Subgroups — C₁₁×#22536 #14599729

Order 2816.

1314 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	C₂×C₁₁×#550	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₁×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₁₁×#1965	C₂	1	6	yes	maximal, complemented
4	C₁₁×#1800	C₂	1	3	yes	maximal, complemented
5	C₄×C₁₁×#100	C₂	1	3	yes	maximal, complemented
6	C₂²×C₁₁×#41	C₄	1	2	yes	—
7	C₁₁×#550	—	2	2	no	—
8	C₁₁×#550	—	2	6	no	—
9	C₂×C₁₁×#127	C₂²	1	3	yes	—
10	C₂×C₁₁×#121	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₁	C₂²	1	3	yes	—
13	C₁₁×#446	—	2	6	no	—
14	C₈×Q₈×C₁₁	—	2	2	no	—
15	C₁₁×#423	C₂²	1	6	yes	—
16	C₂×C₁₁×#100	C₂²	1	6	yes	—
17	C₂×C₁₁×#100	C₂²	1	6	yes	—
18	C₄×C₁₁×M₁₆	C₂²	1	6	yes	—
19	C₁₁×#343	C₂²	1	1	yes	characteristic
20	C₁₁×#343	C₂²	1	3	yes	—
21	C₂×C₁₁×#41	—	2	6	no	—
22	C₂×C₁₁×#41	—	2	2	no	—
23	C₂×C₁₁×#41	—	4	8	no	—
24	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₁₁	C₂×C₄	1	6	yes	—
26	C₂²×D₄×C₁₁	C₂×C₄	1	6	yes	—
27	C₂²×D₄×C₁₁	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
29	C₂×C₁₁×M₁₆	C₂³	1	6	yes	—
30	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
31	C₁₁×#127	—	2	6	no	—
32	C₁₁×#127	—	2	6	no	—
33	C₁₁×#121	—	2	12	no	—
34	C₁₁×#121	—	2	12	no	—
35	C₄×Q₈×C₁₁	D₄	1	2	yes	—
36	C₄×Q₈×C₁₁	D₄	1	6	yes	—
37	C₄×D₄×C₁₁	—	2	6	no	—
38	C₄×D₄×C₁₁	—	2	6	no	—
39	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₂×C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
41	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
43	C₁₁×C₄⋊C₈	—	2	6	no	—
44	C₁₁×C₄⋊C₈	—	2	6	no	—
45	C₁₁×#100	—	2	12	no	—
46	C₁₁×#100	—	4	6	no	—
47	C₁₁×#97	—	2	6	no	—
48	C₄×C₈×C₁₁	—	2	6	no	—
49	#22536	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
51	C₁₁×#41	—	4	24	no	—
52	C₁₁×#41	—	4	8	no	—
53	C₂×Q₈×C₁₁	—	2	4	no	—
54	C₂×Q₈×C₁₁	C₂²⋊C₄	1	3	yes	—
55	C₂×Q₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₄×C₁₁	—	2	6	no	—
57	C₂×D₄×C₁₁	—	2	6	no	—
58	C₂×D₄×C₁₁	—	2	12	no	—
59	C₂×D₄×C₁₁	—	2	6	no	—
60	C₂×D₄×C₁₁	—	2	6	no	—
61	C₂×D₄×C₁₁	—	2	3	no	—
62	C₂×D₄×C₁₁	—	2	3	no	—
63	C₂×D₄×C₁₁	—	4	12	no	—
64	C₂×D₄×C₁₁	—	8	6	no	—
65	C₂²×C₄×C₁₁	—	2	6	no	—
66	C₂²×C₄×C₁₁	—	2	6	no	—
67	C₂²×C₄×C₁₁	—	4	12	no	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
72	C₁₁×M₁₆	—	2	6	no	—
73	C₁₁×M₁₆	—	2	6	no	—
74	C₁₁×M₁₆	—	2	2	no	—
75	C₁₁×M₁₆	#41	1	4	yes	complemented
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	2	6	no	—
78	C₁₁×C₄⋊C₄	—	2	2	no	—
79	C₁₁×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
80	C₁₁×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
81	C₁₁×C₂²⋊C₄	—	2	12	no	—
82	C₄²×C₁₁	C₂×D₄	1	6	yes	—
83	C₄²×C₁₁	C₂×D₄	1	6	yes	—
84	C₂×#550	C₂×C₁₁	1	1	yes	characteristic
85	Q₈×M₁₆	C₂×C₁₁	1	2	yes	complemented
86	#1965	C₂×C₁₁	1	6	yes	complemented
87	#1800	C₂×C₁₁	1	3	yes	complemented
88	C₄×#100	C₂×C₁₁	1	3	yes	complemented
89	C₂³×C₁₁	—	2	6	no	—
90	C₂³×C₁₁	—	2	6	no	—
91	C₂³×C₁₁	—	2	6	no	—
92	C₂³×C₁₁	—	4	6	no	—
93	C₂³×C₁₁	—	4	6	no	—
94	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
95	Q₈×C₁₁	—	2	6	no	—
96	Q₈×C₁₁	#100	1	4	yes	complemented
97	D₄×C₁₁	—	4	12	no	—
98	D₄×C₁₁	—	4	6	no	—
99	D₄×C₁₁	—	4	6	no	—
100	D₄×C₁₁	—	4	12	no	—
101	D₄×C₁₁	—	8	12	no	—
102	C₂×C₄×C₁₁	—	2	6	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	4	3	no	—
105	C₂×C₄×C₁₁	—	4	12	no	—
106	C₂×C₄×C₁₁	—	4	12	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
110	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
111	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₈×C₁₁	—	2	4	no	—
116	C₈×C₁₁	—	4	6	no	—
117	C₂²×#41	C₄×C₁₁	1	2	yes	—
118	#550	—	2	2	no	—
119	#550	—	2	6	no	—
120	C₂×#127	C₂²×C₁₁	1	3	yes	—
121	C₂×#121	C₄×C₁₁	1	6	yes	—
122	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
123	C₂×C₄×D₄	C₂²×C₁₁	1	3	yes	—
124	#446	—	2	6	no	—
125	C₈×Q₈	—	2	2	no	—
126	#423	C₂²×C₁₁	1	6	yes	—
127	C₂×#100	C₂²×C₁₁	1	6	yes	—
128	C₂×#100	C₂²×C₁₁	1	6	yes	—
129	C₄×M₁₆	C₂²×C₁₁	1	6	yes	—
130	#343	C₂²×C₁₁	1	3	yes	—
131	#343	C₂²×C₁₁	1	1	yes	characteristic
132	C₂²×C₁₁	—	2	1	no	—
133	C₂²×C₁₁	—	2	1	no	—
134	C₂²×C₁₁	—	4	6	no	—
135	C₂²×C₁₁	—	4	6	no	—
136	C₂²×C₁₁	—	4	6	no	—
137	C₂²×C₁₁	—	8	6	no	—
138	C₂²×C₁₁	—	8	6	no	—
139	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₁₁	C₄×#41	1	1	yes	characteristic
141	C₂²×C₁₁	#520	1	1	yes	characteristic
142	C₄×C₁₁	—	2	3	no	—
143	C₄×C₁₁	—	2	6	no	—
144	C₄×C₁₁	—	4	2	no	—
145	C₄×C₁₁	C₂×#100	1	6	yes	—
146	C₄×C₁₁	#548	1	2	yes	—
147	C₄×C₁₁	#550	1	2	yes	—
148	C₂×#41	—	2	6	no	—
149	C₂×#41	—	2	2	no	—
150	C₂×#41	—	4	8	no	—
151	C₂²×Q₈	C₂×C₄×C₁₁	1	1	yes	characteristic
152	C₂²×D₄	C₂×C₄×C₁₁	1	6	yes	—
153	C₂²×D₄	C₂×C₄×C₁₁	1	6	yes	—
154	C₂²×D₄	C₂³×C₁₁	1	3	yes	—
155	C₂³×C₄	C₂×C₄×C₁₁	1	6	yes	—
156	C₂×M₁₆	C₂³×C₁₁	1	6	yes	—
157	C₂×M₁₆	C₂³×C₁₁	1	2	yes	—
158	#127	—	2	6	no	—
159	#127	—	2	6	no	—
160	#121	—	2	12	no	—
161	#121	—	2	12	no	—
162	C₄×Q₈	D₄×C₁₁	1	2	yes	—
163	C₄×Q₈	D₄×C₁₁	1	6	yes	—
164	C₄×D₄	—	2	6	no	—
165	C₄×D₄	—	2	6	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	3	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	6	yes	—
169	C₂×C₄²	C₂³×C₁₁	1	3	yes	—
170	C₄⋊C₈	—	2	6	no	—
171	C₄⋊C₈	—	2	6	no	—
172	#100	—	2	12	no	—
173	#100	—	4	6	no	—
174	#97	—	2	6	no	—
175	C₄×C₈	—	2	6	no	—
176	C₂×C₁₁	—	2	1	no	—
177	C₂×C₁₁	—	2	1	no	—
178	C₂×C₁₁	—	8	6	no	—
179	C₂×C₁₁	#2341	1	1	yes	characteristic
180	C₂×C₁₁	C₂²×#100	1	1	yes	characteristic
181	C₂×C₁₁	#2888	1	1	yes	characteristic
182	C₂⁴	C₂²×C₄×C₁₁	1	6	yes	—
183	#41	—	4	24	no	—
184	#41	—	4	8	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	C₁₁×C₂²⋊C₄	1	3	yes	—
187	C₂×Q₈	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×D₄	—	2	6	no	—
189	C₂×D₄	—	2	6	no	—
190	C₂×D₄	—	2	12	no	—
191	C₂×D₄	—	2	6	no	—
192	C₂×D₄	—	2	6	no	—
193	C₂×D₄	—	2	3	no	—
194	C₂×D₄	—	2	3	no	—
195	C₂×D₄	—	4	12	no	—
196	C₂×D₄	—	8	6	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	4	12	no	—
200	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
203	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	6	no	—
206	M₁₆	—	2	6	no	—
207	M₁₆	C₁₁×#41	1	4	yes	complemented
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	6	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	C₁₁×C₂²⋊C₄	1	6	yes	—
212	C₄⋊C₄	C₁₁×C₂²⋊C₄	1	6	yes	—
213	C₂²⋊C₄	—	2	12	no	—
214	C₄²	C₂×D₄×C₁₁	1	6	yes	—
215	C₄²	C₂×D₄×C₁₁	1	6	yes	—
216	C₁₁	#22536	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	2	6	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	2	6	no	—
220	C₂³	—	4	6	no	—
221	C₂³	—	4	6	no	—
222	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
223	Q₈	—	2	6	no	—
224	Q₈	C₁₁×#100	1	4	yes	complemented
225	D₄	—	4	12	no	—
226	D₄	—	4	6	no	—
227	D₄	—	4	12	no	—
228	D₄	—	4	6	no	—
229	D₄	—	8	12	no	—
230	C₂×C₄	—	2	6	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	3	no	—
233	C₂×C₄	—	4	12	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	6	yes	—
237	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
238	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
239	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
240	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
241	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	4	6	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	6	no	—
248	C₂²	—	4	6	no	—
249	C₂²	—	4	6	no	—
250	C₂²	—	8	6	no	—
251	C₂²	—	8	6	no	—
252	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²	C₄×C₁₁×#41	1	1	yes	characteristic
254	C₂²	C₁₁×#520	1	1	yes	characteristic
255	C₄	—	2	3	no	—
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	C₂×C₁₁×#100	1	6	yes	—
259	C₄	C₁₁×#548	1	2	yes	—
260	C₄	C₁₁×#550	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	8	6	no	—
264	C₂	C₁₁×#2341	1	1	yes	characteristic
265	C₂	C₂²×C₁₁×#100	1	1	yes	characteristic
266	C₂	C₁₁×#2888	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#550	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄×Q₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₁×M₁₆	C₂	semidirect
3. C₁₁×#1965	C₂	semidirect
4. C₁₁×#1800	C₂	semidirect
5. C₄×C₁₁×#100	C₂	semidirect
49. #22536	C₁₁	direct (N × H)
75. C₁₁×M₁₆	#41	semidirect
85. Q₈×M₁₆	C₂×C₁₁	semidirect
86. #1965	C₂×C₁₁	semidirect
87. #1800	C₂×C₁₁	semidirect
88. C₄×#100	C₂×C₁₁	semidirect
96. Q₈×C₁₁	#100	semidirect
207. M₁₆	C₁₁×#41	semidirect
216. C₁₁	#22536	direct (N × H)
224. Q₈	C₁₁×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#550	2
Q₈×C₁₁×M₁₆	2
C₁₁×#1965	2
C₁₁×#1800	2
C₄×C₁₁×#100	2
#22536	11

Subgroups — C11×#22536 #14599729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#22536 #14599729