Subgroups — #14662056

Order 2816.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66883	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622171	C₂	1	2	yes	maximal
3	#13068342	C₂	1	2	yes	maximal
4	C₂×#10622190	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
9	C₂²×#4318	C₂²	1	2	yes	—
10	C₂²×#4318	C₂²	1	2	yes	—
11	C₂×#66883	—	2	2	no	—
12	C₂×#66883	—	2	4	no	—
13	C₁₁×#336	—	2	2	no	—
14	#10622196	—	2	4	no	—
15	#10622190	C₄	1	4	yes	complemented
16	#10622171	—	2	4	no	—
17	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₁₁×M₁₆	—	2	1	no	—
19	C₂×C₁₁×M₁₆	—	2	2	no	—
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
22	C₂²×C₈×C₁₁	—	2	2	no	—
23	C₂²×C₈×C₁₁	—	2	2	no	—
24	C₂²×C₈×C₁₁	—	2	1	no	—
25	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₁	D₄	1	2	yes	—
27	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
28	C₁₁×#117	—	2	2	no	—
29	C₁₁×#117	—	2	2	no	—
30	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₁×C₂²⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₁	—	2	1	no	—
33	C₂×#4318	—	2	2	no	—
34	C₂×#4318	—	2	2	no	—
35	C₂×#4318	—	2	2	no	—
36	C₂×#4318	—	2	2	no	—
37	C₂×#4318	—	2	2	no	—
38	C₂×#4318	—	2	2	no	—
39	C₂×#4318	—	2	2	no	—
40	C₂×#4318	—	2	2	no	—
41	C₂×#4318	—	2	2	no	—
42	C₂×#4318	—	2	2	no	—
43	C₂×#4318	C₂×C₄	1	4	yes	—
44	C₂×#4318	D₄	1	4	yes	—
45	C₂²×C₁₁⋊C₈	—	2	2	no	—
46	C₂²×C₁₁⋊C₈	—	2	2	no	—
47	#66883	—	2	8	no	—
48	#11272	—	11	1	no	maximal, Sylow
49	C₂⁴×C₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	—	2	2	no	—
51	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	#41	1	2	yes	—
57	C₂²×C₄×C₁₁	#41	1	2	yes	—
58	C₂²×C₄×C₁₁	#41	1	2	yes	—
59	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
61	C₁₁×M₁₆	—	2	4	no	—
62	C₁₁×M₁₆	—	4	2	no	—
63	C₂×C₈×C₁₁	—	2	4	no	—
64	C₂×C₈×C₁₁	—	2	2	no	—
65	C₂×C₈×C₁₁	—	2	2	no	—
66	C₂×C₈×C₁₁	—	2	4	no	—
67	C₂×C₈×C₁₁	—	2	2	no	—
68	C₂×C₈×C₁₁	—	4	1	no	—
69	C₂×C₈×C₁₁	—	4	1	no	—
70	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
71	C₁₁×C₄⋊C₄	—	4	2	no	—
72	C₁₁×C₂²⋊C₄	—	2	4	no	—
73	C₄²×C₁₁	—	4	1	no	—
74	C₄²×C₁₁	—	4	1	no	—
75	#4318	—	2	4	no	—
76	#4318	—	2	4	no	—
77	#4318	—	2	4	no	—
78	#4318	—	2	4	no	—
79	#4318	—	4	2	no	—
80	#4318	—	4	2	no	—
81	#4318	—	4	2	no	—
82	#4318	—	4	2	no	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	2	4	no	—
85	C₂×C₁₁⋊C₈	—	4	2	no	—
86	C₂×C₁₁⋊C₈	—	4	2	no	—
87	C₂×C₁₁⋊C₈	—	4	2	no	—
88	C₂×C₁₁⋊C₈	—	4	2	no	—
89	C₂²×#108	—	11	1	no	—
90	C₂×#344	—	11	2	no	—
91	#1738	—	11	2	no	—
92	C₂×#341	—	11	1	no	—
93	#1729	D₁₁	1	1	yes	characteristic
94	C₂³×C₁₁	—	4	2	no	—
95	C₂³×C₁₁	C₄×Q₈	1	1	yes	characteristic
96	C₂³×C₁₁	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₁₁	#125	1	2	yes	—
99	C₂³×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₁	—	4	4	no	—
101	C₂×C₄×C₁₁	—	4	2	no	—
102	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	#117	1	2	yes	—
104	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₁	#121	1	2	yes	—
108	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₁	#123	1	2	yes	—
111	C₂×C₄×C₁₁	#123	1	2	yes	—
112	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₁	#124	1	2	yes	—
114	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#125	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#125	1	2	yes	—
118	C₂×C₄×C₁₁	#126	1	2	yes	—
119	C₂×C₄×C₁₁	#126	1	2	yes	—
120	C₈×C₁₁	—	2	4	no	—
121	C₈×C₁₁	—	4	2	no	—
122	C₁₁⋊C₈	—	4	4	no	—
123	C₁₁⋊C₈	—	4	4	no	—
124	C₂²×M₁₆	—	11	2	no	—
125	C₂²×M₁₆	—	11	2	no	—
126	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
128	C₂×#117	C₂×D₁₁	1	1	yes	characteristic
129	C₂×#108	—	22	4	no	—
130	C₂×#108	—	22	2	no	—
131	#344	—	22	4	no	—
132	#341	—	11	4	no	—
133	#341	—	22	4	no	—
134	#336	—	2	2	no	—
135	C₂²×C₁₁	—	2	2	no	—
136	C₂²×C₁₁	—	2	2	no	—
137	C₂²×C₁₁	—	2	2	no	—
138	C₂²×C₁₁	—	4	2	no	—
139	C₂²×C₁₁	—	4	2	no	—
140	C₂²×C₁₁	#387	1	2	yes	—
141	C₂²×C₁₁	#389	1	1	yes	characteristic
142	C₂²×C₁₁	#393	1	1	yes	characteristic
143	C₂²×C₁₁	#398	1	2	yes	—
144	C₂²×C₁₁	#400	1	1	yes	characteristic
145	C₄×C₁₁	—	8	2	no	—
146	C₄×C₁₁	#382	1	1	yes	characteristic
147	C₄×C₁₁	#385	1	1	yes	characteristic
148	C₄×C₁₁	#388	1	2	yes	—
149	C₄×C₁₁	#394	1	1	yes	characteristic
150	C₄×C₁₁	#397	1	1	yes	characteristic
151	C₄×C₁₁	#400	1	2	yes	—
152	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	11	4	no	—
157	C₂×M₁₆	—	11	4	no	—
158	C₂×M₁₆	—	22	2	no	—
159	C₂×M₁₆	—	22	2	no	—
160	C₂×M₁₆	—	22	2	no	—
161	C₂×M₁₆	—	22	2	no	—
162	C₂×M₁₆	—	22	2	no	—
163	C₂×M₁₆	—	22	2	no	—
164	C₂×M₁₆	—	22	2	no	—
165	C₂×M₁₆	—	22	2	no	—
166	C₂×M₁₆	—	22	2	no	—
167	C₂×M₁₆	—	22	2	no	—
168	C₂×M₁₆	D₄₄	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	22	2	no	—
173	C₂²×C₈	—	22	2	no	—
174	C₂²×C₈	Dic₂₂	1	2	yes	—
175	C₂²×C₈	C₄×D₁₁	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	Dic₂₂	1	2	yes	—
181	C₂×C₄²	—	2	1	no	—
182	#108	—	22	8	no	—
183	C₂×C₁₁	—	2	1	no	—
184	C₂×C₁₁	—	2	2	no	—
185	C₂×C₁₁	—	2	2	no	—
186	C₂×C₁₁	—	2	1	no	—
187	C₂×C₁₁	#1442	1	1	yes	characteristic
188	C₂×C₁₁	#1953	1	2	yes	—
189	C₂⁴	C₂×Dic₂₂	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
194	C₂²×C₄	#4339	1	1	yes	characteristic
195	C₂²×C₄	#4339	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₁₁	1	2	yes	—
197	C₂²×C₄	#4341	1	2	yes	—
198	C₂²×C₄	#4341	1	2	yes	—
199	C₂²×C₄	#4343	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	22	4	no	—
204	M₁₆	—	22	4	no	—
205	M₁₆	—	22	4	no	—
206	M₁₆	—	22	4	no	—
207	M₁₆	—	44	2	no	—
208	M₁₆	—	44	2	no	—
209	M₁₆	—	44	2	no	—
210	M₁₆	—	44	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	22	4	no	—
219	C₂×C₈	—	22	4	no	—
220	C₂×C₈	—	44	2	no	—
221	C₂×C₈	—	44	2	no	—
222	C₂×C₈	—	44	2	no	—
223	C₂×C₈	—	44	2	no	—
224	C₂×C₈	Dic₁₁⋊C₄	1	4	yes	—
225	C₄⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	—	4	1	no	—
228	C₄²	—	4	1	no	—
229	C₁₁	#11272	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	4	2	no	—
231	C₂³	C₄×Dic₂₂	1	1	yes	characteristic
232	C₂³	#66922	1	1	yes	characteristic
233	C₂³	#66931	1	2	yes	—
234	C₂³	#66943	1	2	yes	—
235	C₂³	#66977	1	1	yes	characteristic
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#66923	1	1	yes	characteristic
239	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
240	C₂×C₄	#66931	1	2	yes	—
241	C₂×C₄	#66932	1	2	yes	—
242	C₂×C₄	(C₂²⋊D₁₁)⋊C₄	1	2	yes	—
243	C₂×C₄	#66938	1	2	yes	—
244	C₂×C₄	#66939	1	2	yes	—
245	C₂×C₄	#66943	1	2	yes	—
246	C₂×C₄	#66945	1	2	yes	—
247	C₂×C₄	#66947	1	2	yes	—
248	C₂×C₄	#66948	1	2	yes	—
249	C₂×C₄	#66951	1	2	yes	—
250	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#66983	1	1	yes	characteristic
252	C₂×C₄	#66988	1	1	yes	characteristic
253	C₂×C₄	#66989	1	1	yes	characteristic
254	C₂×C₄	#66990	1	1	yes	characteristic
255	C₂×C₄	#67000	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	44	4	no	—
259	C₈	—	44	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#10622268	1	2	yes	—
266	C₂²	#10622274	1	2	yes	—
267	C₂²	#10622541	1	1	yes	characteristic
268	C₂²	#10622560	1	1	yes	characteristic
269	C₂²	#10622599	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	#10622273	1	2	yes	—
272	C₄	#10622283	1	2	yes	—
273	C₄	#10622559	1	1	yes	characteristic
274	C₄	#10622566	1	1	yes	characteristic
275	C₄	#10622590	1	1	yes	characteristic
276	C₄	#10622604	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	#13066872	1	1	yes	characteristic
282	C₂	#13069406	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#66983
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₁
Fitting	C₁₁×#1729	C₂
socle	C₂²×C₁₁	#389
prime radical	C₂⁴×C₁₁	C₂×Q₈
cosocle	C₂³×C₄×C₁₁	C₂³
hypercenter	#1729	D₁₁
hypocenter	C₁₁	#11272
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #10622190	C₄	semidirect
229. C₁₁	#11272	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66883	2
C₂×#10622171	2
#13068342	2
C₂×#10622190	2
C₁₁×#1729	2
#11272	11