Subgroups — #14668692

Order 2816.

586 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622719	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622719	C₂	1	1	yes	maximal, characteristic
3	#13066722	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×#66979	—	2	1	no	—
7	C₂×#66979	—	2	1	no	—
8	C₂×C₁₁×M₃₂	—	2	2	no	—
9	C₂×C₁₁×M₃₂	—	2	1	no	—
10	C₂×C₁₁×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₁×C₁₆	—	2	1	no	—
13	#10622719	—	2	1	no	—
14	#10622719	—	2	1	no	—
15	#10622719	—	2	2	no	—
16	#10622719	C₄	1	2	yes	complemented
17	#10622719	C₄	1	4	yes	complemented
18	#10622719	C₂²	1	2	yes	—
19	C₂×#66882	—	2	1	no	—
20	C₂×#66882	C₄	1	2	yes	complemented
21	C₂×#66881	—	2	1	no	—
22	C₂×#66881	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	D₄	1	2	yes	—
32	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₁	—	4	1	no	—
35	C₂×C₂²⋊Dic₁₁	—	4	1	no	—
36	#66979	—	2	2	no	—
37	#66979	—	2	4	no	—
38	#66979	—	2	2	no	—
39	#66979	—	4	1	no	—
40	#66979	—	4	2	no	—
41	#66979	—	4	1	no	—
42	C₂×C₄⋊Dic₁₁	—	2	2	no	—
43	C₂×C₄⋊Dic₁₁	—	2	2	no	—
44	C₂×C₄×Dic₁₁	—	4	1	no	—
45	C₂×C₄×Dic₁₁	—	4	1	no	—
46	C₁₁×M₃₂	—	2	4	no	—
47	C₁₁×M₃₂	—	2	2	no	—
48	C₁₁×M₃₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₁×C₁₆	—	2	4	no	—
50	C₂×C₁₁×C₁₆	—	2	2	no	—
51	#66882	—	2	4	no	—
52	#66882	—	2	2	no	—
53	#66882	C₂×C₄	1	4	yes	—
54	#66881	—	2	2	no	—
55	#66881	—	2	2	no	—
56	#66881	—	2	2	no	—
57	#66881	C₂×C₄	1	4	yes	complemented
58	#12567	—	11	1	no	maximal, Sylow
59	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₁	—	4	2	no	—
71	C₂²×Dic₁₁	—	4	2	no	—
72	C₂×C₈×C₁₁	C₄²	1	2	yes	—
73	C₂×C₈×C₁₁	C₄²	1	2	yes	—
74	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
90	C₂²⋊Dic₁₁	—	4	2	no	—
91	C₂²⋊Dic₁₁	—	4	4	no	—
92	C₂²⋊Dic₁₁	—	4	2	no	—
93	C₄⋊Dic₁₁	—	2	4	no	—
94	C₄⋊Dic₁₁	—	2	4	no	—
95	C₄⋊Dic₁₁	—	4	2	no	—
96	C₄⋊Dic₁₁	—	4	2	no	—
97	C₄×Dic₁₁	—	4	4	no	—
98	C₄×Dic₁₁	—	4	2	no	—
99	C₄×Dic₁₁	—	4	2	no	—
100	C₁₁×C₁₆	—	2	4	no	—
101	C₂²×M₃₂	D₁₁	1	1	yes	characteristic
102	C₂×#427	—	11	1	no	—
103	C₂×#427	—	11	1	no	—
104	#1386	—	11	4	no	—
105	C₂³×C₁₁	—	4	2	no	—
106	C₂³×C₁₁	#102	1	1	yes	characteristic
107	C₂³×C₁₁	#102	1	1	yes	characteristic
108	C₂³×C₁₁	#102	1	2	yes	—
109	C₂³×C₁₁	#107	1	2	yes	—
110	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#95	1	2	yes	—
112	C₂×C₄×C₁₁	#95	1	2	yes	—
113	C₂×C₄×C₁₁	#102	1	2	yes	—
114	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#103	1	2	yes	—
117	C₂×C₄×C₁₁	#103	1	2	yes	—
118	C₂×C₄×C₁₁	#103	1	2	yes	—
119	C₂×C₄×C₁₁	#103	1	2	yes	—
120	C₂×C₄×C₁₁	#106	1	2	yes	—
121	C₂×C₄×C₁₁	#106	1	2	yes	—
122	C₂×C₄×C₁₁	#107	1	2	yes	—
123	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×Dic₁₁	—	4	4	no	—
130	C₂×Dic₁₁	—	4	4	no	—
131	C₂×Dic₁₁	—	8	4	no	—
132	C₂×Dic₁₁	—	8	4	no	—
133	C₈×C₁₁	#95	1	2	yes	—
134	C₈×C₁₁	#95	1	2	yes	—
135	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
136	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
140	C₂×#117	—	22	1	no	—
141	C₂×#117	—	22	1	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	—	2	1	no	—
144	C₂×M₃₂	Dic₁₁	1	4	yes	complemented
145	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
146	C₂²×C₁₆	—	2	1	no	—
147	#427	—	11	4	no	—
148	#427	—	11	2	no	—
149	#427	—	11	2	no	—
150	#427	—	22	1	no	—
151	#427	—	22	2	no	—
152	#427	—	22	1	no	—
153	C₂×#107	—	11	2	no	—
154	C₂×#107	—	22	1	no	—
155	C₂×#106	—	11	2	no	—
156	C₂×#106	—	22	1	no	—
157	C₂²×C₁₁	—	2	2	no	—
158	C₂²×C₁₁	—	2	2	no	—
159	C₂²×C₁₁	—	2	2	no	—
160	C₂²×C₁₁	—	4	2	no	—
161	C₂²×C₁₁	—	4	2	no	—
162	C₂²×C₁₁	#340	1	2	yes	—
163	C₂²×C₁₁	#340	1	2	yes	—
164	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
165	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
166	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
167	C₄×C₁₁	#340	1	2	yes	—
168	C₄×C₁₁	#340	1	2	yes	—
169	C₄×C₁₁	C₂×#95	1	1	yes	characteristic
170	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
171	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
172	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
173	Dic₁₁	—	8	4	no	—
174	Dic₁₁	—	8	4	no	—
175	C₂³×C₄	D₄₄	1	1	yes	characteristic
176	C₂²×C₈	Dic₂₂	1	2	yes	—
177	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₁₁	1	2	yes	—
180	C₂²×C₈	D₄₄	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
182	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
185	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
186	#117	—	22	4	no	—
187	#117	—	22	2	no	—
188	#117	—	22	2	no	—
189	#117	—	44	2	no	—
190	#117	—	44	1	no	—
191	#117	—	44	1	no	—
192	C₂×C₄⋊C₄	—	22	2	no	—
193	C₂×C₄⋊C₄	—	22	2	no	—
194	C₂×C₂²⋊C₄	—	44	1	no	—
195	C₂×C₂²⋊C₄	—	44	1	no	—
196	C₂×C₄²	—	44	1	no	—
197	C₂×C₄²	—	44	1	no	—
198	M₃₂	—	2	4	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	C₂×Dic₁₁	1	4	yes	complemented
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	2	no	—
203	#107	—	11	4	no	—
204	#107	—	22	2	no	—
205	#107	—	22	4	no	—
206	#106	—	11	4	no	—
207	#106	—	22	2	no	—
208	#106	—	22	2	no	—
209	#106	—	22	2	no	—
210	C₂×C₁₁	—	2	1	no	—
211	C₂×C₁₁	—	2	2	no	—
212	C₂×C₁₁	—	2	2	no	—
213	C₂×C₁₁	—	2	1	no	—
214	C₂×C₁₁	C₂×#340	1	1	yes	characteristic
215	C₂×C₁₁	#2145	1	2	yes	—
216	C₂⁴	D₈₈	1	1	yes	characteristic
217	C₂²×C₄	—	44	2	no	—
218	C₂²×C₄	—	44	2	no	—
219	C₂²×C₄	#4314	1	2	yes	—
220	C₂²×C₄	#4314	1	1	yes	characteristic
221	C₂²×C₄	#4314	1	1	yes	characteristic
222	C₂²×C₄	D₈₈	1	1	yes	characteristic
223	C₂²×C₄	Dic₄₄	1	2	yes	—
224	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
225	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
227	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
229	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
230	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
231	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
232	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
233	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
234	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
235	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
236	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
237	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
239	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
240	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
241	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
247	C₄⋊C₄	—	22	4	no	—
248	C₄⋊C₄	—	22	4	no	—
249	C₄⋊C₄	—	44	2	no	—
250	C₄⋊C₄	—	44	2	no	—
251	C₂²⋊C₄	—	44	2	no	—
252	C₂²⋊C₄	—	44	4	no	—
253	C₂²⋊C₄	—	44	2	no	—
254	C₄²	—	44	4	no	—
255	C₄²	—	44	2	no	—
256	C₄²	—	44	2	no	—
257	C₁₆	—	2	4	no	—
258	C₁₁	#12567	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	4	2	no	—
260	C₂³	#66882	1	2	yes	—
261	C₂³	#66885	1	1	yes	characteristic
262	C₂³	#66885	1	1	yes	characteristic
263	C₂³	#66885	1	2	yes	—
264	C₂³	C₂×D₈₈	1	1	yes	characteristic
265	C₂×C₄	—	44	4	no	—
266	C₂×C₄	—	44	4	no	—
267	C₂×C₄	—	88	4	no	—
268	C₂×C₄	—	88	4	no	—
269	C₂×C₄	#66880	1	2	yes	—
270	C₂×C₄	#66880	1	2	yes	—
271	C₂×C₄	#66880	1	2	yes	—
272	C₂×C₄	#66880	1	2	yes	—
273	C₂×C₄	#66881	1	2	yes	—
274	C₂×C₄	#66881	1	2	yes	—
275	C₂×C₄	#66882	1	2	yes	—
276	C₂×C₄	#66885	1	2	yes	—
277	C₂×C₄	#66885	1	1	yes	characteristic
278	C₂×C₄	#66885	1	1	yes	characteristic
279	C₂×C₄	C₁₁⋊#95	1	2	yes	—
280	C₂×C₄	C₁₁⋊#95	1	2	yes	—
281	C₂×C₄	C₂×#4314	1	1	yes	characteristic
282	C₂×C₄	C₂×#4314	1	1	yes	characteristic
283	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
285	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
287	C₈	C₁₁⋊#95	1	2	yes	—
288	C₈	C₁₁⋊#95	1	2	yes	—
289	C₈	C₂×C₄×Dic₁₁	1	1	yes	characteristic
290	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
291	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
292	C₈	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#10622189	1	2	yes	—
299	C₂²	#10622189	1	2	yes	—
300	C₂²	C₂×#66882	1	1	yes	characteristic
301	C₂²	C₂×#66885	1	1	yes	characteristic
302	C₂²	C₂×#66885	1	1	yes	characteristic
303	C₄	—	88	4	no	—
304	C₄	—	88	4	no	—
305	C₄	#10622189	1	2	yes	—
306	C₄	#10622189	1	2	yes	—
307	C₄	C₂×#66880	1	1	yes	characteristic
308	C₄	C₂×#66880	1	1	yes	characteristic
309	C₄	C₂×#66881	1	1	yes	characteristic
310	C₄	C₂×C₁₁⋊#95	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₂×#10622189	1	1	yes	characteristic
316	C₂	#13070353	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#66885
commutator	C₈×C₁₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂²×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂⁴×C₁₁	D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂²×M₃₂	D₁₁
hypocenter	C₁₁	#12567
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13066722	C₂	semidirect
10. C₂×C₁₁×M₃₂	C₄	semidirect
16. #10622719	C₄	semidirect
17. #10622719	C₄	semidirect
20. C₂×#66882	C₄	semidirect
48. C₁₁×M₃₂	C₂×C₄	semidirect
57. #66881	C₂×C₄	semidirect
144. C₂×M₃₂	Dic₁₁	semidirect
200. M₃₂	C₂×Dic₁₁	semidirect
258. C₁₁	#12567	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622719	2
#13066722	2
C₂²×C₁₁×M₃₂	2
#12567	11