	—	2	1	no	—
75	C₄²×C₁₁	—	2	1	no	—
76	C₄²×C₁₁	—	2	2	no	—
77	C₄²×C₁₁	D₄	1	1	yes	characteristic
78	C₄²×C₁₁	D₄	1	1	yes	characteristic
79	C₄²×C₁₁	C₂³	1	1	yes	characteristic
80	C₄²×C₁₁	C₂³	1	1	yes	characteristic
81	#3252	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₁	—	2	1	no	—
83	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₁	—	2	2	no	—
85	Q₈×C₁₁	—	2	1	no	—
86	Q₈×C₁₁	—	2	1	no	—
87	Q₈×C₁₁	—	2	2	no	—
88	Q₈×C₁₁	—	4	2	no	—
89	D₄×C₁₁	—	2	1	no	—
90	D₄×C₁₁	—	2	1	no	—
91	D₄×C₁₁	—	4	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	1	no	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	4	1	no	—
101	C₂×C₄×C₁₁	—	4	2	no	—
102	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₁	—	2	2	no	—
109	C₈×C₁₁	—	2	2	no	—
110	#533	C₂×C₁₁	1	1	yes	characteristic, complemented
111	#529	C₂×C₁₁	1	1	yes	characteristic
112	#479	C₂×C₁₁	1	2	yes	complemented
113	#477	C₂×C₁₁	1	1	yes	characteristic, complemented
114	#476	C₂×C₁₁	1	1	yes	characteristic, complemented
115	#470	C₂×C₁₁	1	2	yes	complemented
116	#468	C₂×C₁₁	1	2	yes	—
117	#464	C₂×C₁₁	1	2	yes	complemented
118	#462	C₂×C₁₁	1	1	yes	characteristic, complemented
119	#461	C₂×C₁₁	1	1	yes	characteristic, complemented
120	#423	C₂×C₁₁	1	1	yes	characteristic, complemented
121	C₂²×C₁₁	—	2	1	no	—
122	C₂²×C₁₁	—	2	1	no	—
123	C₂²×C₁₁	—	2	1	no	—
124	C₂²×C₁₁	—	4	1	no	—
125	C₂²×C₁₁	—	4	1	no	—
126	C₂²×C₁₁	—	4	1	no	—
127	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₁₁	—	2	1	no	—
129	C₄×C₁₁	—	2	1	no	—
130	C₄×C₁₁	—	2	1	no	—
131	C₄×C₁₁	—	2	1	no	—
132	C₄×C₁₁	—	4	1	no	—
133	C₄×C₁₁	—	4	2	no	—
134	C₄×C₁₁	—	4	1	no	—
135	C₄×C₁₁	—	4	1	no	—
136	C₄×C₁₁	—	4	2	no	—
137	C₄×C₁₁	#137	1	2	yes	—
138	C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×Q₁₆	C₂²×C₁₁	1	2	yes	—
141	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
142	C₂×M₁₆	C₂²×C₁₁	1	2	yes	—
143	C₄⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
144	C₄⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
145	#126	—	2	1	no	—
146	#125	—	2	1	no	—
147	#125	C₂²×C₁₁	1	1	yes	characteristic
148	#124	C₂²×C₁₁	1	1	yes	characteristic
149	#123	—	2	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
152	C₂²⋊Q₈	C₂²×C₁₁	1	2	yes	—
153	#121	C₂²×C₁₁	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
156	C₄×Q₈	C₂²×C₁₁	1	2	yes	—
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	C₂²×C₁₁	1	1	yes	characteristic
159	#117	—	2	1	no	—
160	#117	—	2	1	no	—
161	C₂×C₄²	C₂²×C₁₁	1	1	yes	characteristic
162	#107	C₂²×C₁₁	1	2	yes	—
163	#106	C₂²×C₁₁	1	2	yes	—
164	C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
166	C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
167	#103	C₂²×C₁₁	1	2	yes	—
168	#103	C₂²×C₁₁	1	2	yes	—
169	#103	C₂²×C₁₁	1	2	yes	—
170	#102	C₂²×C₁₁	1	2	yes	—
171	C₂×C₁₁	—	4	1	no	—
172	C₂×C₁₁	—	8	1	no	—
173	C₂×C₁₁	#537	1	1	yes	characteristic
174	C₂×C₁₁	C₂×#137	1	1	yes	characteristic
175	C₂×C₁₁	#575	1	1	yes	characteristic
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₁₁	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	D₄×C₁₁	1	1	yes	characteristic
183	C₂²×C₄	D₄×C₁₁	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
185	Q₁₆	—	4	2	no	—
186	SD₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	C₂³×C₁₁	1	2	yes	—
189	C₂×C₈	C₂³×C₁₁	1	2	yes	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
200	C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	2	no	—
210	C₄²	D₄×C₁₁	1	1	yes	characteristic
211	C₄²	D₄×C₁₁	1	1	yes	characteristic
212	C₄²	C₂³×C₁₁	1	1	yes	characteristic
213	C₄²	C₂³×C₁₁	1	1	yes	characteristic
214	C₁₁	#3252	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	1	no	—
216	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	C₁₁×#137	1	2	yes	—
260	C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
261	C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	8	1	no	—
264	C₂	C₁₁×#537	1	1	yes	characteristic
265	C₂	C₂×C₁₁×#137	1	1	yes	characteristic
266	C₂	C₁₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁
prime radical	C₁₁×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₁×#533	C₂	semidirect
3. C₁₁×#479	C₂	semidirect
4. C₁₁×#477	C₂	semidirect
5. C₁₁×#476	C₂	semidirect
6. C₁₁×#470	C₂	semidirect
8. C₁₁×#464	C₂	semidirect
9. C₁₁×#462	C₂	semidirect
10. C₁₁×#461	C₂	semidirect
11. C₁₁×#423	C₂	semidirect
81. #3252	C₁₁	direct (N × H)
110. #533	C₂×C₁₁	semidirect
112. #479	C₂×C₁₁	semidirect
113. #477	C₂×C₁₁	semidirect
114. #476	C₂×C₁₁	semidirect
115. #470	C₂×C₁₁	semidirect
117. #464	C₂×C₁₁	semidirect
118. #462	C₂×C₁₁	semidirect
119. #461	C₂×C₁₁	semidirect
120. #423	C₂×C₁₁	semidirect
214. C₁₁	#3252	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×#533	2
C₁₁×#529	2
C₁₁×#479	2
C₁₁×#477	2
C₁₁×#476	2
C₁₁×#470	2
C₁₁×#468	2
C₁₁×#464	2
C₁₁×#462	2
C₁₁×#461	2
C₁₁×#423	2
#3252	11

Subgroups — C11×#3252 #13066090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3252 #13066090