Subgroups — #14640334

Order 2816.

448 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	C₄×C₂²⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622162	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
5	C₂³×C₁₁⋊C₈	—	2	1	no	—
6	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
7	C₂×C₁₁×C₂²⋊C₈	—	2	1	no	—
8	C₂×C₂²⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
9	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
10	C₁₁×#336	—	2	1	no	—
11	#10622162	—	2	1	no	—
12	#10622162	—	2	1	no	—
13	#10622162	C₄	1	2	yes	—
14	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	—	2	2	no	—
18	C₂×C₄²×C₁₁	—	4	2	no	—
19	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
23	C₂²×C₁₁⋊C₈	—	2	4	no	—
24	C₂²×C₁₁⋊C₈	—	2	2	no	—
25	C₂²×C₁₁⋊C₈	—	2	2	no	—
26	C₂²×C₁₁⋊C₈	—	4	2	no	—
27	C₂²×C₁₁⋊C₈	—	4	1	no	—
28	C₂²×C₁₁⋊C₈	—	4	1	no	—
29	C₁₁×C₄⋊C₈	—	4	2	no	—
30	C₁₁×C₂²⋊C₈	—	4	1	no	—
31	C₁₁×C₂²⋊C₈	—	4	1	no	—
32	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
33	C₂²⋊(C₁₁⋊C₈)	C₈	1	4	yes	complemented
34	C₄×C₁₁⋊C₈	—	4	2	no	—
35	#7029	—	11	1	no	maximal, Sylow
36	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁	—	2	2	no	—
38	C₂²×C₄×C₁₁	—	2	2	no	—
39	C₂²×C₄×C₁₁	—	2	2	no	—
40	C₂²×C₄×C₁₁	—	2	2	no	—
41	C₂²×C₄×C₁₁	—	4	1	no	—
42	C₂²×C₄×C₁₁	—	4	1	no	—
43	C₂²×C₄×C₁₁	—	4	1	no	—
44	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₁	—	4	2	no	—
57	C₂×C₈×C₁₁	—	4	2	no	—
58	C₂×C₈×C₁₁	—	4	2	no	—
59	C₄²×C₁₁	—	2	2	no	—
60	C₄²×C₁₁	—	2	2	no	—
61	C₄²×C₁₁	—	4	1	no	—
62	C₄²×C₁₁	—	4	1	no	—
63	C₂×C₁₁⋊C₈	—	2	4	no	—
64	C₂×C₁₁⋊C₈	—	2	4	no	—
65	C₂×C₁₁⋊C₈	—	4	2	no	—
66	C₂×C₁₁⋊C₈	—	4	2	no	—
67	C₂×C₁₁⋊C₈	—	4	2	no	—
68	C₂×C₁₁⋊C₈	—	4	4	no	—
69	C₂×C₁₁⋊C₈	—	4	4	no	—
70	C₂×C₁₁⋊C₈	—	4	2	no	—
71	#1797	D₁₁	1	1	yes	characteristic
72	C₄×C₂²⋊C₈	—	11	1	no	—
73	C₂×#336	—	11	1	no	—
74	C₂³×C₁₁	—	2	1	no	—
75	C₂³×C₁₁	—	2	1	no	—
76	C₂³×C₁₁	—	2	1	no	—
77	C₂³×C₁₁	—	2	1	no	—
78	C₂³×C₁₁	#95	1	1	yes	characteristic
79	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
80	C₂³×C₁₁	#106	1	1	yes	characteristic
81	C₂³×C₁₁	#107	1	1	yes	characteristic
82	C₂³×C₁₁	#108	1	2	yes	—
83	C₂×C₄×C₁₁	—	2	1	no	—
84	C₂×C₄×C₁₁	—	2	1	no	—
85	C₂×C₄×C₁₁	—	2	1	no	—
86	C₂×C₄×C₁₁	—	2	1	no	—
87	C₂×C₄×C₁₁	—	2	1	no	—
88	C₂×C₄×C₁₁	—	2	1	no	—
89	C₂×C₄×C₁₁	—	2	1	no	—
90	C₂×C₄×C₁₁	—	2	1	no	—
91	C₂×C₄×C₁₁	—	2	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	4	2	no	—
99	C₂×C₄×C₁₁	—	4	2	no	—
100	C₂×C₄×C₁₁	—	4	4	no	—
101	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₁	#99	1	2	yes	—
105	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
108	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
109	C₂×C₄×C₁₁	#104	1	2	yes	—
110	C₂×C₄×C₁₁	#108	1	2	yes	—
111	C₈×C₁₁	—	8	2	no	—
112	C₁₁⋊C₈	—	4	4	no	—
113	C₁₁⋊C₈	—	4	4	no	—
114	C₂³×C₈	—	22	1	no	—
115	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
116	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
117	C₂×C₂²⋊C₈	—	2	1	no	—
118	C₂×C₂²⋊C₈	—	11	2	no	—
119	C₂×C₄×C₈	—	22	1	no	—
120	#336	—	2	1	no	—
121	#336	—	11	2	no	—
122	#336	—	22	1	no	—
123	#336	—	22	1	no	—
124	C₂²×C₁₁	—	2	2	no	—
125	C₂²×C₁₁	—	2	1	no	—
126	C₂²×C₁₁	—	2	1	no	—
127	C₂²×C₁₁	—	2	2	no	—
128	C₂²×C₁₁	—	2	1	no	—
129	C₂²×C₁₁	—	2	1	no	—
130	C₂²×C₁₁	—	2	2	no	—
131	C₂²×C₁₁	—	2	2	no	—
132	C₂²×C₁₁	#334	1	2	yes	—
133	C₂²×C₁₁	#335	1	2	yes	—
134	C₂²×C₁₁	#336	1	1	yes	characteristic
135	C₂²×C₁₁	#339	1	1	yes	characteristic
136	C₂²×C₁₁	#340	1	1	yes	characteristic
137	C₂²×C₁₁	#341	1	1	yes	characteristic
138	C₂²×C₁₁	#341	1	1	yes	characteristic
139	C₂²×C₁₁	#342	1	1	yes	characteristic
140	C₂²×C₁₁	#344	1	1	yes	characteristic
141	C₄×C₁₁	—	2	1	no	—
142	C₄×C₁₁	—	2	1	no	—
143	C₄×C₁₁	—	2	2	no	—
144	C₄×C₁₁	—	2	1	no	—
145	C₄×C₁₁	—	2	1	no	—
146	C₄×C₁₁	—	2	2	no	—
147	C₄×C₁₁	—	4	1	no	—
148	C₄×C₁₁	#327	1	2	yes	—
149	C₄×C₁₁	#328	1	2	yes	—
150	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
151	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
152	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	22	4	no	—
155	C₂²×C₈	—	22	2	no	—
156	C₂²×C₈	—	22	2	no	—
157	C₂²×C₈	—	44	2	no	—
158	C₂²×C₈	—	44	1	no	—
159	C₂²×C₈	—	44	1	no	—
160	C₂×C₄²	—	4	2	no	—
161	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
162	C₂×C₄²	D₄₄	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
165	C₄⋊C₈	—	4	2	no	—
166	C₂²⋊C₈	—	4	1	no	—
167	C₂²⋊C₈	—	4	1	no	—
168	C₂²⋊C₈	—	11	4	no	—
169	C₂²⋊C₈	—	22	2	no	—
170	C₄×C₈	—	44	2	no	—
171	C₂×C₁₁	—	2	2	no	—
172	C₂×C₁₁	—	2	2	no	—
173	C₂×C₁₁	#1277	1	1	yes	characteristic
174	C₂×C₁₁	#1278	1	1	yes	characteristic
175	C₂×C₁₁	#1282	1	1	yes	characteristic
176	C₂×C₁₁	#1294	1	1	yes	characteristic
177	C₂×C₁₁	#1298	1	1	yes	characteristic
178	C₂×C₁₁	#1301	1	1	yes	characteristic
179	C₂×C₁₁	#1302	1	1	yes	characteristic
180	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	C₈×D₁₁	1	2	yes	—
189	C₂²×C₄	#4313	1	2	yes	—
190	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
191	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
193	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	#4323	1	1	yes	characteristic
196	C₂²×C₄	#4324	1	1	yes	characteristic
197	C₂²×C₄	#4325	1	1	yes	characteristic
198	C₂²×C₄	#4326	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	22	4	no	—
204	C₂×C₈	—	22	4	no	—
205	C₂×C₈	—	44	2	no	—
206	C₂×C₈	—	44	2	no	—
207	C₂×C₈	—	44	4	no	—
208	C₂×C₈	—	44	4	no	—
209	C₂×C₈	—	44	2	no	—
210	C₂×C₈	—	44	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₁₁	#7029	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	#66871	1	1	yes	characteristic
221	C₂³	#66872	1	1	yes	characteristic
222	C₂³	Dic₁₁⋊C₈	1	2	yes	—
223	C₂³	#66886	1	2	yes	—
224	C₂³	C₁₁⋊#95	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	#66873	1	1	yes	characteristic
244	C₂×C₄	#66874	1	1	yes	characteristic
245	C₂×C₄	C₈×Dic₁₁	1	1	yes	characteristic
246	C₂×C₄	#66879	1	1	yes	characteristic
247	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
248	C₂×C₄	#66886	1	2	yes	—
249	C₂×C₄	#66896	1	1	yes	characteristic
250	C₂×C₄	#66898	1	2	yes	—
251	C₂×C₄	#66899	1	1	yes	characteristic
252	C₂×C₄	#66901	1	2	yes	—
253	C₈	—	8	2	no	—
254	C₈	—	44	4	no	—
255	C₈	—	44	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#10622117	1	2	yes	—
265	C₂²	#10622118	1	2	yes	—
266	C₂²	#10622164	1	1	yes	characteristic
267	C₂²	#10622167	1	1	yes	characteristic
268	C₂²	#10622170	1	1	yes	characteristic
269	C₂²	#10622188	1	1	yes	characteristic
270	C₂²	#10622194	1	1	yes	characteristic
271	C₂²	#10622196	1	1	yes	characteristic
272	C₂²	#10622196	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	#10622175	1	2	yes	—
281	C₄	#10622176	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#13066450	1	1	yes	characteristic
285	C₂	#13066452	1	1	yes	characteristic
286	C₂	#13066463	1	1	yes	characteristic
287	C₂	#13066494	1	1	yes	characteristic
288	C₂	#13066503	1	1	yes	characteristic
289	C₂	#13066511	1	1	yes	characteristic
290	C₂	#13066514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₁
commutator	C₂×C₄×C₁₁	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₁
Fitting	C₁₁×#1797	C₂
socle	C₂³×C₁₁	#95
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂²×C₄²×C₁₁	C₂²
hypercenter	#1797	D₁₁
hypocenter	C₁₁	#7029
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
33. C₂²⋊(C₁₁⋊C₈)	C₈	semidirect
215. C₁₁	#7029	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₁⋊C₈)	2
C₂×#10622162	2
C₁₁×#1797	2
#7029	11