Subgroups — #14757347

Order 2816.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622738	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622738	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622737	C₂	1	2	yes	maximal
4	#13069406	C₂	1	8	yes	maximal, complemented
5	C₂×#10622171	C₂	1	2	yes	maximal
6	C₁₁×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×#4318	C₂²	1	2	yes	—
11	C₂²×#4318	C₂²	1	2	yes	—
12	C₁₁×#405	—	2	2	no	—
13	C₁₁×#405	C₂²	1	4	yes	—
14	C₄×C₁₁×M₁₆	—	2	2	no	—
15	#10622738	—	2	2	no	—
16	#10622738	—	2	2	no	—
17	#10622738	C₄	1	8	yes	complemented
18	#10622737	—	2	4	no	—
19	#10622737	C₂²	1	8	yes	—
20	C₂×#66886	—	2	4	no	—
21	C₂×#66883	C₂²	1	4	yes	—
22	C₂×#66883	C₂²	1	4	yes	—
23	#10622171	C₂²	1	8	yes	—
24	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₁₁×M₁₆	—	2	1	no	—
26	C₂×C₁₁×M₁₆	—	2	2	no	—
27	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₁×M₁₆	D₄	1	4	yes	—
29	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
30	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
31	C₂×C₁₁×M₁₆	Q₈	1	4	yes	—
32	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₁	—	2	1	no	—
34	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
35	C₁₁×#117	—	2	1	no	—
36	C₁₁×#117	—	2	1	no	—
37	C₁₁×#117	Q₈	1	2	yes	—
38	C₁₁×#117	C₂³	1	2	yes	—
39	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₁×C₂²⋊C₄	Q₈	1	2	yes	—
41	C₂×C₄²×C₁₁	—	2	1	no	—
42	C₂×#4318	—	2	2	no	—
43	C₂×#4318	—	2	2	no	—
44	C₂×#4318	—	2	2	no	—
45	C₂×#4318	—	2	2	no	—
46	C₂×#4318	—	2	2	no	—
47	C₂×#4318	—	2	2	no	—
48	C₂×#4318	—	2	2	no	—
49	C₂×#4318	—	2	2	no	—
50	C₂×#4318	C₂×C₄	1	4	yes	—
51	C₂×#4318	C₂×C₄	1	4	yes	—
52	C₂×#4318	C₂³	1	4	yes	—
53	C₂×#4318	C₂³	1	4	yes	—
54	C₂²×C₁₁⋊C₈	—	2	2	no	—
55	C₂²×C₁₁⋊C₈	—	2	2	no	—
56	C₁₁×#97	—	2	4	no	—
57	C₄×C₈×C₁₁	—	2	4	no	—
58	#66886	—	2	16	no	—
59	#66883	—	2	8	no	—
60	#66883	C₂×C₄	1	16	yes	complemented
61	#21184	—	11	1	no	maximal, Sylow
62	C₂⁴×C₁₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	—	2	2	no	—
64	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
74	C₁₁×M₁₆	—	2	4	no	—
75	C₁₁×M₁₆	—	2	2	no	—
76	C₁₁×M₁₆	C₄⋊C₄	1	8	yes	—
77	C₁₁×M₁₆	C₄⋊C₄	1	8	yes	—
78	C₁₁×M₁₆	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₁₁	—	2	4	no	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₂²⋊C₄	—	2	2	no	—
87	C₁₁×C₂²⋊C₄	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₁	—	2	2	no	—
89	C₄²×C₁₁	—	2	2	no	—
90	#4318	—	2	4	no	—
91	#4318	—	2	4	no	—
92	#4318	—	2	4	no	—
93	#4318	—	2	4	no	—
94	#4318	—	2	4	no	—
95	#4318	—	4	2	no	—
96	#4318	—	4	2	no	—
97	#4318	C₂²×C₄	1	8	yes	—
98	C₂×C₁₁⋊C₈	—	2	4	no	—
99	C₂×C₁₁⋊C₈	—	2	4	no	—
100	C₂×C₁₁⋊C₈	—	2	4	no	—
101	C₂×C₁₁⋊C₈	—	2	4	no	—
102	C₂×C₁₁⋊C₈	—	4	2	no	—
103	C₂×C₁₁⋊C₈	—	4	2	no	—
104	C₂×#430	—	11	1	no	—
105	C₂×#430	—	11	1	no	—
106	C₂×#430	—	11	2	no	—
107	#2873	D₁₁	1	1	yes	characteristic
108	#1953	—	11	8	no	—
109	C₂×#344	—	11	2	no	—
110	C₂³×C₁₁	—	8	1	no	—
111	C₂³×C₁₁	C₄×Q₈	1	2	yes	—
112	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₁₁	C₄⋊Q₈	1	2	yes	—
115	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	—	2	4	no	—
117	C₂×C₄×C₁₁	—	8	2	no	—
118	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₁₁	#121	1	2	yes	—
124	C₂×C₄×C₁₁	#121	1	2	yes	—
125	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₁	#125	1	2	yes	—
128	C₂×C₄×C₁₁	#125	1	2	yes	—
129	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
136	C₈×C₁₁	—	2	4	no	—
137	C₈×C₁₁	C₂×C₄⋊C₄	1	8	yes	—
138	C₁₁⋊C₈	—	2	8	no	—
139	C₁₁⋊C₈	—	4	4	no	—
140	C₂²×M₁₆	—	11	2	no	—
141	C₂²×M₁₆	—	11	2	no	—
142	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
143	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
144	C₂×#117	C₂×D₁₁	1	1	yes	characteristic
145	#430	—	11	8	no	—
146	#430	—	11	8	no	—
147	#430	—	22	4	no	—
148	#430	—	22	2	no	—
149	#430	—	22	2	no	—
150	C₂×#108	—	11	4	no	—
151	C₂×#108	—	11	4	no	—
152	C₂×#108	—	22	4	no	—
153	#405	—	2	2	no	—
154	#405	C₂×D₁₁	1	4	yes	—
155	C₄×M₁₆	—	2	2	no	—
156	#344	—	11	8	no	—
157	C₂²×C₁₁	—	4	1	no	—
158	C₂²×C₁₁	—	4	2	no	—
159	C₂²×C₁₁	—	4	1	no	—
160	C₂²×C₁₁	—	4	2	no	—
161	C₂²×C₁₁	—	4	1	no	—
162	C₂²×C₁₁	#389	1	2	yes	—
163	C₂²×C₁₁	#389	1	2	yes	—
164	C₂²×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₄×C₁₁	—	4	4	no	—
168	C₄×C₁₁	#389	1	2	yes	—
169	C₄×C₁₁	#389	1	2	yes	—
170	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₁₁	C₂×#121	1	1	yes	characteristic
173	C₄×C₁₁	C₂×#125	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	11	4	no	—
178	C₂×M₁₆	—	11	4	no	—
179	C₂×M₁₆	—	11	4	no	—
180	C₂×M₁₆	—	11	4	no	—
181	C₂×M₁₆	—	22	2	no	—
182	C₂×M₁₆	—	22	2	no	—
183	C₂×M₁₆	—	22	2	no	—
184	C₂×M₁₆	—	22	2	no	—
185	C₂×M₁₆	—	22	2	no	—
186	C₂×M₁₆	—	22	2	no	—
187	C₂×M₁₆	—	22	2	no	—
188	C₂×M₁₆	—	22	2	no	—
189	C₂×M₁₆	Dic₂₂	1	4	yes	—
190	C₂×M₁₆	C₄×D₁₁	1	4	yes	—
191	C₂×M₁₆	D₄₄	1	4	yes	—
192	C₂×M₁₆	C₂²⋊D₁₁	1	2	yes	—
193	C₂×M₁₆	C₂²⋊D₁₁	1	2	yes	—
194	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	22	2	no	—
197	C₂²×C₈	—	22	2	no	—
198	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
199	#117	—	2	1	no	—
200	#117	—	2	1	no	—
201	#117	Dic₂₂	1	2	yes	—
202	#117	C₂²×D₁₁	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	Dic₂₂	1	2	yes	—
205	C₂×C₄²	—	2	1	no	—
206	#108	—	11	16	no	—
207	#108	—	22	16	no	—
208	#108	—	22	8	no	—
209	#97	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	C₂×C₁₁	—	2	1	no	—
212	C₂×C₁₁	—	2	1	no	—
213	C₂×C₁₁	—	2	1	no	—
214	C₂×C₁₁	—	2	2	no	—
215	C₂×C₁₁	—	2	2	no	—
216	C₂×C₁₁	C₂×#389	1	1	yes	characteristic
217	C₂⁴	C₂×Dic₂₂	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
222	C₂²×C₄	C₂×D₄₄	1	2	yes	—
223	C₂²×C₄	#4339	1	2	yes	—
224	C₂²×C₄	#4339	1	1	yes	characteristic
225	C₂²×C₄	#4339	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	11	8	no	—
232	M₁₆	—	22	4	no	—
233	M₁₆	—	22	4	no	—
234	M₁₆	—	22	4	no	—
235	M₁₆	—	22	4	no	—
236	M₁₆	—	22	4	no	—
237	M₁₆	—	44	2	no	—
238	M₁₆	—	44	2	no	—
239	M₁₆	Dic₁₁⋊C₄	1	8	yes	—
240	M₁₆	Dic₁₁⋊C₄	1	8	yes	—
241	M₁₆	C₂×D₄₄	1	4	yes	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	22	4	no	—
245	C₂×C₈	—	22	4	no	—
246	C₂×C₈	—	22	4	no	—
247	C₂×C₈	—	22	4	no	—
248	C₂×C₈	—	44	2	no	—
249	C₂×C₈	—	44	2	no	—
250	C₂×C₈	C₂×Dic₂₂	1	4	yes	—
251	C₂×C₈	C₂×C₄×D₁₁	1	4	yes	—
252	C₂×C₈	C₂×C₂²⋊D₁₁	1	4	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	C₂×Dic₂₂	1	4	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₁₁	#21184	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	8	1	no	—
261	C₂³	C₄×Dic₂₂	1	2	yes	—
262	C₂³	#66922	1	2	yes	—
263	C₂³	#66977	1	1	yes	characteristic
264	C₂³	#66977	1	1	yes	characteristic
265	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₄×Dic₂₂	1	2	yes	—
269	C₂×C₄	#66922	1	2	yes	—
270	C₂×C₄	#66923	1	2	yes	—
271	C₂×C₄	#66923	1	2	yes	—
272	C₂×C₄	C₄×D₄₄	1	2	yes	—
273	C₂×C₄	C₄×D₄₄	1	2	yes	—
274	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
275	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
276	C₂×C₄	#66977	1	1	yes	characteristic
277	C₂×C₄	#66977	1	1	yes	characteristic
278	C₂×C₄	#66983	1	2	yes	—
279	C₂×C₄	#66983	1	2	yes	—
280	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
283	C₂×C₄	C₂×#4339	1	1	yes	characteristic
284	C₂×C₄	C₂×#4339	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	22	8	no	—
288	C₈	—	44	4	no	—
289	C₈	C₂×Dic₁₁⋊C₄	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#10622541	1	2	yes	—
296	C₂²	#10622541	1	2	yes	—
297	C₂²	C₂×C₄×Dic₂₂	1	1	yes	characteristic
298	C₂²	C₂×#66922	1	1	yes	characteristic
299	C₂²	C₂×#66977	1	1	yes	characteristic
300	C₄	—	4	4	no	—
301	C₄	#10622541	1	2	yes	—
302	C₄	#10622541	1	2	yes	—
303	C₄	C₂×#66923	1	1	yes	characteristic
304	C₄	C₂×C₄×D₄₄	1	1	yes	characteristic
305	C₄	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
306	C₄	C₂×#66983	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₂×#10622541	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#66983
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#2873	C₂
socle	C₂×C₁₁	C₂×#389
prime radical	C₂⁴×C₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#2873	D₁₁
hypocenter	C₁₁	#21184
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2873

Complements

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

N	H ≅ G/N	type
4. #13069406	C₂	semidirect
17. #10622738	C₄	semidirect
60. #66883	C₂×C₄	semidirect
259. C₁₁	#21184	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622738	2
C₂×#10622737	2
#13069406	2
C₂×#10622171	2
C₁₁×#2873	2
#21184	11