Subgroups — #14909335

Order 2816.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079202	C₂	1	1	yes	maximal, characteristic, complemented
2	#13079195	C₂	1	1	yes	maximal, characteristic
3	#13070898	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070872	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070844	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070840	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070712	C₂	1	4	yes	maximal, complemented
8	C₄×#66864	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#10623008	—	2	1	no	—
11	#10623001	—	2	1	no	—
12	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622729	—	2	1	no	—
15	#10622729	C₂²	1	2	yes	—
16	#10622728	—	2	1	no	—
17	#10622728	C₂²	1	2	yes	—
18	C₂×#66864	C₂²	1	4	yes	—
19	#10622312	C₂²	1	1	yes	characteristic
20	C₄×Dic₄₄	—	2	1	no	—
21	C₄×Dic₄₄	C₂²	1	2	yes	—
22	#10622308	C₂²	1	1	yes	characteristic
23	#10622307	—	2	1	no	—
24	C₄×D₈₈	—	2	1	no	—
25	C₄×D₈₈	C₂²	1	2	yes	—
26	#10622295	C₂²	1	1	yes	characteristic
27	#10622295	C₂²	1	1	yes	characteristic
28	#10622294	—	2	1	no	—
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
31	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
32	#10622147	C₂²	1	4	yes	—
33	#10622143	C₂²	1	4	yes	—
34	#10622141	C₂²	1	4	yes	—
35	C₂×#4339	—	4	1	no	—
36	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
39	#66983	—	2	2	no	—
40	#66983	—	4	1	no	—
41	#66979	—	4	1	no	—
42	#66977	—	2	2	no	—
43	#66977	—	4	1	no	—
44	C₂×Dic₄₄	—	2	1	no	—
45	C₂×Dic₄₄	C₂³	1	2	yes	—
46	C₂×D₈₈	—	2	1	no	—
47	C₂×D₈₈	C₂³	1	2	yes	—
48	#66928	—	4	1	no	—
49	#66927	—	2	1	no	—
50	C₄×D₄₄	—	2	2	no	—
51	C₄×D₄₄	—	4	1	no	—
52	#66923	—	4	1	no	—
53	#66922	—	2	1	no	—
54	#66922	—	2	1	no	—
55	#66922	—	2	1	no	—
56	C₄×Dic₂₂	—	2	2	no	—
57	C₄×Dic₂₂	—	4	1	no	—
58	C₂×C₁₁×C₁₆	—	2	4	no	—
59	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
61	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	#66885	—	2	2	no	—
66	#66885	—	2	2	no	—
67	#66882	—	2	1	no	—
68	#66882	—	2	1	no	—
69	#66882	C₂³	1	2	yes	—
70	#66882	C₂³	1	2	yes	—
71	#66880	—	2	2	no	—
72	#66880	—	2	2	no	—
73	#66864	—	2	8	no	—
74	#33736	—	11	1	no	maximal, Sylow
75	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₁	—	4	2	no	—
79	#4339	—	8	4	no	—
80	C₂×D₄₄	—	2	2	no	—
81	C₂×D₄₄	—	4	1	no	—
82	C₂×D₄₄	—	4	1	no	—
83	C₂×C₄×D₁₁	—	4	2	no	—
84	C₂×Dic₂₂	—	2	2	no	—
85	C₂×Dic₂₂	—	4	1	no	—
86	C₂×Dic₂₂	—	4	1	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₄²×C₁₁	D₈	1	1	yes	characteristic
94	C₄²×C₁₁	D₈	1	1	yes	characteristic
95	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₁₁	—	4	2	no	—
98	(C₂×D₁₁)⋊C₄	—	4	2	no	—
99	(C₂×D₁₁)⋊C₄	—	4	2	no	—
100	C₄⋊Dic₁₁	—	2	2	no	—
101	C₄⋊Dic₁₁	—	2	2	no	—
102	C₄⋊Dic₁₁	—	4	1	no	—
103	C₄⋊Dic₁₁	—	4	1	no	—
104	C₄⋊Dic₁₁	—	4	1	no	—
105	C₄⋊Dic₁₁	—	4	1	no	—
106	Dic₁₁⋊C₄	—	4	2	no	—
107	Dic₁₁⋊C₄	—	4	2	no	—
108	C₄×Dic₁₁	—	4	2	no	—
109	Dic₄₄	—	2	4	no	—
110	Dic₄₄	—	4	2	no	—
111	D₈₈	—	2	4	no	—
112	D₈₈	—	4	2	no	—
113	C₁₁×C₁₆	—	2	2	no	—
114	C₁₁×C₁₆	#41	1	4	yes	complemented
115	#3170	—	11	1	no	—
116	#3169	—	11	1	no	—
117	#2254	—	11	1	no	—
118	#2248	—	11	1	no	—
119	#2241	—	11	1	no	—
120	#2240	—	11	1	no	—
121	#2217	—	11	4	no	—
122	C₄×SD₃₂	—	11	4	no	—
123	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
124	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₁₁	—	4	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	SD₃₂	1	4	yes	complemented
130	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₁₁	—	8	4	no	—
137	C₂×Dic₁₁	—	4	2	no	—
138	C₂×Dic₁₁	—	4	2	no	—
139	C₂×Dic₁₁	—	4	2	no	—
140	D₄₄	—	4	2	no	—
141	D₄₄	—	4	2	no	—
142	D₄₄	—	8	2	no	—
143	D₄₄	—	8	2	no	—
144	C₄×D₁₁	—	8	4	no	—
145	Dic₂₂	—	4	2	no	—
146	Dic₂₂	—	4	2	no	—
147	Dic₂₂	—	8	2	no	—
148	Dic₂₂	—	8	2	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	#135	1	2	yes	—
152	C₈×C₁₁	C₂×#41	1	2	yes	—
153	#533	—	22	1	no	—
154	#532	—	22	1	no	—
155	C₂×SD₃₂	—	11	4	no	—
156	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
157	#500	—	11	1	no	—
158	#500	—	11	1	no	—
159	#494	—	11	1	no	—
160	#493	—	11	1	no	—
161	#487	—	22	1	no	—
162	#486	—	22	1	no	—
163	#467	—	11	2	no	—
164	#467	—	22	1	no	—
165	#466	—	11	2	no	—
166	#466	—	22	1	no	—
167	C₄×Q₁₆	—	11	2	no	—
168	C₄×Q₁₆	—	22	1	no	—
169	C₄×D₈	—	11	2	no	—
170	C₄×D₈	—	22	1	no	—
171	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
172	#367	—	11	4	no	—
173	#358	—	11	4	no	—
174	#357	—	11	4	no	—
175	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
176	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
177	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	C₂×SD₃₂	1	2	yes	—
181	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
182	C₂×D₁₁	—	8	2	no	—
183	C₂×D₁₁	—	8	2	no	—
184	C₂×D₁₁	—	8	2	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	C₂×SD₃₂	1	2	yes	—
188	C₄×C₁₁	C₂×SD₃₂	1	2	yes	—
189	C₄×C₁₁	#508	1	2	yes	—
190	C₄×C₁₁	#532	1	1	yes	characteristic
191	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
192	Dic₁₁	—	8	2	no	—
193	Dic₁₁	—	8	2	no	—
194	Dic₁₁	—	8	2	no	—
195	C₂×#41	—	44	1	no	—
196	C₂×Q₁₆	—	11	2	no	—
197	C₂×Q₁₆	—	22	1	no	—
198	C₂×D₈	—	11	2	no	—
199	C₂×D₈	—	22	1	no	—
200	C₂²×C₈	D₄₄	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	22	1	no	—
203	C₄⋊Q₈	—	22	1	no	—
204	C₄⋊Q₈	—	22	1	no	—
205	#127	—	22	1	no	—
206	#125	—	44	1	no	—
207	#124	—	44	1	no	—
208	C₂²⋊Q₈	—	22	2	no	—
209	C₂²⋊Q₈	—	44	1	no	—
210	#121	—	22	2	no	—
211	#121	—	44	1	no	—
212	C₄×Q₈	—	22	2	no	—
213	C₄×Q₈	—	44	1	no	—
214	C₄×D₄	—	22	2	no	—
215	C₄×D₄	—	44	1	no	—
216	#117	—	44	1	no	—
217	C₂×C₄²	D₄₄	1	1	yes	characteristic
218	SD₃₂	—	22	8	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
221	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
222	#107	—	11	2	no	—
223	#107	—	11	2	no	—
224	#107	—	22	1	no	—
225	#107	—	22	1	no	—
226	#103	—	22	2	no	—
227	#103	—	22	2	no	—
228	#102	—	22	2	no	—
229	#102	—	22	2	no	—
230	C₄×C₈	D₄₄	1	1	yes	characteristic
231	C₄×C₈	D₄₄	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
234	C₂×C₁₁	—	2	2	no	—
235	C₂×C₁₁	#3169	1	1	yes	characteristic
236	C₂×C₁₁	C₂²×SD₃₂	1	1	yes	characteristic
237	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
238	D₁₁	—	16	2	no	—
239	#41	—	88	4	no	—
240	C₂×Q₈	—	22	2	no	—
241	C₂×Q₈	—	44	1	no	—
242	C₂×Q₈	—	44	1	no	—
243	C₂×D₄	—	22	2	no	—
244	C₂×D₄	—	44	2	no	—
245	C₂×D₄	—	44	1	no	—
246	C₂×D₄	—	44	1	no	—
247	C₂²×C₄	—	44	2	no	—
248	C₂²×C₄	D₈₈	1	1	yes	characteristic
249	C₂²×C₄	D₈₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
251	Q₁₆	—	22	4	no	—
252	Q₁₆	—	44	2	no	—
253	D₈	—	22	4	no	—
254	D₈	—	44	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
261	C₄⋊C₄	—	22	2	no	—
262	C₄⋊C₄	—	22	2	no	—
263	C₄⋊C₄	—	44	1	no	—
264	C₄⋊C₄	—	44	1	no	—
265	C₄⋊C₄	—	44	1	no	—
266	C₄⋊C₄	—	44	2	no	—
267	C₄⋊C₄	—	44	2	no	—
268	C₄⋊C₄	—	44	1	no	—
269	C₂²⋊C₄	—	44	2	no	—
270	C₂²⋊C₄	—	44	2	no	—
271	C₂²⋊C₄	—	44	2	no	—
272	C₄²	—	44	2	no	—
273	C₄²	D₈₈	1	1	yes	characteristic
274	C₄²	D₈₈	1	1	yes	characteristic
275	C₄²	C₂×D₄₄	1	1	yes	characteristic
276	C₄²	C₂×D₄₄	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	#4339	1	4	yes	complemented
279	C₁₁	#33736	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	44	2	no	—
281	C₂³	C₂×D₈₈	1	1	yes	characteristic
282	Q₈	—	44	2	no	—
283	Q₈	—	44	2	no	—
284	Q₈	—	88	2	no	—
285	Q₈	—	88	2	no	—
286	D₄	—	44	2	no	—
287	D₄	—	44	2	no	—
288	D₄	—	88	4	no	—
289	D₄	—	88	2	no	—
290	D₄	—	88	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	44	2	no	—
295	C₂×C₄	—	44	2	no	—
296	C₂×C₄	—	44	2	no	—
297	C₂×C₄	—	88	4	no	—
298	C₂×C₄	#66864	1	4	yes	complemented
299	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	#66957	1	2	yes	—
308	C₈	C₂×#4339	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	88	2	no	—
312	C₂²	—	88	2	no	—
313	C₂²	—	88	2	no	—
314	C₂²	C₂×#66864	1	2	yes	—
315	C₂²	C₂²×D₈₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	88	2	no	—
319	C₄	—	88	2	no	—
320	C₄	—	88	2	no	—
321	C₄	C₂×#66864	1	2	yes	—
322	C₄	C₂×#66864	1	2	yes	—
323	C₄	#10622517	1	2	yes	—
324	C₄	#10623008	1	1	yes	characteristic
325	C₄	C₂×#66957	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	176	2	no	—
328	C₂	#13079195	1	1	yes	characteristic
329	C₂	C₂²×#66864	1	1	yes	characteristic
330	C₂	C₂×#10622517	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	#13079195	C₂
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#33736
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #13079202	C₂	semidirect
3. #13070898	C₂	semidirect
4. #13070872	C₂	semidirect
5. #13070844	C₂	semidirect
6. #13070840	C₂	semidirect
7. #13070712	C₂	semidirect
8. C₄×#66864	C₂	semidirect
9. C₂×C₄×C₁₁×C₁₆	C₂	semidirect
114. C₁₁×C₁₆	#41	semidirect
123. C₂×C₄×C₁₆	D₁₁	semidirect
129. C₂×C₄×C₁₁	SD₃₂	semidirect
278. C₁₆	#4339	semidirect
279. C₁₁	#33736	semidirect
298. C₂×C₄	#66864	semidirect

Maximal subgroups

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

subgroup	index
#13079202	2
#13079195	2
#13070898	2
#13070872	2
#13070844	2
#13070840	2
#13070712	2
C₄×#66864	2
C₂×C₄×C₁₁×C₁₆	2
#33736	11