Subgroups — #14672188

Order 2816.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070400	C₂	1	1	yes	maximal, characteristic, complemented
2	#13070352	C₂	1	2	yes	maximal, complemented
3	#13070212	C₂	1	1	yes	maximal, characteristic, complemented
4	#13069515	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×#508	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#135	C₂²	1	2	yes	—
7	C₁₁×#508	—	2	1	no	—
8	C₁₁×#508	—	2	1	no	—
9	C₁₁×#508	—	2	1	no	—
10	C₁₁×#508	—	2	1	no	—
11	C₂×C₁₁×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₁×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₁×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
15	#10622853	—	2	2	no	—
16	#10622849	—	2	2	no	—
17	#10622719	C₂²	1	2	yes	—
18	#10622714	C₂²	1	1	yes	characteristic
19	C₂×#66876	C₂²	1	1	yes	characteristic
20	#10622204	—	2	4	no	—
21	#10622143	—	2	1	no	—
22	#10622142	—	2	1	no	—
23	#10622140	—	2	1	no	—
24	C₁₆×Dic₁₁	—	2	1	no	—
25	C₂×C₁₁×#41	—	2	2	no	—
26	C₁₁×#135	—	2	2	no	—
27	C₁₁×#135	—	2	2	no	—
28	C₁₁×#135	—	2	2	no	—
29	C₁₁×#135	—	2	2	no	—
30	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₁×SD₁₆	—	2	2	no	—
32	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	#66979	—	2	2	no	—
35	#66979	D₄	1	1	yes	characteristic
36	C₂×#4318	D₄	1	1	yes	characteristic
37	C₁₁×Q₃₂	—	2	1	no	—
38	C₁₁×Q₃₂	—	2	1	no	—
39	C₁₁×SD₃₂	—	2	2	no	—
40	C₁₁×SD₃₂	—	2	2	no	—
41	C₁₁×D₁₆	—	2	1	no	—
42	C₁₁×D₁₆	—	2	1	no	—
43	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
46	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
47	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
48	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
49	#66901	—	4	4	no	—
50	#66898	—	4	4	no	—
51	#66882	—	2	2	no	—
52	#66881	—	2	2	no	—
53	#66879	—	2	1	no	—
54	C₈×Dic₁₁	—	2	1	no	—
55	#66876	—	2	1	no	—
56	#66876	D₄	1	2	yes	—
57	C₂×C₁₁⋊C₁₆	—	2	1	no	—
58	#13243	—	11	1	no	maximal, Sylow
59	C₁₁×#41	—	4	2	no	—
60	C₁₁×#41	—	4	2	no	—
61	C₁₁×#41	—	4	2	no	—
62	C₁₁×#41	—	4	2	no	—
63	C₂×Q₈×C₁₁	—	2	2	no	—
64	C₂×D₄×C₁₁	—	2	2	no	—
65	C₂×D₄×C₁₁	—	4	2	no	—
66	C₂²×C₄×C₁₁	—	4	2	no	—
67	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
68	C₁₁×Q₁₆	—	2	2	no	—
69	C₁₁×Q₁₆	—	2	2	no	—
70	C₁₁×SD₁₆	—	4	2	no	—
71	C₁₁×SD₁₆	—	4	2	no	—
72	C₁₁×D₈	—	2	2	no	—
73	C₁₁×D₈	—	2	2	no	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
80	C₂²⋊Dic₁₁	—	4	2	no	—
81	C₂²⋊Dic₁₁	D₈	1	2	yes	—
82	C₄⋊Dic₁₁	—	2	1	no	—
83	C₄⋊Dic₁₁	—	4	2	no	—
84	C₄×Dic₁₁	—	2	1	no	—
85	C₄×Dic₁₁	—	4	2	no	—
86	#4318	—	2	1	no	—
87	#4318	D₈	1	2	yes	—
88	C₂×C₁₁⋊C₈	—	2	1	no	—
89	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
92	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
93	C₁₁⋊C₁₆	—	2	2	no	—
94	C₂×#508	D₁₁	1	1	yes	characteristic
95	#2158	—	11	1	no	—
96	#2145	—	11	2	no	—
97	#2108	—	11	1	no	—
98	#1976	—	11	2	no	—
99	C₂³×C₁₁	—	4	2	no	—
100	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
101	Q₈×C₁₁	—	4	2	no	—
102	Q₈×C₁₁	—	4	2	no	—
103	D₄×C₁₁	—	4	2	no	—
104	D₄×C₁₁	—	4	2	no	—
105	D₄×C₁₁	—	8	2	no	—
106	D₄×C₁₁	—	8	2	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	—	8	2	no	—
109	C₂×C₄×C₁₁	—	8	2	no	—
110	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
116	C₂×Dic₁₁	—	2	2	no	—
117	C₂×Dic₁₁	—	8	2	no	—
118	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₁	#121	1	1	yes	characteristic
121	C₈×C₁₁	#124	1	1	yes	characteristic
122	C₁₁⋊C₈	—	2	2	no	—
123	C₂×#135	C₂×D₁₁	1	2	yes	—
124	#508	—	2	1	no	—
125	#508	—	2	1	no	—
126	#508	—	2	1	no	—
127	#508	—	2	1	no	—
128	C₂×Q₃₂	Dic₁₁	1	1	yes	characteristic, complemented
129	C₂×SD₃₂	Dic₁₁	1	2	yes	complemented
130	C₂×D₁₆	Dic₁₁	1	1	yes	characteristic, complemented
131	C₂×M₃₂	—	11	1	no	—
132	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
133	#427	—	11	2	no	—
134	#421	—	22	2	no	—
135	#410	—	22	2	no	—
136	#405	—	11	1	no	—
137	#367	—	22	1	no	—
138	#366	—	22	1	no	—
139	#360	—	22	4	no	—
140	#346	—	22	1	no	—
141	C₄×C₁₆	—	22	1	no	—
142	C₂²×C₁₁	—	4	1	no	—
143	C₂²×C₁₁	—	8	2	no	—
144	C₂²×C₁₁	—	16	2	no	—
145	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₁₁	#466	1	1	yes	characteristic
147	C₂²×C₁₁	#493	1	1	yes	characteristic
148	C₄×C₁₁	—	8	2	no	—
149	C₄×C₁₁	#390	1	1	yes	characteristic
150	C₄×C₁₁	C₄×D₈	1	1	yes	characteristic
151	C₄×C₁₁	#466	1	1	yes	characteristic
152	C₄×C₁₁	#495	1	1	yes	characteristic
153	Dic₁₁	—	4	2	no	—
154	Dic₁₁	—	16	2	no	—
155	C₂×#41	—	2	2	no	—
156	#135	—	2	2	no	—
157	#135	—	2	2	no	—
158	#135	—	2	2	no	—
159	#135	—	2	2	no	—
160	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
163	C₂×M₁₆	—	11	1	no	—
164	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
165	#117	—	11	1	no	—
166	#117	—	22	2	no	—
167	Q₃₂	—	2	1	no	—
168	Q₃₂	—	2	1	no	—
169	SD₃₂	—	2	2	no	—
170	SD₃₂	—	2	2	no	—
171	D₁₆	—	2	1	no	—
172	D₁₆	—	2	1	no	—
173	M₃₂	—	11	2	no	—
174	M₃₂	—	22	1	no	—
175	C₂×C₁₆	—	22	1	no	—
176	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
177	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
178	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
179	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
180	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
181	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
182	#107	—	22	2	no	—
183	#106	—	22	2	no	—
184	#104	—	44	4	no	—
185	#99	—	44	4	no	—
186	#97	—	22	1	no	—
187	C₄×C₈	—	22	1	no	—
188	C₂×C₁₁	—	2	1	no	—
189	C₂×C₁₁	—	2	1	no	—
190	C₂×C₁₁	—	2	1	no	—
191	C₂×C₁₁	—	16	2	no	—
192	C₂×C₁₁	#1970	1	1	yes	characteristic
193	#41	—	4	2	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
202	Q₁₆	—	2	2	no	—
203	Q₁₆	—	2	2	no	—
204	SD₁₆	—	4	2	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	2	2	no	—
207	D₈	—	2	2	no	—
208	M₁₆	—	11	2	no	—
209	M₁₆	—	22	1	no	—
210	C₂×C₈	—	22	1	no	—
211	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
212	C₂×C₈	#4341	1	1	yes	characteristic
213	C₂×C₈	#4341	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
217	C₄⋊C₄	—	22	1	no	—
218	C₄⋊C₄	—	44	2	no	—
219	C₂²⋊C₄	—	11	2	no	—
220	C₂²⋊C₄	—	44	2	no	—
221	C₄²	—	22	1	no	—
222	C₄²	—	44	2	no	—
223	C₁₆	—	22	2	no	—
224	C₁₆	C₂²⋊Dic₁₁	1	1	yes	characteristic
225	C₁₆	C₂²⋊Dic₁₁	1	1	yes	characteristic
226	C₁₆	C₂²⋊Dic₁₁	1	1	yes	characteristic
227	C₁₆	C₂²⋊Dic₁₁	1	1	yes	characteristic
228	C₁₁	#13243	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	4	2	no	—
230	C₂³	D₈×D₁₁	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	22	2	no	—
241	C₂×C₄	—	88	2	no	—
242	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
243	C₂×C₄	#66965	1	1	yes	characteristic
244	C₂×C₄	#66965	1	1	yes	characteristic
245	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
246	C₂×C₄	#66991	1	1	yes	characteristic
247	C₂×C₄	#66993	1	1	yes	characteristic
248	C₈	—	22	2	no	—
249	C₈	D₄×Dic₁₁	1	1	yes	characteristic
250	C₈	#66989	1	1	yes	characteristic
251	C₈	#66991	1	1	yes	characteristic
252	C₈	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	D₈×Dic₁₁	1	1	yes	characteristic
257	C₂²	#10622764	1	1	yes	characteristic
258	C₂²	#10622770	1	1	yes	characteristic
259	C₄	—	8	2	no	—
260	C₄	—	44	2	no	—
261	C₄	—	176	2	no	—
262	C₄	D₈×Dic₁₁	1	1	yes	characteristic
263	C₄	#10622768	1	1	yes	characteristic
264	C₄	#10622770	1	1	yes	characteristic
265	C₄	#10622834	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#13069483	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10622770
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting, prime radical	C₂×C₁₁×#508	C₂
socle	C₂×C₁₁	#1970
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#508	D₁₁
hypocenter	C₁₁	#13243
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #13070400	C₂	semidirect
2. #13070352	C₂	semidirect
3. #13070212	C₂	semidirect
4. #13069515	C₂	semidirect
11. C₂×C₁₁×Q₃₂	C₄	semidirect
12. C₂×C₁₁×SD₃₂	C₄	semidirect
13. C₂×C₁₁×D₁₆	C₄	semidirect
128. C₂×Q₃₂	Dic₁₁	semidirect
129. C₂×SD₃₂	Dic₁₁	semidirect
130. C₂×D₁₆	Dic₁₁	semidirect
228. C₁₁	#13243	semidirect

Maximal subgroups

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

subgroup	index
#13070400	2
#13070352	2
#13070212	2
#13069515	2
C₂×C₁₁×#508	2
#13243	11