Subgroups — C₁₁×#23198 #14600391

Order 2816.

946 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#2949	C₂	1	4	yes	maximal, complemented
3	C₁₁×#1977	C₂	1	4	yes	maximal, complemented
4	C₁₁×#1887	C₂	1	2	yes	maximal
5	C₁₁×#1886	C₂	1	2	yes	maximal, complemented
6	C₁₁×#1772	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₁×SD₁₆	C₂²	1	4	yes	—
8	C₂×C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#127	C₂²	1	1	yes	characteristic
10	C₁₁×#519	C₂²	1	2	yes	—
11	C₁₁×#518	C₂²	1	2	yes	—
12	C₁₁×#492	C₄	1	8	yes	complemented
13	C₁₁×#440	—	2	8	no	—
14	C₄×C₁₁×SD₁₆	—	2	8	no	—
15	C₂×C₁₁×#108	C₂²	1	2	yes	—
16	C₁₁×#427	C₂²	1	2	yes	—
17	C₁₁×#419	C₂²	1	4	yes	—
18	C₁₁×#418	C₂²	1	4	yes	—
19	C₂×C₁₁×#101	C₂²	1	2	yes	—
20	C₂×C₁₁×#100	C₂²	1	2	yes	—
21	C₁₁×#405	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
23	C₁₁×#339	C₂²	1	4	yes	—
24	C₂²×Q₈×C₁₁	C₂³	1	2	yes	—
25	C₂²×D₄×C₁₁	—	4	1	no	—
26	C₂²×D₄×C₁₁	C₂³	1	2	yes	—
27	C₂×C₁₁×SD₁₆	—	2	4	no	—
28	C₂×C₁₁×SD₁₆	—	2	4	no	—
29	C₂×C₁₁×SD₁₆	—	2	8	no	—
30	C₂×C₁₁×SD₁₆	C₂×C₄	1	16	yes	complemented
31	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
33	C₁₁×C₄⋊Q₈	—	4	1	no	—
34	C₁₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
35	C₁₁×#127	—	4	1	no	—
36	C₁₁×#127	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₁	—	2	8	no	—
38	C₄×D₄×C₁₁	—	2	8	no	—
39	C₁₁×#117	—	4	2	no	—
40	C₁₁×#117	C₂³	1	4	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
42	C₂×C₁₁×C₂²⋊C₄	—	4	2	no	—
43	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
44	C₁₁×#108	—	2	4	no	—
45	C₁₁×#107	D₄	1	4	yes	complemented
46	C₁₁×#106	D₄	1	4	yes	complemented
47	C₁₁×#103	—	2	8	no	—
48	C₁₁×#102	—	2	8	no	—
49	C₁₁×#101	—	4	2	no	—
50	C₁₁×#100	—	4	2	no	—
51	C₁₁×#97	—	2	4	no	—
52	C₄×C₈×C₁₁	—	2	4	no	—
53	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	complemented
54	#23198	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₁	—	2	2	no	—
56	C₂×Q₈×C₁₁	—	2	2	no	—
57	C₂×Q₈×C₁₁	—	2	4	no	—
58	C₂×Q₈×C₁₁	—	2	4	no	—
59	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₁₁	—	2	4	no	—
61	C₂×D₄×C₁₁	—	2	4	no	—
62	C₂×D₄×C₁₁	—	2	2	no	—
63	C₂×D₄×C₁₁	—	4	4	no	—
64	C₂×D₄×C₁₁	—	8	4	no	—
65	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₁	—	2	1	no	—
67	C₂²×C₄×C₁₁	—	2	2	no	—
68	C₂²×C₄×C₁₁	—	4	4	no	—
69	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
70	C₁₁×SD₁₆	—	2	16	no	—
71	C₁₁×SD₁₆	—	2	16	no	—
72	C₁₁×M₁₆	—	2	4	no	—
73	C₁₁×M₁₆	—	2	4	no	—
74	C₂×C₈×C₁₁	—	2	4	no	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₁	#41	1	4	yes	—
79	C₁₁×C₄⋊C₄	—	4	4	no	—
80	C₁₁×C₄⋊C₄	—	4	4	no	—
81	C₁₁×C₄⋊C₄	C₂×D₄	1	8	yes	—
82	C₁₁×C₂²⋊C₄	—	2	4	no	—
83	C₁₁×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₁₁	—	2	4	no	—
85	C₄²×C₁₁	—	2	1	no	—
86	C₄²×C₁₁	—	4	4	no	—
87	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
88	C₂×#492	C₂×C₁₁	1	1	yes	characteristic
89	#2949	C₂×C₁₁	1	4	yes	complemented
90	#1977	C₂×C₁₁	1	4	yes	complemented
91	#1887	C₂×C₁₁	1	2	yes	—
92	#1886	C₂×C₁₁	1	2	yes	complemented
93	#1772	C₂×C₁₁	1	2	yes	complemented
94	C₂³×C₁₁	—	2	4	no	—
95	C₂³×C₁₁	—	4	2	no	—
96	C₂³×C₁₁	—	4	2	no	—
97	C₂³×C₁₁	—	8	4	no	—
98	C₂³×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
99	Q₈×C₁₁	—	2	8	no	—
100	Q₈×C₁₁	—	4	4	no	—
101	D₄×C₁₁	—	2	8	no	—
102	D₄×C₁₁	—	4	4	no	—
103	D₄×C₁₁	—	8	8	no	—
104	C₂×C₄×C₁₁	—	2	4	no	—
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	4	no	—
107	C₂×C₄×C₁₁	—	2	8	no	—
108	C₂×C₄×C₁₁	—	4	2	no	—
109	C₂×C₄×C₁₁	—	4	2	no	—
110	C₂×C₄×C₁₁	—	16	4	no	—
111	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₈×C₁₁	—	4	4	no	—
116	C₈×C₁₁	C₄×D₄	1	4	yes	—
117	C₈×C₁₁	C₄×D₄	1	4	yes	—
118	C₂²×SD₁₆	C₂²×C₁₁	1	4	yes	—
119	C₂×C₄⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
120	C₂×#127	C₂²×C₁₁	1	1	yes	characteristic
121	#519	C₂²×C₁₁	1	2	yes	—
122	#518	C₂²×C₁₁	1	2	yes	—
123	#492	C₄×C₁₁	1	8	yes	complemented
124	#440	—	2	8	no	—
125	C₄×SD₁₆	—	2	8	no	—
126	C₂×#108	C₂²×C₁₁	1	2	yes	—
127	#427	C₂²×C₁₁	1	2	yes	—
128	#419	C₂²×C₁₁	1	4	yes	—
129	#418	C₂²×C₁₁	1	4	yes	—
130	C₂×#101	C₂²×C₁₁	1	2	yes	—
131	C₂×#100	C₂²×C₁₁	1	2	yes	—
132	#405	C₂²×C₁₁	1	4	yes	—
133	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
134	#339	C₂²×C₁₁	1	4	yes	—
135	C₂²×C₁₁	—	2	2	no	—
136	C₂²×C₁₁	—	4	4	no	—
137	C₂²×C₁₁	—	8	4	no	—
138	C₂²×C₁₁	—	8	4	no	—
139	C₂²×C₁₁	—	16	2	no	—
140	C₂²×C₁₁	#518	1	1	yes	characteristic
141	C₂²×C₁₁	D₄²	1	1	yes	characteristic
142	C₂²×C₁₁	#559	1	1	yes	characteristic
143	C₄×C₁₁	—	2	4	no	—
144	C₄×C₁₁	—	4	4	no	—
145	C₄×C₁₁	—	4	8	no	—
146	C₄×C₁₁	C₂×C₄×D₄	1	2	yes	—
147	C₄×C₁₁	#550	1	2	yes	—
148	C₂²×Q₈	C₂³×C₁₁	1	2	yes	—
149	C₂²×D₄	—	4	1	no	—
150	C₂²×D₄	C₂³×C₁₁	1	2	yes	—
151	C₂×SD₁₆	—	2	4	no	—
152	C₂×SD₁₆	—	2	4	no	—
153	C₂×SD₁₆	—	2	8	no	—
154	C₂×SD₁₆	C₂×C₄×C₁₁	1	16	yes	complemented
155	C₂×M₁₆	C₂³×C₁₁	1	4	yes	—
156	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
157	C₄⋊Q₈	—	4	1	no	—
158	C₄⋊Q₈	C₂×C₄×C₁₁	1	4	yes	complemented
159	#127	—	4	1	no	—
160	#127	C₂×C₄×C₁₁	1	4	yes	—
161	C₄×Q₈	—	2	8	no	—
162	C₄×D₄	—	2	8	no	—
163	#117	—	4	2	no	—
164	#117	C₂³×C₁₁	1	4	yes	—
165	C₂×C₄⋊C₄	—	4	1	no	—
166	C₂×C₂²⋊C₄	—	4	2	no	—
167	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
168	#108	—	2	4	no	—
169	#107	D₄×C₁₁	1	4	yes	complemented
170	#106	D₄×C₁₁	1	4	yes	complemented
171	#103	—	2	8	no	—
172	#102	—	2	8	no	—
173	#101	—	4	2	no	—
174	#100	—	4	2	no	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	C₂×C₄×C₁₁	1	4	yes	complemented
178	C₂×C₁₁	—	2	1	no	—
179	C₂×C₁₁	—	2	1	no	—
180	C₂×C₁₁	—	8	4	no	—
181	C₂×C₁₁	#2366	1	1	yes	characteristic
182	C₂×C₁₁	#3291	1	2	yes	—
183	C₂⁴	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂²×C₄×C₁₁	1	4	yes	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	8	4	no	—
193	C₂×D₄	C₂²×C₄×C₁₁	1	4	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
198	SD₁₆	—	2	16	no	—
199	SD₁₆	—	2	16	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²×C₄×C₁₁	1	4	yes	—
204	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
206	C₂×C₈	C₁₁×#41	1	4	yes	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₂×D₄×C₁₁	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂²×C₄×C₁₁	1	2	yes	—
216	C₁₁	#23198	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	8	4	no	—
221	C₂³	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
222	Q₈	—	2	8	no	—
223	Q₈	—	4	4	no	—
224	D₄	—	2	8	no	—
225	D₄	—	4	4	no	—
226	D₄	—	8	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
236	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
237	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
238	C₈	—	4	4	no	—
239	C₈	C₄×D₄×C₁₁	1	4	yes	—
240	C₈	C₄×D₄×C₁₁	1	4	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₁₁×#518	1	1	yes	characteristic
247	C₂²	D₄²×C₁₁	1	1	yes	characteristic
248	C₂²	C₁₁×#559	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	4	no	—
252	C₄	C₂×C₄×D₄×C₁₁	1	2	yes	—
253	C₄	C₁₁×#550	1	2	yes	—
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	8	4	no	—
257	C₂	C₁₁×#2366	1	1	yes	characteristic
258	C₂	C₁₁×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁×#2949	C₂	semidirect
3. C₁₁×#1977	C₂	semidirect
5. C₁₁×#1886	C₂	semidirect
6. C₁₁×#1772	C₂	semidirect
12. C₁₁×#492	C₄	semidirect
30. C₂×C₁₁×SD₁₆	C₂×C₄	semidirect
34. C₁₁×C₄⋊Q₈	C₂×C₄	semidirect
45. C₁₁×#107	D₄	semidirect
46. C₁₁×#106	D₄	semidirect
53. C₄×C₈×C₁₁	C₂×C₄	semidirect
54. #23198	C₁₁	direct (N × H)
89. #2949	C₂×C₁₁	semidirect
90. #1977	C₂×C₁₁	semidirect
92. #1886	C₂×C₁₁	semidirect
93. #1772	C₂×C₁₁	semidirect
123. #492	C₄×C₁₁	semidirect
154. C₂×SD₁₆	C₂×C₄×C₁₁	semidirect
158. C₄⋊Q₈	C₂×C₄×C₁₁	semidirect
169. #107	D₄×C₁₁	semidirect
170. #106	D₄×C₁₁	semidirect
177. C₄×C₈	C₂×C₄×C₁₁	semidirect
216. C₁₁	#23198	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#492	2
C₁₁×#2949	2
C₁₁×#1977	2
C₁₁×#1887	2
C₁₁×#1886	2
C₁₁×#1772	2
#23198	11

Subgroups — C11×#23198 #14600391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23198 #14600391