Subgroups — #14907249

Order 2816.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077533	C₂	1	1	yes	maximal, characteristic, complemented
2	#13077468	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070515	C₂	1	8	yes	maximal, complemented
4	#13070389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622201	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070352	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070220	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#509	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#135	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×D₈	C₂²	1	1	yes	characteristic
11	#10623350	—	2	1	no	—
12	#10623329	—	2	1	no	—
13	C₁₁×#509	C₄	1	4	yes	complemented
14	C₁₁×#509	C₂²	1	4	yes	—
15	C₂×C₁₁×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₁₁×D₁₆	C₄	1	2	yes	complemented
17	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
18	#10622849	—	2	1	no	—
19	#10622828	—	2	1	no	—
20	#10622810	C₂²	1	4	yes	complemented
21	#10622809	—	2	2	no	—
22	#10622765	C₂²	1	4	yes	complemented
23	D₈×Dic₁₁	—	2	2	no	—
24	C₂×#66883	C₂²	1	1	yes	characteristic
25	#10622719	C₂²	1	1	yes	characteristic
26	#10622714	C₂²	1	1	yes	characteristic
27	C₂×#66876	C₂²	1	1	yes	characteristic
28	#10622204	C₂²	1	4	yes	complemented
29	#10622201	C₂²	1	4	yes	complemented
30	#10622151	C₂²	1	4	yes	complemented
31	#10622150	C₂²	1	4	yes	complemented
32	C₂×C₁₁×#41	—	2	1	no	—
33	C₂²×D₄×C₁₁	—	2	1	no	—
34	C₁₁×#135	—	2	1	no	—
35	C₁₁×#135	—	2	1	no	—
36	C₁₁×#135	C₂×C₄	1	2	yes	complemented
37	C₁₁×#135	C₂³	1	2	yes	—
38	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×C₁₁×SD₁₆	—	2	1	no	—
40	C₂×C₁₁×D₈	—	2	1	no	—
41	C₂×C₁₁×D₈	—	2	1	no	—
42	C₂×C₁₁×D₈	—	2	1	no	—
43	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
44	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
45	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×C₁₁×D₈	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₁₁	—	4	2	no	—
49	#67001	—	2	4	no	—
50	#67001	—	4	2	no	—
51	D₄×Dic₁₁	—	2	4	no	—
52	D₄×Dic₁₁	—	4	2	no	—
53	#66979	—	2	1	no	—
54	#66979	D₄	1	1	yes	characteristic
55	C₂×#4318	—	2	1	no	—
56	C₂×#4318	—	4	2	no	—
57	C₂×#4318	D₄	1	1	yes	characteristic
58	C₁₁×SD₃₂	C₂×C₄	1	8	yes	complemented
59	C₁₁×D₁₆	C₂×C₄	1	8	yes	complemented
60	C₁₁×M₃₂	C₂×C₄	1	2	yes	complemented
61	C₁₁×M₃₂	C₂³	1	2	yes	—
62	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	complemented
63	#66901	—	2	4	no	—
64	#66896	—	2	4	no	—
65	#66883	—	2	1	no	—
66	#66883	C₂³	1	2	yes	—
67	#66882	—	2	1	no	—
68	#66881	C₂³	1	2	yes	—
69	#66879	C₂³	1	2	yes	—
70	C₈×Dic₁₁	D₄	1	2	yes	—
71	#66876	—	2	1	no	—
72	#66876	C₂³	1	2	yes	—
73	C₂×C₁₁⋊C₁₆	—	2	1	no	—
74	#33471	—	11	1	no	maximal, Sylow
75	C₂⁴×C₁₁	—	4	1	no	—
76	C₁₁×#41	—	2	2	no	—
77	C₁₁×#41	—	2	2	no	—
78	C₁₁×#41	—	4	1	no	—
79	C₁₁×#41	—	4	1	no	—
80	C₂×Q₈×C₁₁	—	2	1	no	—
81	C₂×D₄×C₁₁	—	2	2	no	—
82	C₂×D₄×C₁₁	—	2	2	no	—
83	C₂×D₄×C₁₁	—	2	1	no	—
84	C₂×D₄×C₁₁	—	2	2	no	—
85	C₂×D₄×C₁₁	—	4	1	no	—
86	C₂×D₄×C₁₁	—	4	1	no	—
87	C₂×D₄×C₁₁	—	4	1	no	—
88	C₂×D₄×C₁₁	—	4	1	no	—
89	C₂²×C₄×C₁₁	—	4	1	no	—
90	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₁₁	—	4	4	no	—
92	C₁₁×Q₁₆	—	2	1	no	—
93	C₁₁×Q₁₆	C₂²×C₄	1	2	yes	—
94	C₁₁×SD₁₆	—	2	2	no	—
95	C₁₁×SD₁₆	—	2	2	no	—
96	C₁₁×D₈	—	2	2	no	—
97	C₁₁×D₈	—	2	2	no	—
98	C₁₁×D₈	—	2	1	no	—
99	C₁₁×D₈	—	2	2	no	—
100	C₁₁×D₈	C₂²×C₄	1	2	yes	—
101	C₁₁×D₈	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₁₁	—	2	1	no	—
109	C₂²⋊Dic₁₁	—	4	4	no	—
110	C₂²⋊Dic₁₁	—	4	1	no	—
111	C₄⋊Dic₁₁	—	2	1	no	—
112	C₄⋊Dic₁₁	—	2	2	no	—
113	C₄×Dic₁₁	—	4	1	no	—
114	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
115	#4318	—	2	1	no	—
116	#4318	—	2	1	no	—
117	#4318	—	2	2	no	—
118	#4318	—	4	4	no	—
119	#4318	—	4	1	no	—
120	C₂×C₁₁⋊C₈	—	4	4	no	—
121	C₂×C₁₁⋊C₈	—	4	1	no	—
122	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
123	C₁₁×C₁₆	C₂²×C₄	1	4	yes	—
124	C₁₁⋊C₁₆	—	2	2	no	—
125	C₂×#509	D₁₁	1	1	yes	characteristic
126	#2957	—	11	1	no	—
127	#2948	—	11	1	no	—
128	AGL(1,ℤ/16)	—	11	8	no	—
129	#2155	—	11	1	no	—
130	C₂×#361	—	11	1	no	—
131	#2145	—	11	1	no	—
132	#2110	—	11	1	no	—
133	C₂³×C₁₁	—	4	2	no	—
134	C₂³×C₁₁	—	4	1	no	—
135	C₂³×C₁₁	—	4	2	no	—
136	C₂³×C₁₁	—	4	2	no	—
137	C₂³×C₁₁	—	16	2	no	—
138	C₂³×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
139	Q₈×C₁₁	—	2	2	no	—
140	Q₈×C₁₁	—	4	1	no	—
141	D₄×C₁₁	—	2	4	no	—
142	D₄×C₁₁	—	2	2	no	—
143	D₄×C₁₁	—	4	2	no	—
144	D₄×C₁₁	—	4	2	no	—
145	D₄×C₁₁	—	4	2	no	—
146	D₄×C₁₁	—	4	2	no	—
147	D₄×C₁₁	—	4	2	no	—
148	D₄×C₁₁	—	4	1	no	—
149	C₂×C₄×C₁₁	—	4	1	no	—
150	C₂×C₄×C₁₁	—	4	2	no	—
151	C₂×C₄×C₁₁	—	4	2	no	—
152	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×Dic₁₁	—	2	2	no	—
159	C₂×Dic₁₁	—	4	2	no	—
160	C₂×Dic₁₁	—	16	4	no	—
161	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
162	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
163	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
164	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
165	C₁₁⋊C₈	—	2	2	no	—
166	C₁₁⋊C₈	—	4	2	no	—
167	C₂×#135	C₂×D₁₁	1	1	yes	characteristic
168	C₂²×D₈	C₂×D₁₁	1	1	yes	characteristic
169	#568	—	22	1	no	—
170	#518	—	22	1	no	—
171	#509	Dic₁₁	1	4	yes	complemented
172	#509	C₂×D₁₁	1	4	yes	—
173	C₂×SD₃₂	Dic₁₁	1	2	yes	complemented
174	C₂×D₁₆	Dic₁₁	1	2	yes	complemented
175	C₂×M₃₂	—	11	1	no	—
176	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
177	#444	—	11	4	no	—
178	#443	—	22	2	no	—
179	#442	—	11	4	no	—
180	C₄×D₈	—	22	2	no	—
181	C₂×#108	—	11	1	no	—
182	#427	—	11	1	no	—
183	#421	—	22	1	no	—
184	#418	—	22	1	no	—
185	#405	—	11	1	no	—
186	#365	—	11	4	no	—
187	#361	—	11	4	no	—
188	#360	—	11	4	no	—
189	#347	—	11	4	no	—
190	C₂²×C₁₁	—	4	1	no	—
191	C₂²×C₁₁	—	4	2	no	—
192	C₂²×C₁₁	—	4	4	no	—
193	C₂²×C₁₁	—	8	4	no	—
194	C₂²×C₁₁	—	16	1	no	—
195	C₂²×C₁₁	—	16	2	no	—
196	C₂²×C₁₁	—	16	2	no	—
197	C₂²×C₁₁	#442	1	1	yes	characteristic
198	C₂²×C₁₁	#442	1	1	yes	characteristic
199	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₁₁	—	4	2	no	—
201	C₄×C₁₁	#442	1	1	yes	characteristic
202	C₄×C₁₁	#442	1	1	yes	characteristic
203	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	Dic₁₁	—	4	2	no	—
206	Dic₁₁	—	8	2	no	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	#135	—	2	1	no	—
210	#135	—	2	1	no	—
211	#135	C₂×Dic₁₁	1	2	yes	complemented
212	#135	C₂²×D₁₁	1	2	yes	—
213	C₂×Q₁₆	C₂×Dic₁₁	1	1	yes	characteristic, complemented
214	C₂×SD₁₆	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
219	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
220	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic, complemented
221	C₂×D₈	C₂²×D₁₁	1	2	yes	—
222	#131	—	22	4	no	—
223	#131	—	44	2	no	—
224	C₂×M₁₆	—	11	1	no	—
225	C₂×M₁₆	—	22	1	no	—
226	C₂×M₁₆	—	44	2	no	—
227	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
228	C₄×D₄	—	22	4	no	—
229	C₄×D₄	—	44	2	no	—
230	#117	—	11	1	no	—
231	#117	—	22	1	no	—
232	C₂×C₂²⋊C₄	—	44	2	no	—
233	SD₃₂	C₂×Dic₁₁	1	8	yes	complemented
234	D₁₆	C₂×Dic₁₁	1	8	yes	complemented
235	M₃₂	—	11	2	no	—
236	M₃₂	—	22	1	no	—
237	M₃₂	C₂×Dic₁₁	1	2	yes	complemented
238	M₃₂	C₂²×D₁₁	1	2	yes	—
239	C₂×C₁₆	—	22	1	no	—
240	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	complemented
241	#108	—	11	2	no	—
242	#108	—	22	1	no	—
243	#107	—	22	1	no	—
244	#106	—	11	2	no	—
245	#104	—	22	4	no	—
246	#102	—	22	4	no	—
247	#97	—	11	2	no	—
248	C₄×C₈	—	11	2	no	—
249	C₂×C₁₁	—	2	1	no	—
250	C₂×C₁₁	—	2	1	no	—
251	C₂×C₁₁	—	2	1	no	—
252	C₂×C₁₁	—	8	4	no	—
253	C₂×C₁₁	—	8	2	no	—
254	C₂×C₁₁	C₂×#442	1	1	yes	characteristic
255	C₂⁴	—	4	1	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	44	4	no	—
271	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂²×Dic₁₁	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	2	no	—
280	D₈	C₂²×Dic₁₁	1	2	yes	—
281	D₈	C₂²×Dic₁₁	1	4	yes	—
282	M₁₆	—	22	1	no	—
283	M₁₆	—	22	1	no	—
284	M₁₆	—	22	2	no	—
285	M₁₆	—	44	4	no	—
286	M₁₆	—	44	1	no	—
287	C₂×C₈	—	11	2	no	—
288	C₂×C₈	—	44	4	no	—
289	C₂×C₈	—	44	1	no	—
290	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
291	C₂×C₈	#4341	1	1	yes	characteristic
292	C₂×C₈	#4341	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
296	C₄⋊C₄	—	22	1	no	—
297	C₄⋊C₄	—	22	2	no	—
298	C₂²⋊C₄	—	22	1	no	—
299	C₂²⋊C₄	—	44	4	no	—
300	C₂²⋊C₄	—	44	1	no	—
301	C₄²	—	11	2	no	—
302	C₄²	—	44	1	no	—
303	C₁₆	—	22	2	no	—
304	C₁₆	C₂²×Dic₁₁	1	4	yes	—
305	C₁₁	#33471	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	#66964	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	22	2	no	—
326	C₂×C₄	—	44	2	no	—
327	C₂×C₄	—	176	4	no	—
328	C₂×C₄	#66964	1	1	yes	characteristic
329	C₂×C₄	#66964	1	1	yes	characteristic
330	C₂×C₄	#66964	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
332	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
334	C₈	—	22	2	no	—
335	C₈	—	44	2	no	—
336	C₈	D₄×Dic₁₁	1	1	yes	characteristic
337	C₈	D₄×Dic₁₁	1	1	yes	characteristic
338	C₈	C₂×#4341	1	1	yes	characteristic
339	C₈	C₂³×Dic₁₁	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	16	1	no	—
347	C₂²	#10622765	1	1	yes	characteristic
348	C₂²	#10622765	1	1	yes	characteristic
349	C₂²	C₂×#66964	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	44	2	no	—
352	C₄	—	88	2	no	—
353	C₄	#10622765	1	1	yes	characteristic
354	C₄	#10622765	1	1	yes	characteristic
355	C₄	C₂×#66964	1	1	yes	characteristic
356	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#10622765	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#10622765
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting, prime radical	C₂×C₁₁×#509	C₂
socle	C₂×C₁₁	C₂×#442
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#509	D₁₁
hypocenter	C₁₁	#33471
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#509

Complements

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

N	H ≅ G/N	type
1. #13077533	C₂	semidirect
2. #13077468	C₂	semidirect
3. #13070515	C₂	semidirect
4. #13070389	C₂	semidirect
5. C₂×#10622201	C₂	semidirect
6. #13070352	C₂	semidirect
7. #13070220	C₂	semidirect
13. C₁₁×#509	C₄	semidirect
15. C₂×C₁₁×SD₃₂	C₄	semidirect
16. C₂×C₁₁×D₁₆	C₄	semidirect
20. #10622810	C₂²	semidirect
22. #10622765	C₂²	semidirect
28. #10622204	C₂²	semidirect
29. #10622201	C₂²	semidirect
30. #10622151	C₂²	semidirect
31. #10622150	C₂²	semidirect
36. C₁₁×#135	C₂×C₄	semidirect
38. C₂×C₁₁×Q₁₆	C₂×C₄	semidirect
45. C₂×C₁₁×D₈	C₂×C₄	semidirect
58. C₁₁×SD₃₂	C₂×C₄	semidirect
59. C₁₁×D₁₆	C₂×C₄	semidirect
60. C₁₁×M₃₂	C₂×C₄	semidirect
62. C₂×C₁₁×C₁₆	C₂×C₄	semidirect
171. #509	Dic₁₁	semidirect
173. C₂×SD₃₂	Dic₁₁	semidirect
174. C₂×D₁₆	Dic₁₁	semidirect
211. #135	C₂×Dic₁₁	semidirect
213. C₂×Q₁₆	C₂×Dic₁₁	semidirect
220. C₂×D₈	C₂×Dic₁₁	semidirect
233. SD₃₂	C₂×Dic₁₁	semidirect
234. D₁₆	C₂×Dic₁₁	semidirect
237. M₃₂	C₂×Dic₁₁	semidirect
240. C₂×C₁₆	C₂×Dic₁₁	semidirect
305. C₁₁	#33471	semidirect

Maximal subgroups

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

subgroup	index
#13077533	2
#13077468	2
#13070515	2
#13070389	2
C₂×#10622201	2
#13070352	2
#13070220	2
C₂×C₁₁×#509	2
#33471	11