Subgroups — #14670172

Order 2816.

342 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
2	#13066694	C₂	1	1	yes	maximal, characteristic
3	#13066693	C₂	1	2	yes	maximal
4	#13066693	C₂	1	2	yes	maximal
5	C₁₁×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
13	C₄²×Dic₁₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₁×C₁₆	—	2	2	no	—
27	C₂×C₁₁×C₁₆	—	2	2	no	—
28	C₂×C₁₁×C₁₆	—	2	2	no	—
29	C₄×C₈×C₁₁	—	2	2	no	—
30	C₈×Dic₁₁	—	2	2	no	—
31	C₈×Dic₁₁	—	2	2	no	—
32	C₈×Dic₁₁	—	2	1	no	—
33	C₈×Dic₁₁	—	2	1	no	—
34	C₈×Dic₁₁	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	—	2	2	no	—
37	C₂×C₁₁⋊C₁₆	—	2	2	no	—
38	C₄×C₁₁⋊C₈	—	2	2	no	—
39	#12894	—	11	1	no	maximal, Sylow
40	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₁	—	2	1	no	—
48	C₂×C₈×C₁₁	—	2	1	no	—
49	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
50	C₂×C₈×C₁₁	#41	1	2	yes	—
51	C₂×C₈×C₁₁	#41	1	2	yes	—
52	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
53	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
54	C₄²×C₁₁	—	2	1	no	—
55	C₄²×C₁₁	—	2	1	no	—
56	C₄×Dic₁₁	—	2	2	no	—
57	C₄×Dic₁₁	—	2	2	no	—
58	C₄×Dic₁₁	—	2	2	no	—
59	C₄×Dic₁₁	—	2	1	no	—
60	C₄×Dic₁₁	—	2	1	no	—
61	C₄×Dic₁₁	—	2	1	no	—
62	C₄×Dic₁₁	—	2	1	no	—
63	C₄×Dic₁₁	C₁₆	1	4	yes	complemented
64	C₂×C₁₁⋊C₈	—	2	1	no	—
65	C₂×C₁₁⋊C₈	—	2	1	no	—
66	C₂×C₁₁⋊C₈	—	2	1	no	—
67	C₂×C₁₁⋊C₈	—	2	1	no	—
68	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
69	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
70	C₁₁×C₁₆	—	4	2	no	—
71	C₁₁⋊C₁₆	—	4	2	no	—
72	C₄²×C₈	—	11	1	no	—
73	#1374	—	11	2	no	—
74	#1374	—	11	2	no	—
75	#1374	—	11	1	no	—
76	#1374	D₁₁	1	1	yes	characteristic
77	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₁	—	2	1	no	—
79	C₂×C₄×C₁₁	—	2	1	no	—
80	C₂×C₄×C₁₁	—	2	1	no	—
81	C₂×C₄×C₁₁	—	2	1	no	—
82	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
83	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
84	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
85	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×Dic₁₁	—	2	1	no	—
91	C₂×Dic₁₁	—	2	1	no	—
92	C₂×Dic₁₁	—	2	1	no	—
93	C₂×Dic₁₁	—	2	1	no	—
94	C₂×Dic₁₁	—	2	1	no	—
95	C₂×Dic₁₁	—	2	1	no	—
96	C₂×Dic₁₁	—	2	1	no	—
97	C₂×Dic₁₁	—	2	1	no	—
98	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
99	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
100	C₂×Dic₁₁	M₃₂	1	2	yes	—
101	C₂×Dic₁₁	M₃₂	1	2	yes	—
102	C₈×C₁₁	—	2	2	no	—
103	C₈×C₁₁	#126	1	2	yes	—
104	C₈×C₁₁	#126	1	2	yes	—
105	C₁₁⋊C₈	—	2	2	no	—
106	C₁₁⋊C₈	—	2	2	no	—
107	C₂²×C₁₆	—	11	2	no	—
108	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
109	C₂×C₄×C₈	—	11	1	no	—
110	C₂×C₄×C₈	—	11	1	no	—
111	C₂×C₄×C₈	—	11	1	no	—
112	C₂×C₄×C₈	—	11	1	no	—
113	C₂×C₄×C₈	—	11	1	no	—
114	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
115	C₄³	—	11	1	no	—
116	C₂²×C₁₁	#431	1	1	yes	characteristic
117	C₂²×C₁₁	#431	1	1	yes	characteristic
118	C₂²×C₁₁	#431	1	1	yes	characteristic
119	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
120	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
121	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
122	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
123	C₄×C₁₁	—	2	2	no	—
124	C₄×C₁₁	—	2	2	no	—
125	C₄×C₁₁	#383	1	1	yes	characteristic
126	C₄×C₁₁	#432	1	1	yes	characteristic
127	C₄×C₁₁	#432	1	1	yes	characteristic
128	C₄×C₁₁	#432	1	1	yes	characteristic
129	Dic₁₁	—	2	2	no	—
130	Dic₁₁	—	2	2	no	—
131	Dic₁₁	—	2	2	no	—
132	Dic₁₁	—	2	2	no	—
133	C₂²×C₈	—	11	1	no	—
134	C₂²×C₈	—	11	1	no	—
135	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
137	C₂×C₄²	—	11	1	no	—
138	C₂×C₄²	—	11	1	no	—
139	C₂×C₄²	—	11	1	no	—
140	C₂×C₄²	—	11	1	no	—
141	C₂×C₄²	—	11	1	no	—
142	C₂×C₄²	—	11	1	no	—
143	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	22	2	no	—
148	C₂×C₁₆	—	22	2	no	—
149	C₂×C₁₆	—	22	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	22	2	no	—
152	C₄×C₈	—	22	2	no	—
153	C₄×C₈	—	22	2	no	—
154	C₄×C₈	—	22	2	no	—
155	C₄×C₈	—	22	1	no	—
156	C₄×C₈	—	22	1	no	—
157	C₂×C₁₁	#1839	1	1	yes	characteristic
158	C₂×C₁₁	#2162	1	1	yes	characteristic
159	C₂×C₁₁	#2162	1	1	yes	characteristic
160	C₂×C₁₁	#2162	1	1	yes	characteristic
161	C₂×C₁₁	#2163	1	1	yes	characteristic
162	C₂×C₁₁	#2163	1	1	yes	characteristic
163	C₂×C₁₁	#2163	1	1	yes	characteristic
164	C₂²×C₄	—	11	1	no	—
165	C₂²×C₄	—	11	1	no	—
166	C₂²×C₄	—	11	1	no	—
167	C₂²×C₄	—	11	1	no	—
168	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
169	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	11	2	no	—
174	C₂×C₈	—	11	2	no	—
175	C₂×C₈	—	22	1	no	—
176	C₂×C₈	—	22	1	no	—
177	C₂×C₈	—	22	1	no	—
178	C₂×C₈	—	22	1	no	—
179	C₂×C₈	#4313	1	2	yes	—
180	C₂×C₈	#4339	1	2	yes	—
181	C₂×C₈	#4341	1	1	yes	characteristic
182	C₂×C₈	#4341	1	2	yes	—
183	C₂×C₈	#4343	1	1	yes	characteristic
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	11	4	no	—
187	C₄²	—	22	2	no	—
188	C₄²	—	22	2	no	—
189	C₄²	—	22	2	no	—
190	C₄²	—	22	1	no	—
191	C₄²	—	22	1	no	—
192	C₄²	—	22	1	no	—
193	C₄²	—	22	1	no	—
194	C₁₆	—	4	2	no	—
195	C₁₆	—	44	2	no	—
196	C₁₁	#12894	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	11	2	no	—
203	C₂×C₄	—	11	2	no	—
204	C₂×C₄	—	11	2	no	—
205	C₂×C₄	—	11	2	no	—
206	C₂×C₄	—	22	1	no	—
207	C₂×C₄	—	22	1	no	—
208	C₂×C₄	—	22	1	no	—
209	C₂×C₄	—	22	1	no	—
210	C₂×C₄	—	22	1	no	—
211	C₂×C₄	—	22	1	no	—
212	C₂×C₄	—	22	1	no	—
213	C₂×C₄	—	22	1	no	—
214	C₂×C₄	C₁₆×D₁₁	1	2	yes	—
215	C₂×C₄	#66862	1	2	yes	—
216	C₂×C₄	#66925	1	1	yes	characteristic
217	C₂×C₄	#66930	1	1	yes	characteristic
218	C₂×C₄	#66945	1	1	yes	characteristic
219	C₂×C₄	C₂×#4313	1	1	yes	characteristic
220	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
221	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	22	2	no	—
224	C₈	—	22	2	no	—
225	C₈	#66932	1	2	yes	—
226	C₈	#66951	1	2	yes	—
227	C₂²	#10622298	1	1	yes	characteristic
228	C₂²	#10622331	1	1	yes	characteristic
229	C₂²	#10622446	1	1	yes	characteristic
230	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
231	C₂²	C₂×#66862	1	1	yes	characteristic
232	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
233	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	22	2	no	—
237	C₄	—	22	2	no	—
238	C₄	—	22	2	no	—
239	C₄	—	22	2	no	—
240	C₄	#10622264	1	1	yes	characteristic
241	C₄	#10622318	1	1	yes	characteristic
242	C₄	#10622332	1	1	yes	characteristic
243	C₄	#10622448	1	1	yes	characteristic
244	C₂	#13068758	1	1	yes	characteristic
245	C₂	#13070412	1	1	yes	characteristic
246	C₂	#13070413	1	1	yes	characteristic
247	C₂	#13070414	1	1	yes	characteristic
248	C₂	#13070416	1	1	yes	characteristic
249	C₂	#13070417	1	1	yes	characteristic
250	C₂	#13070418	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₁₁×#1374	C₂
socle, prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1374	D₁₁
hypocenter	C₁₁	#12894
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
63. C₄×Dic₁₁	C₁₆	semidirect
196. C₁₁	#12894	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₁₁	2
#13066694	2
#13066693	2
C₁₁×#1374	2
#12894	11