Subgroups — #14772045

Order 2816.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#4318	C₂	1	2	yes	maximal
2	C₂×#10622738	C₂	1	2	yes	maximal, complemented
3	#13069392	C₂	1	2	yes	maximal, complemented
4	#13068823	C₂	1	4	yes	maximal, complemented
5	#13068788	C₂	1	1	yes	maximal, characteristic, complemented
6	#13068658	C₂	1	2	yes	maximal, complemented
7	C₄×#66883	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#436	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²×#4318	C₂²	1	4	yes	—
12	C₁₁×#436	C₄	1	4	yes	—
13	C₁₁×#436	C₄	1	4	yes	—
14	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	#10622820	C₂²	1	4	yes	—
18	#10622738	—	2	8	no	—
19	C₂×#66883	C₂²	1	4	yes	—
20	C₂×#66883	C₂²	1	1	yes	characteristic
21	C₂×#66883	C₂²	1	2	yes	—
22	C₂×#66883	C₂²	1	2	yes	—
23	C₂×#66883	C₂²	1	1	yes	characteristic
24	#10622625	—	2	2	no	—
25	#10622624	—	2	4	no	—
26	D₄×C₁₁⋊C₈	—	2	2	no	—
27	#10622537	C₂²	1	2	yes	—
28	C₄×#4318	C₂²	1	2	yes	—
29	#10622190	C₂²	1	2	yes	—
30	#10622190	C₂²	1	4	yes	—
31	C₂²×D₄×C₁₁	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₁₁	C₂³	1	2	yes	—
33	C₂×C₁₁×M₁₆	—	2	4	no	—
34	C₂×C₁₁×M₁₆	—	2	4	no	—
35	C₂×C₁₁×M₁₆	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₁	D₄	1	2	yes	—
42	C₄×D₄×C₁₁	D₄	1	2	yes	—
43	C₂×C₁₁×C₄⋊C₄	Q₈	1	1	yes	characteristic
44	C₂×C₁₁×C₂²⋊C₄	Q₈	1	2	yes	—
45	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×#4318	—	2	4	no	—
47	C₂×#4318	—	2	8	no	—
48	C₂×#4318	—	4	4	no	—
49	C₂×#4318	C₂³	1	4	yes	—
50	C₂×#4318	C₂³	1	2	yes	—
51	C₂×#4318	C₂³	1	2	yes	—
52	C₂²×C₁₁⋊C₈	—	2	4	no	—
53	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₁₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
59	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
60	#66883	—	2	4	no	—
61	#66883	—	2	2	no	—
62	#66883	—	2	8	no	—
63	#66883	—	4	2	no	—
64	#66883	D₄	1	4	yes	complemented
65	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
66	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
67	#66867	—	2	2	no	—
68	C₄×C₁₁⋊C₈	—	2	2	no	—
69	#22669	—	11	1	no	maximal, Sylow
70	C₂⁴×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂×D₄×C₁₁	—	4	2	no	—
72	C₂×D₄×C₁₁	C₄⋊C₄	1	2	yes	—
73	C₂×D₄×C₁₁	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₁	—	2	2	no	—
75	C₂²×C₄×C₁₁	—	2	2	no	—
76	C₂²×C₄×C₁₁	—	2	2	no	—
77	C₂²×C₄×C₁₁	—	2	2	no	—
78	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
84	C₁₁×M₁₆	—	2	16	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	1	no	—
88	C₂×C₈×C₁₁	—	2	1	no	—
89	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₁	#41	1	2	yes	—
94	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
95	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
96	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
97	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
98	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
102	#4318	—	2	4	no	—
103	#4318	—	2	2	no	—
104	#4318	—	2	2	no	—
105	#4318	—	2	4	no	—
106	#4318	—	2	2	no	—
107	#4318	—	4	8	no	—
108	#4318	C₂×D₄	1	4	yes	—
109	C₂×C₁₁⋊C₈	—	2	2	no	—
110	C₂×C₁₁⋊C₈	—	2	2	no	—
111	C₂×C₁₁⋊C₈	—	2	4	no	—
112	C₂×C₁₁⋊C₈	—	4	8	no	—
113	D₄×M₁₆	—	11	2	no	—
114	C₂×#436	D₁₁	1	1	yes	characteristic
115	C₂×#430	—	11	2	no	—
116	#1950	—	11	2	no	—
117	#1855	—	11	4	no	—
118	#1848	—	11	1	no	—
119	#1811	—	11	2	no	—
120	C₄×#108	—	11	1	no	—
121	C₂³×C₁₁	—	2	2	no	—
122	C₂³×C₁₁	—	2	2	no	—
123	C₂³×C₁₁	—	4	2	no	—
124	C₂³×C₁₁	—	4	2	no	—
125	C₂³×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
126	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
127	D₄×C₁₁	—	2	4	no	—
128	D₄×C₁₁	—	4	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	4	no	—
132	C₂×C₄×C₁₁	—	2	4	no	—
133	C₂×C₄×C₁₁	—	2	4	no	—
134	C₂×C₄×C₁₁	—	2	4	no	—
135	C₂×C₄×C₁₁	—	2	2	no	—
136	C₂×C₄×C₁₁	—	4	1	no	—
137	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₁₁	—	2	4	no	—
149	C₈×C₁₁	—	2	2	no	—
150	C₈×C₁₁	C₄×D₄	1	4	yes	—
151	C₁₁⋊C₈	—	2	4	no	—
152	C₁₁⋊C₈	—	4	4	no	—
153	C₁₁⋊C₈	—	4	2	no	—
154	C₂²×M₁₆	—	11	4	no	—
155	C₂²×M₁₆	C₂×D₁₁	1	2	yes	—
156	C₂×C₄×D₄	C₂×D₁₁	1	1	yes	characteristic
157	#436	—	22	2	no	—
158	#436	Dic₁₁	1	4	yes	—
159	#436	Dic₁₁	1	4	yes	—
160	#435	—	22	4	no	—
161	C₈×D₄	—	22	2	no	—
162	#430	—	22	8	no	—
163	C₂×#108	—	11	1	no	—
164	C₂×#108	—	11	1	no	—
165	C₂×#108	—	11	2	no	—
166	C₂×#108	—	11	4	no	—
167	C₂×#108	—	11	2	no	—
168	#423	—	11	2	no	—
169	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
170	#407	—	11	4	no	—
171	C₂×C₂²⋊C₈	C₂×D₁₁	1	2	yes	—
172	C₄×M₁₆	—	11	2	no	—
173	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
174	#341	—	11	4	no	—
175	#341	—	11	2	no	—
176	C₂²×C₁₁	—	2	1	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	4	no	—
179	C₂²×C₁₁	—	2	4	no	—
180	C₂²×C₁₁	—	2	4	no	—
181	C₂²×C₁₁	—	8	2	no	—
182	C₂²×C₁₁	—	8	2	no	—
183	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₁₁	D₄×Q₈	1	1	yes	characteristic
185	C₂²×C₁₁	#554	1	1	yes	characteristic
186	C₄×C₁₁	—	2	4	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	#430	1	2	yes	complemented
191	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₁₁	#520	1	1	yes	characteristic
193	C₄×C₁₁	D₄²	1	1	yes	characteristic
194	C₄×C₁₁	#547	1	1	yes	characteristic
195	C₂²×D₄	Dic₂₂	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₁₁	1	2	yes	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	11	2	no	—
200	C₂×M₁₆	—	11	4	no	—
201	C₂×M₁₆	—	11	2	no	—
202	C₂×M₁₆	—	22	4	no	—
203	C₂×M₁₆	—	22	8	no	—
204	C₂×M₁₆	—	44	4	no	—
205	C₂×M₁₆	C₂×Dic₁₁	1	8	yes	—
206	C₂²×C₈	—	22	4	no	—
207	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
208	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	C₄×D₄	D₄₄	1	2	yes	—
211	C₄×D₄	D₄₄	1	2	yes	—
212	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
213	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
214	C₂×C₄⋊C₄	Dic₂₂	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	Dic₂₂	1	2	yes	—
216	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
217	#108	—	11	4	no	—
218	#108	—	22	8	no	—
219	#108	—	22	4	no	—
220	#108	—	22	2	no	—
221	#108	—	44	2	no	—
222	C₄⋊C₈	—	22	2	no	—
223	C₄⋊C₈	—	22	2	no	—
224	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
225	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
226	C₂²⋊C₈	—	22	4	no	—
227	C₂²⋊C₈	—	22	4	no	—
228	C₂²⋊C₈	C₂×Dic₁₁	1	8	yes	—
229	#97	—	22	2	no	—
230	C₄×C₈	—	22	2	no	—
231	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
232	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
233	C₂×C₁₁	—	2	1	no	—
234	C₂×C₁₁	—	2	1	no	—
235	C₂×C₁₁	—	4	4	no	—
236	C₂×C₁₁	D₄×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
238	C₂×C₁₁	#2916	1	1	yes	characteristic
239	C₂⁴	C₂×Dic₂₂	1	2	yes	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	C₄⋊Dic₁₁	1	2	yes	—
242	C₂×D₄	C₄⋊Dic₁₁	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
248	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
249	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄₄	1	4	yes	—
251	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
252	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
253	M₁₆	—	2	16	no	—
254	M₁₆	—	11	4	no	—
255	M₁₆	—	22	2	no	—
256	M₁₆	—	22	4	no	—
257	M₁₆	—	22	2	no	—
258	M₁₆	—	22	4	no	—
259	M₁₆	—	22	2	no	—
260	M₁₆	—	44	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	22	2	no	—
266	C₂×C₈	—	22	2	no	—
267	C₂×C₈	—	22	4	no	—
268	C₂×C₈	—	44	8	no	—
269	C₂×C₈	D₄×D₁₁	1	2	yes	—
270	C₂×C₈	#4341	1	2	yes	—
271	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
272	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
273	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
274	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
275	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
276	C₂²⋊C₄	C₄⋊Dic₁₁	1	4	yes	—
277	C₂²⋊C₄	C₄⋊Dic₁₁	1	4	yes	—
278	C₄²	C₂×D₄₄	1	1	yes	characteristic
279	C₄²	C₂×D₄₄	1	1	yes	characteristic
280	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
281	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
282	C₁₁	#22669	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×C₄⋊Dic₁₁	1	4	yes	—
288	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#4341	1	1	yes	characteristic
307	C₂×C₄	#67042	1	1	yes	characteristic
308	C₂×C₄	#67043	1	1	yes	characteristic
309	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	2	2	no	—
312	C₈	—	22	4	no	—
313	C₈	—	44	4	no	—
314	C₈	—	44	2	no	—
315	C₈	D₄×Dic₁₁	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	D₄×Dic₂₂	1	1	yes	characteristic
324	C₂²	#10623074	1	1	yes	characteristic
325	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	#10622738	1	2	yes	complemented
331	C₄	D₄×D₄₄	1	1	yes	characteristic
332	C₄	#10623086	1	1	yes	characteristic
333	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
334	C₄	#10623358	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	4	4	no	—
338	C₂	D₄×C₄⋊Dic₁₁	1	1	yes	characteristic
339	C₂	C₂×#10622738	1	1	yes	characteristic
340	C₂	#13077227	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×#436	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₁₁	Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×#436	D₁₁
hypocenter	C₁₁	#22669
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. C₂×#10622738	C₂	semidirect
3. #13069392	C₂	semidirect
4. #13068823	C₂	semidirect
5. #13068788	C₂	semidirect
6. #13068658	C₂	semidirect
7. C₄×#66883	C₂	semidirect
64. #66883	D₄	semidirect
190. C₄×C₁₁	#430	semidirect
282. C₁₁	#22669	semidirect
330. C₄	#10622738	semidirect

Maximal subgroups

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

subgroup	index
D₄×#4318	2
C₂×#10622738	2
#13069392	2
#13068823	2
#13068788	2
#13068658	2
C₄×#66883	2
C₂×C₁₁×#436	2
#22669	11