Subgroups — #14724304

Order 2816.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079202	C₂	1	1	yes	maximal, characteristic
2	#13077546	C₂	1	2	yes	maximal
3	#13067512	C₂	1	4	yes	maximal, complemented
4	#13067422	C₂	1	4	yes	maximal, complemented
5	#13067070	C₂	1	2	yes	maximal
6	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₁₁×#1567	C₂	1	1	yes	maximal, characteristic
8	#10623001	C₂²	1	2	yes	—
9	C₁₁×#423	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	#10622735	—	2	2	no	—
12	#10622732	—	2	2	no	—
13	#10622729	C₄	1	4	yes	—
14	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
16	#10622444	—	2	2	no	—
17	#10622444	C₂²	1	4	yes	—
18	#10622443	—	2	2	no	—
19	#10622443	C₂²	1	4	yes	—
20	#10622312	C₄	1	1	yes	characteristic
21	C₄×Dic₄₄	C₂²	1	4	yes	—
22	#10622295	C₄	1	1	yes	characteristic
23	#10622294	C₄	1	2	yes	—
24	C₁₁×#335	C₂²	1	4	yes	—
25	#10622122	C₂²	1	8	yes	—
26	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
27	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
28	C₂×C₁₁×M₁₆	—	2	2	no	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
31	#66979	—	2	2	no	—
32	#66977	—	2	2	no	—
33	#66977	C₂×C₄	1	4	yes	—
34	C₂²×C₁₁⋊C₈	—	2	1	no	—
35	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
37	C₂×Dic₄₄	C₂×C₄	1	4	yes	—
38	#66923	—	2	2	no	—
39	#66922	C₂×C₄	1	2	yes	—
40	#66922	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₂	—	2	2	no	—
42	C₄×Dic₂₂	C₂³	1	4	yes	—
43	C₁₁×C₄⋊C₈	—	2	2	no	—
44	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
45	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁	C₂³	1	2	yes	—
48	#66882	C₂×C₄	1	4	yes	—
49	#66880	C₂×C₄	1	8	yes	—
50	#66879	—	2	4	no	—
51	Dic₁₁⋊C₈	—	2	4	no	—
52	Dic₁₁⋊C₈	—	2	4	no	—
53	C₈×Dic₁₁	—	2	4	no	—
54	C₄×C₁₁⋊C₈	—	2	1	no	—
55	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
56	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
57	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
58	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
59	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
60	#17958	—	11	1	no	maximal, Sylow
61	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
64	C₂×Dic₂₂	—	2	2	no	—
65	C₂×Dic₂₂	C₂²×C₄	1	4	yes	—
66	C₁₁×M₁₆	—	4	4	no	—
67	C₂×C₈×C₁₁	—	2	4	no	—
68	C₂×C₈×C₁₁	—	2	4	no	—
69	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₁	#41	1	1	yes	characteristic
74	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
75	C₂²⋊Dic₁₁	—	2	4	no	—
76	C₄⋊Dic₁₁	—	2	2	no	—
77	C₄⋊Dic₁₁	—	2	2	no	—
78	C₄⋊Dic₁₁	—	2	2	no	—
79	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
80	Dic₁₁⋊C₄	—	2	4	no	—
81	Dic₁₁⋊C₄	—	2	4	no	—
82	C₄×Dic₁₁	—	2	4	no	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	2	2	no	—
85	C₂×C₁₁⋊C₈	—	2	2	no	—
86	C₂×C₁₁⋊C₈	Q₁₆	1	4	yes	complemented
87	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
91	Dic₄₄	—	2	8	no	—
92	#3170	—	11	1	no	—
93	#2960	—	11	2	no	—
94	#1591	—	11	4	no	—
95	C₈×Q₁₆	—	11	4	no	—
96	#1567	D₁₁	1	1	yes	characteristic
97	#1493	—	11	2	no	—
98	C₂×C₈²	—	11	1	no	—
99	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁	—	2	2	no	—
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₁₁	#135	1	2	yes	—
107	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₁₁	—	2	4	no	—
111	C₂×Dic₁₁	—	2	4	no	—
112	Dic₂₂	—	2	8	no	—
113	Dic₂₂	—	4	4	no	—
114	C₈×C₁₁	—	2	2	no	—
115	C₈×C₁₁	—	4	4	no	—
116	C₈×C₁₁	#131	1	4	yes	—
117	C₁₁⋊C₈	—	2	4	no	—
118	C₁₁⋊C₈	—	2	1	no	—
119	C₁₁⋊C₈	—	2	1	no	—
120	C₁₁⋊C₈	C₂×Q₁₆	1	2	yes	—
121	C₁₁⋊C₈	C₂×Q₁₆	1	2	yes	—
122	#533	—	11	2	no	—
123	#500	—	11	1	no	—
124	#494	—	11	1	no	—
125	#487	—	11	2	no	—
126	#467	—	11	4	no	—
127	#446	—	11	4	no	—
128	#446	—	22	2	no	—
129	C₈×Q₈	—	11	4	no	—
130	C₈×Q₈	—	22	2	no	—
131	C₄×Q₁₆	—	11	4	no	—
132	#424	—	22	2	no	—
133	#423	C₂×D₁₁	1	2	yes	—
134	#405	—	22	2	no	—
135	C₂×C₄×C₈	—	11	1	no	—
136	C₂×C₄×C₈	—	11	1	no	—
137	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
138	#335	C₂×D₁₁	1	4	yes	—
139	Q₈⋊C₈	—	11	8	no	—
140	C₈²	—	11	2	no	—
141	C₈²	—	11	2	no	—
142	C₂²×C₁₁	—	2	1	no	—
143	C₂²×C₁₁	—	2	1	no	—
144	C₂²×C₁₁	C₄×Q₁₆	1	2	yes	—
145	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₁₁	—	2	1	no	—
147	C₄×C₁₁	—	2	1	no	—
148	C₄×C₁₁	C₄×Q₁₆	1	2	yes	—
149	C₄×C₁₁	#443	1	2	yes	—
150	C₄×C₁₁	C₂×#131	1	2	yes	—
151	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
152	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
153	Dic₁₁	—	4	4	no	—
154	Dic₁₁	—	4	4	no	—
155	C₂×Q₁₆	—	11	4	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	11	1	no	—
158	C₂²×C₈	—	11	1	no	—
159	C₂²×C₈	—	22	1	no	—
160	C₂²×C₈	C₄×D₁₁	1	2	yes	—
161	C₄⋊Q₈	—	11	2	no	—
162	C₄⋊Q₈	—	11	2	no	—
163	#125	—	22	2	no	—
164	C₂²⋊Q₈	—	11	4	no	—
165	C₂²⋊Q₈	—	22	2	no	—
166	C₄×Q₈	—	11	4	no	—
167	C₄×Q₈	—	22	2	no	—
168	#117	—	22	2	no	—
169	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
170	#107	—	11	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	22	4	no	—
173	C₄⋊C₈	—	22	4	no	—
174	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
175	#103	—	11	8	no	—
176	#97	—	22	4	no	—
177	C₄×C₈	—	11	1	no	—
178	C₄×C₈	—	11	1	no	—
179	C₄×C₈	—	11	1	no	—
180	C₄×C₈	—	11	1	no	—
181	C₄×C₈	—	11	2	no	—
182	C₄×C₈	—	22	4	no	—
183	C₄×C₈	—	22	1	no	—
184	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
185	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₁₁	1	2	yes	—
187	C₂×C₁₁	—	2	2	no	—
188	C₂×C₁₁	#2931	1	1	yes	characteristic
189	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
190	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
191	C₂×Q₈	—	11	4	no	—
192	C₂×Q₈	—	22	2	no	—
193	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
194	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
195	C₂²×C₄	#4343	1	1	yes	characteristic
196	Q₁₆	—	22	8	no	—
197	M₁₆	—	4	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	11	4	no	—
201	C₂×C₈	—	11	1	no	—
202	C₂×C₈	—	11	1	no	—
203	C₂×C₈	—	11	1	no	—
204	C₂×C₈	—	11	1	no	—
205	C₂×C₈	—	22	4	no	—
206	C₂×C₈	—	22	2	no	—
207	C₂×C₈	—	22	2	no	—
208	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
210	C₄⋊C₄	—	11	4	no	—
211	C₄⋊C₄	—	22	2	no	—
212	C₄⋊C₄	—	22	2	no	—
213	C₄⋊C₄	—	22	4	no	—
214	C₄⋊C₄	—	22	2	no	—
215	C₄⋊C₄	—	22	4	no	—
216	C₂²⋊C₄	—	22	4	no	—
217	C₄²	—	22	4	no	—
218	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
219	C₄²	D₄×D₁₁	1	1	yes	characteristic
220	C₄²	#4343	1	1	yes	characteristic
221	C₄²	C₂³×D₁₁	1	1	yes	characteristic
222	C₁₁	#17958	1	1	yes	characteristic, complemented, Sylow
223	C₂³	#66946	1	1	yes	characteristic
224	Q₈	—	22	8	no	—
225	Q₈	—	44	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	22	4	no	—
229	C₂×C₄	—	22	4	no	—
230	C₂×C₄	#66946	1	1	yes	characteristic
231	C₂×C₄	#66946	1	1	yes	characteristic
232	C₂×C₄	#66946	1	1	yes	characteristic
233	C₂×C₄	#66965	1	2	yes	—
234	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
237	C₂×C₄	C₂×#4343	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	4	4	no	—
240	C₈	—	11	2	no	—
241	C₈	—	11	2	no	—
242	C₈	—	22	4	no	—
243	C₈	—	22	1	no	—
244	C₈	—	22	1	no	—
245	C₈	#66960	1	4	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#10622486	1	2	yes	—
249	C₂²	C₂×#66946	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	44	4	no	—
253	C₄	—	44	4	no	—
254	C₄	#10622486	1	2	yes	—
255	C₄	#10622508	1	2	yes	—
256	C₄	C₂×#66960	1	2	yes	—
257	C₄	C₂×#66965	1	1	yes	characteristic
258	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#13077327	1	1	yes	characteristic
261	C₂	C₂×#10622486	1	1	yes	characteristic
262	C₂	C₂×#10622508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1567	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1567	D₁₁
hypocenter	C₁₁	#17958
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #13067512	C₂	semidirect
4. #13067422	C₂	semidirect
86. C₂×C₁₁⋊C₈	Q₁₆	semidirect
222. C₁₁	#17958	semidirect

Maximal subgroups

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

subgroup	index
#13079202	2
#13077546	2
#13067512	2
#13067422	2
#13067070	2
C₂×C₈×C₁₁⋊C₈	2
C₁₁×#1567	2
#17958	11