Subgroups — #14643236

Order 2816.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#4318	C₂	1	1	yes	maximal, characteristic
3	#13066450	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×#4318	C₄	1	2	yes	—
7	C₂³×C₁₁⋊C₈	—	2	1	no	—
8	C₁₁×#431	—	2	2	no	—
9	C₁₁×#431	C₄	1	4	yes	—
10	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₁×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₈×C₁₁	—	2	1	no	—
13	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
15	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	4	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₄×#4318	—	2	2	no	—
18	C₄×#4318	—	2	2	no	—
19	C₂×#66867	C₄	1	2	yes	—
20	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	—	2	2	no	—
25	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₁	D₄	1	2	yes	—
29	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	D₄	1	2	yes	—
31	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×#4318	—	2	4	no	—
34	C₂×#4318	—	2	4	no	—
35	C₂×#4318	C₈	1	8	yes	complemented
36	C₂²×C₁₁⋊C₈	—	2	4	no	—
37	C₂²×C₁₁⋊C₈	—	2	2	no	—
38	C₂²×C₁₁⋊C₈	—	4	4	no	—
39	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
40	C₁₁×C₄⋊C₈	—	2	2	no	—
41	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₁₁×C₂²⋊C₈	—	2	4	no	—
43	C₄×C₈×C₁₁	—	2	4	no	—
44	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
47	#66867	—	2	4	no	—
48	C₄×C₁₁⋊C₈	—	2	2	no	—
49	C₄×C₁₁⋊C₈	—	2	2	no	—
50	#7873	—	11	1	no	maximal, Sylow
51	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	—	2	2	no	—
53	C₂²×C₄×C₁₁	—	2	2	no	—
54	C₂²×C₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₁₁	M₁₆	1	4	yes	—
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₁₁	—	2	4	no	—
70	C₂×C₈×C₁₁	—	4	2	no	—
71	C₂×C₈×C₁₁	—	4	2	no	—
72	C₄²×C₁₁	C₄²	1	2	yes	—
73	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
82	#4318	—	2	16	no	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	2	4	no	—
85	C₂×C₁₁⋊C₈	—	2	4	no	—
86	C₂×C₁₁⋊C₈	—	4	8	no	—
87	C₂×C₁₁⋊C₈	—	4	4	no	—
88	C₂×C₁₁⋊C₈	C₂×C₈	1	8	yes	complemented
89	C₂×#431	D₁₁	1	1	yes	characteristic
90	C₂²×C₄⋊C₈	—	11	1	no	—
91	C₂×C₄×M₁₆	—	11	1	no	—
92	#1277	—	11	4	no	—
93	C₂³×C₁₁	—	2	2	no	—
94	C₂³×C₁₁	—	2	2	no	—
95	C₂³×C₁₁	#95	1	2	yes	—
96	C₂³×C₁₁	C₄⋊C₈	1	4	yes	—
97	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	2	no	—
100	C₂×C₄×C₁₁	—	2	2	no	—
101	C₂×C₄×C₁₁	—	2	4	no	—
102	C₂×C₄×C₁₁	—	2	2	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	4	2	no	—
105	C₂×C₄×C₁₁	—	4	2	no	—
106	C₂×C₄×C₁₁	#95	1	2	yes	—
107	C₂×C₄×C₁₁	#95	1	2	yes	—
108	C₂×C₄×C₁₁	#95	1	2	yes	—
109	C₂×C₄×C₁₁	C₄×C₈	1	4	yes	—
110	C₂×C₄×C₁₁	#97	1	4	yes	—
111	C₂×C₄×C₁₁	C₂²⋊C₈	1	8	yes	—
112	C₂×C₄×C₁₁	C₄⋊C₈	1	4	yes	—
113	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₁₁	#137	1	2	yes	—
123	C₈×C₁₁	—	4	4	no	—
124	C₁₁⋊C₈	—	2	8	no	—
125	C₁₁⋊C₈	—	4	4	no	—
126	C₂²×M₁₆	—	11	2	no	—
127	C₂³×C₈	—	22	1	no	—
128	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
129	#431	—	2	2	no	—
130	#431	Dic₁₁	1	4	yes	—
131	C₂×C₄⋊C₈	—	11	2	no	—
132	C₂×C₄⋊C₈	—	11	4	no	—
133	C₂×C₄⋊C₈	—	22	2	no	—
134	C₂×C₄⋊C₈	—	22	1	no	—
135	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
136	C₂×C₂²⋊C₈	—	2	1	no	—
137	C₄×M₁₆	—	22	2	no	—
138	C₄×M₁₆	—	22	2	no	—
139	C₂×#97	—	11	2	no	—
140	C₂×C₄×C₈	—	2	1	no	—
141	C₂×C₄×C₈	—	11	2	no	—
142	C₂²×C₁₁	—	2	8	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	—	2	2	no	—
145	C₂²×C₁₁	#336	1	4	yes	—
146	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
147	C₂²×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
148	C₂²×C₁₁	#428	1	2	yes	—
149	C₂²×C₁₁	#430	1	2	yes	—
150	C₄×C₁₁	—	2	2	no	—
151	C₄×C₁₁	—	2	2	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	—	2	1	no	—
154	C₄×C₁₁	#336	1	4	yes	—
155	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
156	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
157	C₄×C₁₁	C₂×C₂²⋊C₈	1	2	yes	—
158	C₄×C₁₁	#417	1	2	yes	—
159	C₄×C₁₁	#418	1	1	yes	characteristic
160	C₄×C₁₁	#419	1	1	yes	characteristic
161	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
162	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
163	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
164	C₂×M₁₆	—	11	8	no	—
165	C₂×M₁₆	—	22	4	no	—
166	C₂×M₁₆	—	22	4	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	11	4	no	—
169	C₂²×C₈	—	22	4	no	—
170	C₂²×C₈	—	22	2	no	—
171	C₂²×C₈	—	44	4	no	—
172	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₁₁	1	2	yes	—
175	C₂×C₄²	D₄₄	1	2	yes	—
176	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	11	4	no	—
182	C₄⋊C₈	—	22	4	no	—
183	C₄⋊C₈	—	22	2	no	—
184	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	22	4	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	22	2	no	—
189	C₄×C₈	—	22	2	no	—
190	C₂×C₁₁	—	2	4	no	—
191	C₂×C₁₁	#1565	1	4	yes	—
192	C₂×C₁₁	C₂×#336	1	1	yes	characteristic
193	C₂×C₁₁	#1736	1	1	yes	characteristic
194	C₂×C₁₁	#1738	1	1	yes	characteristic
195	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₈×D₁₁	1	4	yes	—
200	C₂²×C₄	#4313	1	4	yes	—
201	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
202	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
205	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
207	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
208	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
209	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
210	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
213	M₁₆	—	22	16	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	11	8	no	—
218	C₂×C₈	—	22	4	no	—
219	C₂×C₈	—	22	4	no	—
220	C₂×C₈	—	22	4	no	—
221	C₂×C₈	—	44	8	no	—
222	C₂×C₈	—	44	4	no	—
223	C₄²	C₄×Dic₁₁	1	2	yes	—
224	C₄²	Dic₁₁⋊C₄	1	2	yes	—
225	C₄²	C₄⋊Dic₁₁	1	2	yes	—
226	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
227	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
228	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
229	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₄²	C₂×D₄₄	1	1	yes	characteristic
231	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
233	C₁₁	#7873	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	Dic₁₁⋊C₈	1	4	yes	—
237	C₂³	C₁₁⋊#95	1	2	yes	—
238	C₂³	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₈×Dic₁₁	1	4	yes	—
248	C₂×C₄	Dic₁₁⋊C₈	1	4	yes	—
249	C₂×C₄	#66879	1	4	yes	—
250	C₂×C₄	(C₂×D₁₁)⋊C₈	1	8	yes	—
251	C₂×C₄	C₁₁⋊#95	1	2	yes	—
252	C₂×C₄	C₁₁⋊#95	1	2	yes	—
253	C₂×C₄	C₁₁⋊#95	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
255	C₂×C₄	C₂×#4313	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
257	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
261	C₂×C₄	#67002	1	2	yes	—
262	C₂×C₄	#67004	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	22	8	no	—
266	C₈	—	44	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	8	no	—
270	C₂²	#10622188	1	4	yes	—
271	C₂²	#10622608	1	2	yes	—
272	C₂²	C₂×Dic₁₁⋊C₈	1	2	yes	—
273	C₂²	#10622737	1	2	yes	—
274	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	#10622188	1	4	yes	—
280	C₄	#10622614	1	1	yes	characteristic
281	C₄	#10622617	1	1	yes	characteristic
282	C₄	C₂×C₈×Dic₁₁	1	1	yes	characteristic
283	C₄	C₂×#66879	1	1	yes	characteristic
284	C₄	C₂×(C₂×D₁₁)⋊C₈	1	2	yes	—
285	C₄	#10622845	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	#13067369	1	4	yes	—
288	C₂	C₂×#10622188	1	1	yes	characteristic
289	C₂	#13068337	1	1	yes	characteristic
290	C₂	#13068342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#431	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	#7873
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13066450	C₂	semidirect
35. C₂×#4318	C₈	semidirect
88. C₂×C₁₁⋊C₈	C₂×C₈	semidirect
233. C₁₁	#7873	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂×C₄×#4318	2
#13066450	2
C₂×C₁₁×#431	2
#7873	11