Subgroups — #14653003

Order 2816.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13069540	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
4	C₂×#10622185	C₂	1	2	yes	maximal
5	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₁×#500	—	2	2	no	—
8	C₁₁×#500	C₄	1	4	yes	complemented
9	C₂×C₁₁×#107	—	2	1	no	—
10	C₂×C₁₁×#107	—	2	1	no	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66899	—	2	4	no	—
13	C₂×C₈×Dic₁₁	—	2	1	no	—
14	C₄×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#10622188	—	2	1	no	—
18	#10622185	—	2	4	no	—
19	#10622096	—	2	2	no	—
20	C₂²×Q₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
23	C₁₁×C₄⋊Q₈	—	2	2	no	—
24	C₁₁×C₄⋊Q₈	—	2	2	no	—
25	C₁₁×C₄⋊Q₈	—	2	2	no	—
26	C₁₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₁	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₁	—	2	1	no	—
32	C₂²×C₁₁⋊C₈	—	2	1	no	—
33	C₂²×C₁₁⋊C₈	—	2	2	no	—
34	C₁₁×#107	—	2	4	no	—
35	C₁₁×#107	—	2	4	no	—
36	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₁	D₄	1	2	yes	—
38	#66899	—	4	8	no	—
39	C₁₁⋊#95	—	2	1	no	—
40	C₈×Dic₁₁	—	4	2	no	—
41	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
42	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
45	#9731	—	11	1	no	maximal, Sylow
46	C₂×Q₈×C₁₁	—	2	4	no	—
47	C₂×Q₈×C₁₁	—	4	2	no	—
48	C₂×Q₈×C₁₁	—	4	4	no	—
49	C₂×Q₈×C₁₁	—	4	4	no	—
50	C₂²×C₄×C₁₁	—	4	2	no	—
51	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
54	C₂²×Dic₁₁	—	2	2	no	—
55	C₂×C₈×C₁₁	—	2	1	no	—
56	C₂×C₈×C₁₁	—	2	1	no	—
57	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
60	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
61	C₁₁×C₄⋊C₄	—	2	4	no	—
62	C₁₁×C₄⋊C₄	—	2	4	no	—
63	C₁₁×C₄⋊C₄	—	4	2	no	—
64	C₁₁×C₄⋊C₄	—	4	2	no	—
65	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₁	#41	1	1	yes	characteristic
69	C₄⋊Dic₁₁	—	2	2	no	—
70	C₄⋊Dic₁₁	D₈	1	4	yes	—
71	C₄×Dic₁₁	—	4	1	no	—
72	C₄×Dic₁₁	—	4	1	no	—
73	C₂×C₁₁⋊C₈	—	2	2	no	—
74	C₂×C₁₁⋊C₈	—	4	2	no	—
75	C₂×C₁₁⋊C₈	—	4	4	no	—
76	C₂×C₁₁⋊C₈	—	4	1	no	—
77	C₂×C₁₁⋊C₈	—	4	1	no	—
78	C₂×#500	D₁₁	1	1	yes	characteristic
79	#1981	—	11	2	no	—
80	C₈×C₄⋊C₄	—	11	1	no	—
81	C₂×#335	—	11	1	no	—
82	C₂×#332	—	11	2	no	—
83	C₂³×C₁₁	#121	1	1	yes	characteristic
84	Q₈×C₁₁	—	4	8	no	—
85	Q₈×C₁₁	—	8	2	no	—
86	Q₈×C₁₁	—	8	2	no	—
87	C₂×C₄×C₁₁	—	2	2	no	—
88	C₂×C₄×C₁₁	—	4	4	no	—
89	C₂×C₄×C₁₁	—	8	4	no	—
90	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₁	#135	1	2	yes	—
100	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
101	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
102	C₂×Dic₁₁	—	2	4	no	—
103	C₂×Dic₁₁	—	4	2	no	—
104	C₂×Dic₁₁	—	4	2	no	—
105	C₈×C₁₁	—	2	2	no	—
106	C₈×C₁₁	#103	1	4	yes	complemented
107	C₁₁⋊C₈	—	4	2	no	—
108	C₁₁⋊C₈	—	8	2	no	—
109	C₂×C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
110	#500	—	2	2	no	—
111	#500	Dic₁₁	1	4	yes	complemented
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×C₄⋊C₈	—	11	1	no	—
115	C₂×C₄⋊C₈	—	11	2	no	—
116	C₂×#103	—	22	4	no	—
117	C₂×C₄×C₈	—	22	1	no	—
118	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	11	1	no	—
120	#336	—	22	1	no	—
121	#335	—	22	2	no	—
122	#332	—	22	4	no	—
123	C₂²×C₁₁	#390	1	1	yes	characteristic
124	C₂²×C₁₁	#461	1	1	yes	characteristic
125	C₂²×C₁₁	#462	1	1	yes	characteristic
126	C₂²×C₁₁	#466	1	2	yes	—
127	C₂²×C₁₁	#473	1	2	yes	—
128	C₄×C₁₁	—	2	1	no	—
129	C₄×C₁₁	—	2	1	no	—
130	C₄×C₁₁	—	8	4	no	—
131	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
132	C₄×C₁₁	#419	1	1	yes	characteristic
133	C₄×C₁₁	C₄×D₈	1	2	yes	—
134	C₄×C₁₁	#461	1	1	yes	characteristic
135	C₄×C₁₁	#462	1	1	yes	characteristic
136	C₄×C₁₁	#493	1	1	yes	characteristic
137	C₄×C₁₁	#495	1	1	yes	characteristic
138	Dic₁₁	—	4	4	no	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂²×C₈	—	22	1	no	—
141	C₂²×C₈	—	22	2	no	—
142	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₂×Dic₁₁	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	11	2	no	—
151	C₂×C₄²	—	22	1	no	—
152	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
153	#107	—	2	4	no	—
154	#107	—	2	4	no	—
155	C₄⋊C₈	—	11	2	no	—
156	C₄⋊C₈	—	22	2	no	—
157	C₄⋊C₈	—	22	1	no	—
158	C₄⋊C₈	—	22	2	no	—
159	#103	—	44	8	no	—
160	C₄×C₈	—	44	2	no	—
161	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
162	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
163	#95	—	22	1	no	—
164	C₂×C₁₁	#1673	1	2	yes	—
165	C₂×C₁₁	#1674	1	2	yes	—
166	C₂×C₁₁	#1970	1	1	yes	characteristic
167	C₂×C₁₁	#1979	1	1	yes	characteristic
168	C₂×C₁₁	#1981	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	22	2	no	—
175	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
176	C₂²×C₄	#4341	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	22	2	no	—
181	C₂×C₈	—	44	1	no	—
182	C₂×C₈	—	44	1	no	—
183	C₂×C₈	—	44	4	no	—
184	C₂×C₈	—	44	2	no	—
185	C₂×C₈	#4325	1	2	yes	—
186	C₂×C₈	#4326	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	11	4	no	—
194	C₄⋊C₄	—	22	2	no	—
195	C₄²	—	44	1	no	—
196	C₄²	—	44	1	no	—
197	C₄²	D₄×D₁₁	1	1	yes	characteristic
198	C₄²	#4341	1	1	yes	characteristic
199	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
201	C₁₁	#9731	1	1	yes	characteristic, complemented, Sylow
202	C₂³	#66991	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	22	4	no	—
210	C₂×C₄	—	44	2	no	—
211	C₂×C₄	—	44	2	no	—
212	C₂×C₄	D₈×D₁₁	1	2	yes	—
213	C₂×C₄	#66965	1	2	yes	—
214	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
216	C₂×C₄	#66989	1	1	yes	characteristic
217	C₂×C₄	#66991	1	1	yes	characteristic
218	C₂×C₄	#66993	1	1	yes	characteristic
219	C₂×C₄	C₂×#4325	1	1	yes	characteristic
220	C₂×C₄	#66995	1	1	yes	characteristic
221	C₂×C₄	#66995	1	1	yes	characteristic
222	C₂×C₄	C₂×#4326	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	44	2	no	—
226	C₈	—	88	2	no	—
227	C₈	#66899	1	4	yes	complemented
228	C₂²	#10622696	1	1	yes	characteristic
229	C₂²	#10622703	1	1	yes	characteristic
230	C₂²	#10622770	1	2	yes	—
231	C₂²	#10622805	1	2	yes	—
232	C₂²	#10622834	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	44	4	no	—
237	C₄	#10622696	1	1	yes	characteristic
238	C₄	#10622703	1	1	yes	characteristic
239	C₄	D₈×Dic₁₁	1	2	yes	—
240	C₄	#10622764	1	1	yes	characteristic
241	C₄	#10622768	1	1	yes	characteristic
242	C₄	C₂×#66899	1	1	yes	characteristic
243	C₄	#10622838	1	1	yes	characteristic
244	C₂	#13068002	1	2	yes	—
245	C₂	#13068007	1	2	yes	—
246	C₂	#13069483	1	1	yes	characteristic
247	C₂	#13069532	1	1	yes	characteristic
248	C₂	#13069540	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66991
commutator	C₄²×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#500	C₂
socle, prime radical	C₂³×C₁₁	#121
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#500	D₁₁
hypocenter	C₁₁	#9731
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
8. C₁₁×#500	C₄	semidirect
106. C₈×C₁₁	#103	semidirect
111. #500	Dic₁₁	semidirect
201. C₁₁	#9731	semidirect
227. C₈	#66899	semidirect

Maximal subgroups

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

subgroup	index
#13069540	2
C₈×C₄⋊Dic₁₁	2
C₂×#10622096	2
C₂×#10622185	2
C₂×C₁₁×#500	2
#9731	11