Subgroups — #14909684

Order 2816.

636 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	#13079195	C₂	1	1	yes	maximal, characteristic, complemented
2	#13070873	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070865	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070843	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070839	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070668	C₂	1	4	yes	maximal, complemented
7	#13070539	C₂	1	4	yes	maximal, complemented
8	#13070415	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₁×#135	—	2	1	no	—
11	C₂×C₁₁×#135	C₂²	1	2	yes	—
12	C₁₁×#532	—	2	1	no	—
13	#10623008	—	2	1	no	—
14	C₁₁×#495	—	2	1	no	—
15	C₁₁×#495	C₂²	1	2	yes	—
16	C₁₁×#494	C₂²	1	1	yes	characteristic
17	C₁₁×#493	C₂²	1	1	yes	characteristic
18	C₁₁×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66892	C₂²	1	4	yes	—
21	C₂×#66890	C₂²	1	4	yes	—
22	#10622728	—	2	1	no	—
23	#10622728	C₂²	1	2	yes	—
24	#10622308	C₂²	1	1	yes	characteristic
25	#10622307	—	2	1	no	—
26	C₄×D₈₈	—	2	1	no	—
27	C₄×D₈₈	C₂²	1	2	yes	—
28	#10622295	C₂²	1	1	yes	characteristic
29	#10622202	C₂²	1	4	yes	—
30	#10622200	C₂²	1	4	yes	—
31	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
32	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₁×#41	—	2	2	no	—
36	C₂×#4339	—	4	1	no	—
37	C₁₁×#135	—	4	4	no	—
38	C₁₁×#135	—	4	4	no	—
39	C₂×C₁₁×Q₁₆	—	2	1	no	—
40	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₁×SD₁₆	—	2	2	no	—
42	C₂×C₁₁×SD₁₆	—	2	2	no	—
43	C₂×C₁₁×D₈	—	2	1	no	—
44	C₂×C₁₁×D₈	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
47	C₁₁×C₄⋊Q₈	—	2	1	no	—
48	C₁₁×#127	—	2	1	no	—
49	C₁₁×#124	—	2	2	no	—
50	C₁₁×#121	—	4	1	no	—
51	C₁₁×#121	—	4	1	no	—
52	C₄×D₄×C₁₁	—	4	1	no	—
53	C₄×D₄×C₁₁	—	4	1	no	—
54	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
55	#66983	—	2	2	no	—
56	#66983	—	4	1	no	—
57	C₂×D₈₈	—	2	1	no	—
58	C₂×D₈₈	C₂³	1	2	yes	—
59	#66928	—	4	1	no	—
60	#66927	—	2	1	no	—
61	C₄×D₄₄	—	2	2	no	—
62	C₄×D₄₄	—	4	1	no	—
63	#66922	—	2	1	no	—
64	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	#66892	—	4	4	no	—
69	#66890	—	4	4	no	—
70	#66885	—	2	2	no	—
71	#66885	—	2	2	no	—
72	#66882	—	2	1	no	—
73	#66882	C₂³	1	2	yes	—
74	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
75	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
76	#33773	—	11	1	no	maximal, Sylow
77	C₁₁×#41	—	4	4	no	—
78	C₁₁×#41	—	4	4	no	—
79	C₂×Q₈×C₁₁	—	2	2	no	—
80	C₂×D₄×C₁₁	—	2	2	no	—
81	C₂×D₄×C₁₁	—	4	2	no	—
82	C₂×D₄×C₁₁	—	4	1	no	—
83	C₂×D₄×C₁₁	—	4	1	no	—
84	C₂²×C₄×C₁₁	—	4	2	no	—
85	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₁	—	4	2	no	—
89	#4339	—	8	4	no	—
90	C₂×D₄₄	—	2	2	no	—
91	C₂×D₄₄	—	4	1	no	—
92	C₂×D₄₄	—	4	1	no	—
93	C₂×C₄×D₁₁	—	4	2	no	—
94	C₂×Dic₂₂	—	4	1	no	—
95	C₁₁×Q₁₆	—	2	4	no	—
96	C₁₁×Q₁₆	—	4	2	no	—
97	C₁₁×SD₁₆	—	4	4	no	—
98	C₁₁×SD₁₆	—	4	4	no	—
99	C₁₁×D₈	—	2	4	no	—
100	C₁₁×D₈	—	4	2	no	—
101	C₂×C₈×C₁₁	—	2	2	no	—
102	C₂×C₈×C₁₁	—	2	2	no	—
103	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
107	C₁₁×C₄⋊C₄	—	4	1	no	—
108	C₁₁×C₄⋊C₄	—	4	1	no	—
109	C₁₁×C₂²⋊C₄	—	4	2	no	—
110	C₁₁×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₁₁	D₈	1	1	yes	characteristic
112	C₄²×C₁₁	D₈	1	1	yes	characteristic
113	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
115	(C₂×D₁₁)⋊C₄	—	4	2	no	—
116	(C₂×D₁₁)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₁₁	—	2	2	no	—
118	C₄⋊Dic₁₁	—	4	1	no	—
119	D₈₈	—	2	4	no	—
120	D₈₈	—	4	2	no	—
121	C₁₁⋊C₁₆	—	2	2	no	—
122	C₁₁⋊C₁₆	—	2	2	no	—
123	#3169	—	11	1	no	—
124	#3147	D₁₁	1	1	yes	characteristic, complemented
125	#2248	—	11	1	no	—
126	#2246	—	11	1	no	—
127	#2241	—	11	1	no	—
128	#2240	—	11	1	no	—
129	#2210	—	11	4	no	—
130	#2186	—	11	4	no	—
131	#2162	—	11	1	no	—
132	C₂³×C₁₁	—	4	2	no	—
133	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₁₁	—	4	2	no	—
135	Q₈×C₁₁	—	4	2	no	—
136	Q₈×C₁₁	—	4	2	no	—
137	D₄×C₁₁	—	4	2	no	—
138	D₄×C₁₁	—	4	2	no	—
139	D₄×C₁₁	—	8	4	no	—
140	D₄×C₁₁	—	8	2	no	—
141	D₄×C₁₁	—	8	2	no	—
142	C₂×C₄×C₁₁	—	2	1	no	—
143	C₂×C₄×C₁₁	—	2	1	no	—
144	C₂×C₄×C₁₁	—	2	2	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	—	4	2	no	—
147	C₂×C₄×C₁₁	—	8	4	no	—
148	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₁₁	—	8	4	no	—
155	C₂×Dic₁₁	—	4	2	no	—
156	D₄₄	—	4	2	no	—
157	D₄₄	—	4	2	no	—
158	D₄₄	—	8	2	no	—
159	D₄₄	—	8	2	no	—
160	C₄×D₁₁	—	8	4	no	—
161	Dic₂₂	—	8	2	no	—
162	C₈×C₁₁	—	2	1	no	—
163	C₈×C₁₁	—	2	1	no	—
164	C₈×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
166	C₂×#135	—	2	1	no	—
167	C₂×#135	C₂×D₁₁	1	2	yes	—
168	#532	—	2	1	no	—
169	#532	—	22	1	no	—
170	C₂×SD₃₂	—	11	4	no	—
171	C₂×D₁₆	—	11	4	no	—
172	#500	—	11	1	no	—
173	#495	—	2	1	no	—
174	#495	C₂×D₁₁	1	2	yes	—
175	#494	C₂×D₁₁	1	1	yes	characteristic
176	#493	—	11	1	no	—
177	#493	C₂×D₁₁	1	1	yes	characteristic
178	#492	—	2	1	no	—
179	#486	—	22	1	no	—
180	#466	—	11	2	no	—
181	#466	—	22	1	no	—
182	C₄×D₈	—	11	2	no	—
183	C₄×D₈	—	22	1	no	—
184	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
185	C₄⋊C₁₆	—	11	2	no	—
186	#358	—	11	4	no	—
187	#357	—	11	4	no	—
188	C₂²⋊C₁₆	—	11	2	no	—
189	C₄×C₁₆	—	11	1	no	—
190	C₄×C₁₆	—	11	1	no	—
191	C₂²×C₁₁	—	2	1	no	—
192	C₂²×C₁₁	—	2	1	no	—
193	C₂²×C₁₁	—	2	1	no	—
194	C₂²×C₁₁	—	8	2	no	—
195	C₂²×C₁₁	—	8	2	no	—
196	C₂²×C₁₁	—	8	2	no	—
197	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
198	C₂×D₁₁	—	8	2	no	—
199	C₂×D₁₁	—	8	2	no	—
200	C₂×D₁₁	—	8	2	no	—
201	C₄×C₁₁	—	2	1	no	—
202	C₄×C₁₁	—	2	1	no	—
203	C₄×C₁₁	—	2	1	no	—
204	C₄×C₁₁	—	8	2	no	—
205	C₄×C₁₁	#508	1	2	yes	—
206	C₄×C₁₁	#509	1	2	yes	—
207	C₄×C₁₁	#535	1	1	yes	characteristic
208	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	Dic₁₁	—	8	2	no	—
210	C₂×#41	—	2	2	no	—
211	C₂×#41	—	44	1	no	—
212	#135	—	4	4	no	—
213	#135	—	4	4	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	11	2	no	—
220	C₂×D₈	—	22	1	no	—
221	C₂×D₈	C₂²×D₁₁	1	2	yes	—
222	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	22	1	no	—
226	#127	—	2	1	no	—
227	#127	—	22	1	no	—
228	#124	—	2	2	no	—
229	#124	—	44	1	no	—
230	#121	—	4	1	no	—
231	#121	—	4	1	no	—
232	#121	—	22	2	no	—
233	#121	—	44	1	no	—
234	C₄×D₄	—	4	1	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	—	22	2	no	—
237	C₄×D₄	—	44	1	no	—
238	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
239	SD₃₂	—	44	4	no	—
240	D₁₆	—	44	4	no	—
241	C₂×C₁₆	—	11	2	no	—
242	C₂×C₁₆	—	11	2	no	—
243	#107	—	11	2	no	—
244	#107	—	22	1	no	—
245	#102	—	22	2	no	—
246	#102	—	22	2	no	—
247	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
248	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
251	C₂×C₁₁	—	4	1	no	—
252	C₂×C₁₁	—	16	2	no	—
253	C₂×C₁₁	#3205	1	1	yes	characteristic
254	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
255	C₂×C₁₁	C₂×#509	1	1	yes	characteristic
256	D₁₁	—	16	2	no	—
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	#41	—	88	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	44	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	22	2	no	—
267	C₂×D₄	—	44	2	no	—
268	C₂×D₄	—	44	1	no	—
269	C₂×D₄	—	44	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	44	2	no	—
272	C₂²×C₄	#4323	1	1	yes	characteristic
273	C₂²×C₄	#4323	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
275	Q₁₆	—	2	4	no	—
276	Q₁₆	—	4	2	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	2	4	no	—
280	D₈	—	4	2	no	—
281	D₈	—	22	4	no	—
282	D₈	—	44	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	22	2	no	—
292	C₄⋊C₄	—	44	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	44	2	no	—
296	C₂²⋊C₄	—	44	2	no	—
297	C₄²	#4323	1	1	yes	characteristic
298	C₄²	#4323	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
301	C₁₆	—	22	2	no	—
302	C₁₆	—	22	2	no	—
303	C₁₁	#33773	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	4	2	no	—
305	C₂³	—	44	2	no	—
306	C₂³	C₂×#4323	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	88	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	44	2	no	—
316	D₄	—	44	2	no	—
317	D₄	—	88	4	no	—
318	D₄	—	88	2	no	—
319	D₄	—	88	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	44	2	no	—
327	C₂×C₄	—	88	4	no	—
328	C₂×C₄	C₂×#4323	1	1	yes	characteristic
329	C₂×C₄	C₂×#4323	1	1	yes	characteristic
330	C₂×C₄	C₂×#4323	1	1	yes	characteristic
331	C₂×C₄	C₂×#4323	1	1	yes	characteristic
332	C₂×C₄	C₂×#4323	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	#66985	1	2	yes	—
337	C₈	#67037	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	88	2	no	—
345	C₂²	—	88	2	no	—
346	C₂²	—	88	2	no	—
347	C₂²	C₂²×#4323	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	88	2	no	—
353	C₄	#10622806	1	2	yes	—
354	C₄	#10622807	1	2	yes	—
355	C₄	C₂×#66985	1	1	yes	characteristic
356	C₄	#10623336	1	1	yes	characteristic
357	C₂	—	4	1	no	—
358	C₂	—	16	2	no	—
359	C₂	—	176	2	no	—
360	C₂	#13079534	1	1	yes	characteristic
361	C₂	C₂×#10622807	1	1	yes	characteristic
362	C₂	C₂×#10622806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3147	C₂
socle	C₂²×C₁₁	C₂²×D₈
hypercenter	#3147	D₁₁
hypocenter	C₁₁	#33773
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #13079195	C₂	semidirect
2. #13070873	C₂	semidirect
3. #13070865	C₂	semidirect
4. #13070843	C₂	semidirect
5. #13070839	C₂	semidirect
6. #13070668	C₂	semidirect
7. #13070539	C₂	semidirect
8. #13070415	C₂	semidirect
9. C₁₁×#3147	C₂	semidirect
124. #3147	D₁₁	semidirect
303. C₁₁	#33773	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13070873	2
#13070865	2
#13070843	2
#13070839	2
#13070668	2
#13070539	2
#13070415	2
C₁₁×#3147	2
#33773	11