Subgroups — #14724465

Order 2816.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079189	C₂	1	1	yes	maximal, characteristic
2	#13077764	C₂	1	2	yes	maximal, complemented
3	#13067521	C₂	1	2	yes	maximal, complemented
4	#13067513	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067500	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067426	C₂	1	2	yes	maximal, complemented
7	C₈×D₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067373	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067080	C₂	1	2	yes	maximal, complemented
10	#13066898	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623111	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
14	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
15	#10622804	C₄	1	1	yes	characteristic
16	#10622801	C₄	1	1	yes	characteristic
17	C₄×#4325	C₂²	1	2	yes	—
18	#10622637	C₂²	1	2	yes	—
19	#10622496	C₄	1	1	yes	characteristic
20	#10622491	C₄	1	1	yes	characteristic
21	#10622452	—	2	2	no	—
22	#10622451	C₂²	1	2	yes	—
23	#10622450	C₂²	1	2	yes	—
24	#10622449	—	2	2	no	—
25	#10622420	C₄	1	2	yes	—
26	#10622413	C₄	1	2	yes	—
27	C₄×D₈₈	C₂²	1	1	yes	characteristic
28	#10622303	—	2	2	no	—
29	#10622301	—	2	2	no	—
30	C₈×D₄₄	C₂²	1	2	yes	—
31	#10622298	C₂²	1	1	yes	characteristic
32	#10622298	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
34	C₁₁×#335	C₂²	1	1	yes	characteristic
35	C₁₁×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₁	C₂²	1	1	yes	characteristic
37	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
38	#10622121	C₂²	1	2	yes	—
39	#10622117	C₂²	1	2	yes	—
40	#10622097	C₂²	1	2	yes	—
41	#10622096	C₂²	1	1	yes	characteristic
42	#10622092	C₂²	1	1	yes	characteristic
43	#10622092	C₂²	1	1	yes	characteristic
44	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
47	#67000	C₂×C₄	1	2	yes	—
48	#66999	C₂×C₄	1	2	yes	—
49	C₂×#4325	C₂×C₄	1	2	yes	—
50	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
51	C₂×#4313	—	2	2	no	—
52	C₂×#4313	—	2	2	no	—
53	C₂×C₈×D₁₁	—	2	2	no	—
54	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
56	#66951	—	2	2	no	—
57	#66948	C₂×C₄	1	2	yes	—
58	#66947	—	2	2	no	—
59	#66943	C₂×C₄	1	2	yes	—
60	C₄×D₄₄	—	2	2	no	—
61	C₄×D₄₄	C₂³	1	2	yes	—
62	#66925	—	2	2	no	—
63	C₄²×D₁₁	C₂³	1	1	yes	characteristic
64	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₁×C₄⋊C₈	—	2	2	no	—
66	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₁×#103	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₁	—	2	2	no	—
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	#66899	C₂×C₄	1	2	yes	—
73	#66885	C₂×C₄	1	2	yes	—
74	(C₂×D₁₁)⋊C₈	—	2	1	no	—
75	(C₂×D₁₁)⋊C₈	—	2	2	no	—
76	(C₂×D₁₁)⋊C₈	—	2	2	no	—
77	(C₂×D₁₁)⋊C₈	D₄	1	1	yes	characteristic
78	(C₂×D₁₁)⋊C₈	D₄	1	1	yes	characteristic
79	#66882	C₂×C₄	1	1	yes	characteristic
80	Dic₁₁⋊C₈	—	2	2	no	—
81	Dic₁₁⋊C₈	—	2	1	no	—
82	Dic₁₁⋊C₈	D₄	1	1	yes	characteristic
83	Dic₁₁⋊C₈	D₄	1	1	yes	characteristic
84	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
86	#66873	C₂×C₄	1	2	yes	—
87	#66872	C₂×C₄	1	2	yes	—
88	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
89	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
90	#66867	—	2	2	no	—
91	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
93	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
94	#17973	—	11	1	no	maximal, Sylow
95	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄₄	—	2	2	no	—
97	C₂×D₄₄	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₁	—	2	2	no	—
99	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₁₁	#41	1	1	yes	characteristic
102	C₁₁×Q₁₆	—	2	2	no	—
103	C₂×C₈×C₁₁	—	2	2	no	—
104	C₂×C₈×C₁₁	—	2	1	no	—
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₁₁×C₄⋊C₄	—	2	2	no	—
110	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₁	—	2	2	no	—
112	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
113	#4325	—	2	4	no	—
114	(C₂×D₁₁)⋊C₄	—	2	2	no	—
115	(C₂×D₁₁)⋊C₄	—	2	2	no	—
116	(C₂×D₁₁)⋊C₄	—	2	2	no	—
117	(C₂×D₁₁)⋊C₄	—	2	2	no	—
118	(C₂×D₁₁)⋊C₄	—	2	2	no	—
119	C₄⋊Dic₁₁	—	2	2	no	—
120	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
121	Dic₁₁⋊C₄	—	2	2	no	—
122	Dic₁₁⋊C₄	—	2	2	no	—
123	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₁	#41	1	1	yes	characteristic
126	C₂×C₁₁⋊C₈	—	2	2	no	—
127	C₂×C₁₁⋊C₈	—	2	1	no	—
128	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
132	D₈₈	—	2	2	no	—
133	#4313	—	8	2	no	—
134	#4313	—	8	2	no	—
135	C₈×D₁₁	—	2	2	no	—
136	C₈×D₁₁	—	2	2	no	—
137	C₈×D₁₁	—	8	2	no	—
138	#3168	—	11	1	no	—
139	#2993	—	11	2	no	—
140	#1592	—	11	2	no	—
141	#1591	—	11	1	no	—
142	#1589	—	11	1	no	—
143	SD₁₆⋊C₈	—	11	2	no	—
144	C₈×Q₁₆	D₁₁	1	1	yes	characteristic, complemented
145	C₈×D₈	—	11	1	no	—
146	#1567	—	11	1	no	—
147	#1495	—	11	2	no	—
148	#1450	—	11	1	no	—
149	C₂²×D₁₁	—	2	2	no	—
150	C₂²×D₁₁	C₄×D₄	1	1	yes	characteristic
151	Q₈×C₁₁	—	2	2	no	—
152	Q₈×C₁₁	—	2	2	no	—
153	C₂×C₄×C₁₁	—	2	2	no	—
154	C₂×C₄×C₁₁	—	2	2	no	—
155	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₁₁	—	2	2	no	—
159	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
160	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
162	D₄₄	—	2	2	no	—
163	D₄₄	—	2	2	no	—
164	D₄₄	—	8	2	no	—
165	C₄×D₁₁	—	2	1	no	—
166	C₄×D₁₁	—	2	1	no	—
167	C₄×D₁₁	—	2	1	no	—
168	C₄×D₁₁	—	2	1	no	—
169	C₄×D₁₁	—	4	1	no	—
170	C₄×D₁₁	—	4	2	no	—
171	C₄×D₁₁	—	4	2	no	—
172	C₈×C₁₁	—	2	1	no	—
173	C₈×C₁₁	—	2	1	no	—
174	C₈×C₁₁	—	2	1	no	—
175	C₈×C₁₁	—	4	1	no	—
176	C₈×C₁₁	—	4	2	no	—
177	C₈×C₁₁	#131	1	2	yes	—
178	C₁₁⋊C₈	—	2	1	no	—
179	C₁₁⋊C₈	—	2	1	no	—
180	C₁₁⋊C₈	—	2	1	no	—
181	C₁₁⋊C₈	—	4	1	no	—
182	C₁₁⋊C₈	—	4	2	no	—
183	C₁₁⋊C₈	#131	1	2	yes	—
184	#529	—	11	2	no	—
185	#499	—	11	1	no	—
186	#495	—	11	1	no	—
187	#488	—	11	2	no	—
188	#467	—	11	1	no	—
189	#466	—	11	1	no	—
190	#465	—	11	2	no	—
191	#446	—	11	2	no	—
192	C₈×Q₈	C₂×D₁₁	1	2	yes	—
193	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
194	C₄×SD₁₆	—	11	2	no	—
195	C₄×D₈	—	11	1	no	—
196	#436	—	11	2	no	—
197	#436	—	22	2	no	—
198	#435	—	22	2	no	—
199	C₈×D₄	—	11	2	no	—
200	#433	—	22	2	no	—
201	#431	—	11	1	no	—
202	#431	—	11	1	no	—
203	#424	—	22	2	no	—
204	#423	—	11	2	no	—
205	C₂×C₄×C₈	—	11	1	no	—
206	#335	—	11	2	no	—
207	#335	—	11	1	no	—
208	#335	C₂×D₁₁	1	1	yes	characteristic
209	Q₈⋊C₈	—	11	2	no	—
210	Q₈⋊C₈	C₂×D₁₁	1	2	yes	—
211	D₄⋊C₈	—	11	2	no	—
212	D₄⋊C₈	—	11	2	no	—
213	#322	—	11	1	no	—
214	#322	—	11	1	no	—
215	C₈²	—	11	1	no	—
216	C₈²	C₂×D₁₁	1	1	yes	characteristic
217	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×D₁₁	—	2	1	no	—
219	C₂×D₁₁	—	2	1	no	—
220	C₂×D₁₁	—	2	1	no	—
221	C₂×D₁₁	—	4	2	no	—
222	C₂×D₁₁	—	4	2	no	—
223	C₂×D₁₁	—	4	2	no	—
224	C₄×C₁₁	—	2	1	no	—
225	C₄×C₁₁	—	4	2	no	—
226	C₄×C₁₁	—	4	2	no	—
227	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
228	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
229	C₄×C₁₁	#576	1	1	yes	characteristic
230	C₄×C₁₁	#578	1	1	yes	characteristic
231	Dic₁₁	—	2	1	no	—
232	Dic₁₁	—	2	1	no	—
233	Dic₁₁	—	2	1	no	—
234	Dic₁₁	—	4	2	no	—
235	C₂×Q₁₆	C₄×D₁₁	1	1	yes	characteristic
236	C₂×SD₁₆	—	11	2	no	—
237	C₂×D₈	—	11	1	no	—
238	C₂×M₁₆	—	22	2	no	—
239	C₂×M₁₆	—	22	2	no	—
240	C₂²×C₈	—	11	1	no	—
241	C₂²×C₈	—	11	1	no	—
242	C₂²×C₈	—	22	2	no	—
243	#126	—	22	2	no	—
244	#125	—	11	2	no	—
245	#124	—	11	2	no	—
246	#123	—	22	2	no	—
247	C₂²⋊Q₈	—	11	2	no	—
248	#121	—	11	2	no	—
249	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
250	C₄×D₄	—	11	2	no	—
251	C₄×D₄	—	22	2	no	—
252	#117	—	22	2	no	—
253	C₂×C₄²	—	11	1	no	—
254	#107	—	11	1	no	—
255	#107	C₄×D₁₁	1	1	yes	characteristic
256	#106	—	11	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	11	1	no	—
259	C₄⋊C₈	—	11	1	no	—
260	C₄⋊C₈	—	11	2	no	—
261	C₄⋊C₈	—	22	2	no	—
262	C₄⋊C₈	—	22	2	no	—
263	C₄⋊C₈	—	22	1	no	—
264	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
265	#103	—	11	2	no	—
266	#103	C₄×D₁₁	1	2	yes	—
267	#102	—	11	2	no	—
268	#102	—	11	2	no	—
269	C₂²⋊C₈	—	11	1	no	—
270	C₂²⋊C₈	—	11	1	no	—
271	C₂²⋊C₈	—	22	2	no	—
272	C₂²⋊C₈	—	22	1	no	—
273	C₂²⋊C₈	—	22	2	no	—
274	#97	—	22	2	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	11	1	no	—
277	C₄×C₈	—	11	1	no	—
278	C₄×C₈	—	11	1	no	—
279	C₄×C₈	—	11	1	no	—
280	C₄×C₈	—	11	1	no	—
281	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
284	C₂×C₁₁	#2931	1	1	yes	characteristic
285	C₂×C₁₁	#2951	1	1	yes	characteristic
286	C₂×C₁₁	#2957	1	1	yes	characteristic
287	D₁₁	—	4	1	no	—
288	D₁₁	—	8	2	no	—
289	C₂×Q₈	C₂×C₄×D₁₁	1	2	yes	—
290	C₂×D₄	—	11	2	no	—
291	C₂×D₄	—	22	2	no	—
292	C₂²×C₄	—	11	1	no	—
293	C₂²×C₄	—	11	1	no	—
294	C₂²×C₄	—	11	1	no	—
295	C₂²×C₄	—	22	2	no	—
296	Q₁₆	—	2	2	no	—
297	SD₁₆	—	22	4	no	—
298	D₈	—	22	2	no	—
299	M₁₆	—	88	2	no	—
300	M₁₆	—	88	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	11	1	no	—
304	C₂×C₈	—	11	1	no	—
305	C₂×C₈	—	11	1	no	—
306	C₂×C₈	—	11	1	no	—
307	C₂×C₈	—	22	2	no	—
308	C₂×C₈	—	22	2	no	—
309	C₂×C₈	—	22	2	no	—
310	C₂×C₈	—	22	1	no	—
311	C₂×C₈	—	88	2	no	—
312	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
314	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
315	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	11	2	no	—
318	C₄⋊C₄	—	22	2	no	—
319	C₄⋊C₄	—	22	2	no	—
320	C₄⋊C₄	—	22	2	no	—
321	C₄⋊C₄	C₂×C₄×D₁₁	1	2	yes	—
322	C₂²⋊C₄	—	22	2	no	—
323	C₂²⋊C₄	—	22	2	no	—
324	C₂²⋊C₄	—	22	2	no	—
325	C₂²⋊C₄	—	22	2	no	—
326	C₂²⋊C₄	—	22	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	11	1	no	—
329	C₄²	—	11	1	no	—
330	C₄²	—	11	1	no	—
331	C₄²	C₂³×D₁₁	1	1	yes	characteristic
332	C₁₁	#17973	1	1	yes	characteristic, complemented, Sylow
333	C₂³	—	11	1	no	—
334	C₂³	—	22	2	no	—
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	D₄	—	22	2	no	—
338	D₄	—	22	2	no	—
339	D₄	—	88	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	11	1	no	—
343	C₂×C₄	—	11	1	no	—
344	C₂×C₄	—	11	1	no	—
345	C₂×C₄	—	22	1	no	—
346	C₂×C₄	—	22	1	no	—
347	C₂×C₄	—	22	1	no	—
348	C₂×C₄	—	22	1	no	—
349	C₂×C₄	—	22	2	no	—
350	C₂×C₄	—	44	1	no	—
351	C₂×C₄	—	44	2	no	—
352	C₂×C₄	—	44	2	no	—
353	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
355	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	11	2	no	—
362	C₈	—	22	1	no	—
363	C₈	—	22	1	no	—
364	C₈	—	22	1	no	—
365	C₈	—	44	1	no	—
366	C₈	—	44	2	no	—
367	C₈	#66954	1	2	yes	—
368	C₂²	—	22	1	no	—
369	C₂²	—	22	1	no	—
370	C₂²	—	22	1	no	—
371	C₂²	—	44	2	no	—
372	C₂²	—	44	2	no	—
373	C₂²	—	44	2	no	—
374	C₂²	C₄×D₄×D₁₁	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	22	1	no	—
379	C₄	—	22	1	no	—
380	C₄	—	22	1	no	—
381	C₄	—	44	2	no	—
382	C₄	C₂×#66954	1	1	yes	characteristic
383	C₄	D₁₁×#131	1	1	yes	characteristic
384	C₄	#10623297	1	1	yes	characteristic
385	C₄	#10623300	1	1	yes	characteristic
386	C₂	—	44	1	no	—
387	C₂	—	88	2	no	—
388	C₂	#13077323	1	1	yes	characteristic
389	C₂	#13077491	1	1	yes	characteristic
390	C₂	#13077528	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×Q₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#10622496	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×Q₁₆	D₁₁
hypocenter	C₁₁	#17973
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13077764	C₂	semidirect
3. #13067521	C₂	semidirect
4. #13067513	C₂	semidirect
5. #13067500	C₂	semidirect
6. #13067426	C₂	semidirect
7. C₈×D₈₈	C₂	semidirect
8. #13067373	C₂	semidirect
9. #13067080	C₂	semidirect
10. #13066898	C₂	semidirect
11. C₈×C₁₁×Q₁₆	C₂	semidirect
144. C₈×Q₁₆	D₁₁	semidirect
332. C₁₁	#17973	semidirect

Maximal subgroups

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

subgroup	index
#13079189	2
#13077764	2
#13067521	2
#13067513	2
#13067500	2
#13067426	2
C₈×D₈₈	2
#13067373	2
#13067080	2
#13066898	2
C₈×C₁₁×Q₁₆	2
#17973	11