Subgroups — #14757196

Order 2816.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66883	C₂	1	1	yes	maximal, characteristic
2	#13077180	C₂	1	2	yes	maximal
3	C₂×#10622714	C₂	1	1	yes	maximal, characteristic
4	#13069370	C₂	1	4	yes	maximal, complemented
5	#13069365	C₂	1	4	yes	maximal, complemented
6	#13068325	C₂	1	2	yes	maximal
7	C₂×C₁₁×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	2	yes	—
10	C₂×#66979	C₂²	1	1	yes	characteristic
11	C₂²×#4318	C₂²	1	2	yes	—
12	C₂²×#4318	C₂²	1	1	yes	characteristic
13	C₁₁×#427	C₄	1	2	yes	complemented
14	C₁₁×#427	C₄	1	2	yes	complemented
15	C₁₁×#427	C₂²	1	2	yes	—
16	C₁₁×#427	C₂²	1	2	yes	—
17	C₂×C₁₁×#107	C₄	1	2	yes	complemented
18	C₂×C₁₁×#106	C₄	1	2	yes	complemented
19	C₂×#66883	—	2	4	no	—
20	C₂×#66883	C₂²	1	2	yes	—
21	C₂×#66883	C₂²	1	2	yes	—
22	#10622714	—	2	4	no	—
23	C₂×#66879	C₂²	1	2	yes	—
24	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
25	#10622612	—	2	4	no	—
26	#10622612	C₂²	1	8	yes	—
27	#10622537	—	2	4	no	—
28	#10622170	C₂²	1	8	yes	—
29	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	D₄	1	4	yes	—
35	C₂²×C₈×C₁₁	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₁₁×#117	—	2	2	no	—
39	C₁₁×#117	—	2	2	no	—
40	C₁₁×#117	C₂×C₄	1	4	yes	—
41	C₁₁×#117	C₂³	1	4	yes	—
42	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₄²×C₁₁	—	2	2	no	—
45	C₂×C₂²⋊Dic₁₁	—	2	1	no	—
46	#66979	—	2	2	no	—
47	#66979	Q₈	1	4	yes	—
48	C₂×C₄⋊Dic₁₁	—	2	1	no	—
49	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
50	C₂×#4318	—	2	2	no	—
51	C₂×#4318	—	2	2	no	—
52	C₂×#4318	—	2	2	no	—
53	C₂×#4318	—	2	4	no	—
54	C₂×#4318	—	2	4	no	—
55	C₂×#4318	D₄	1	2	yes	—
56	C₂×#4318	D₄	1	2	yes	—
57	C₂×#4318	C₂³	1	4	yes	—
58	C₂²×C₁₁⋊C₈	—	2	2	no	—
59	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
60	C₁₁×#107	C₂×C₄	1	4	yes	—
61	C₁₁×#107	C₂×C₄	1	4	yes	—
62	C₁₁×#106	C₂×C₄	1	4	yes	—
63	C₁₁×#106	C₂×C₄	1	4	yes	—
64	#66883	—	2	4	no	—
65	#66883	—	2	4	no	—
66	#66879	—	2	4	no	—
67	C₈×Dic₁₁	—	2	4	no	—
68	C₄⋊(C₁₁⋊C₈)	—	2	16	no	—
69	#21163	—	11	1	no	maximal, Sylow
70	C₂⁴×C₁₁	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	—	2	4	no	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
80	C₂²×C₄×C₁₁	#41	1	2	yes	—
81	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
85	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
86	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
95	C₁₁×C₄⋊C₄	—	2	4	no	—
96	C₁₁×C₄⋊C₄	C₂²×C₄	1	8	yes	—
97	C₁₁×C₂²⋊C₄	—	2	4	no	—
98	C₁₁×C₂²⋊C₄	—	2	4	no	—
99	C₄²×C₁₁	—	2	4	no	—
100	C₄²×C₁₁	—	2	4	no	—
101	C₂²⋊Dic₁₁	—	2	4	no	—
102	C₄⋊Dic₁₁	—	2	4	no	—
103	C₄×Dic₁₁	—	2	2	no	—
104	C₄×Dic₁₁	C₂×Q₈	1	4	yes	—
105	#4318	—	2	4	no	—
106	#4318	—	2	4	no	—
107	#4318	—	2	4	no	—
108	#4318	—	2	4	no	—
109	#4318	—	4	4	no	—
110	C₂×C₁₁⋊C₈	—	2	2	no	—
111	C₂×C₁₁⋊C₈	—	2	4	no	—
112	C₂×C₁₁⋊C₈	—	2	2	no	—
113	C₂×C₁₁⋊C₈	—	4	4	no	—
114	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
115	C₂²×#108	—	11	1	no	—
116	C₂×#427	D₁₁	1	1	yes	characteristic
117	#2905	—	11	2	no	—
118	C₂×#405	—	11	1	no	—
119	#1946	—	11	4	no	—
120	#1945	—	11	4	no	—
121	#1733	—	11	2	no	—
122	C₂³×C₁₁	—	4	2	no	—
123	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
125	C₂³×C₁₁	C₂²⋊Q₈	1	2	yes	—
126	C₂³×C₁₁	#125	1	2	yes	—
127	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₁	—	2	8	no	—
129	C₂×C₄×C₁₁	—	8	4	no	—
130	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₁	#121	1	2	yes	—
137	C₂×C₄×C₁₁	#121	1	2	yes	—
138	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₁	#125	1	2	yes	—
140	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
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	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₁₁	—	2	4	no	—
149	C₂×Dic₁₁	—	4	4	no	—
150	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
151	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
152	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
153	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
154	C₁₁⋊C₈	—	2	4	no	—
155	C₁₁⋊C₈	—	4	4	no	—
156	C₂²×M₁₆	—	11	1	no	—
157	C₂²×M₁₆	—	11	2	no	—
158	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
159	C₂×#117	—	11	1	no	—
160	C₂×#117	C₂×D₁₁	1	2	yes	—
161	C₂×#108	—	11	2	no	—
162	C₂×#108	—	11	2	no	—
163	C₂×#108	—	22	4	no	—
164	#427	Dic₁₁	1	2	yes	complemented
165	#427	Dic₁₁	1	2	yes	complemented
166	#427	C₂×D₁₁	1	2	yes	—
167	#427	C₂×D₁₁	1	2	yes	—
168	C₂×#107	Dic₁₁	1	2	yes	complemented
169	C₂×#106	Dic₁₁	1	2	yes	complemented
170	#424	—	11	8	no	—
171	#424	—	22	4	no	—
172	#423	—	22	4	no	—
173	#405	—	22	4	no	—
174	C₂×#97	—	11	2	no	—
175	C₂×C₄×C₈	—	11	2	no	—
176	#339	—	11	8	no	—
177	C₂²×C₁₁	—	2	2	no	—
178	C₂²×C₁₁	—	2	2	no	—
179	C₂²×C₁₁	—	2	2	no	—
180	C₂²×C₁₁	—	4	2	no	—
181	C₂²×C₁₁	—	4	2	no	—
182	C₂²×C₁₁	#389	1	2	yes	—
183	C₂²×C₁₁	#389	1	2	yes	—
184	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₂²×C₁₁	C₂×#125	1	1	yes	characteristic
187	C₄×C₁₁	—	4	8	no	—
188	C₄×C₁₁	#389	1	2	yes	—
189	C₄×C₁₁	#389	1	2	yes	—
190	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₄×C₁₁	C₂×#121	1	1	yes	characteristic
193	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
194	Dic₁₁	—	4	4	no	—
195	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
196	C₂×M₁₆	—	11	2	no	—
197	C₂×M₁₆	—	11	2	no	—
198	C₂×M₁₆	—	11	4	no	—
199	C₂×M₁₆	—	22	2	no	—
200	C₂×M₁₆	—	22	2	no	—
201	C₂×M₁₆	—	22	2	no	—
202	C₂×M₁₆	—	22	4	no	—
203	C₂×M₁₆	—	22	4	no	—
204	C₂²×C₈	—	11	2	no	—
205	C₂²×C₈	—	22	2	no	—
206	C₂²×C₈	Dic₂₂	1	4	yes	—
207	C₂²×C₈	D₄₄	1	4	yes	—
208	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
210	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	#117	—	11	4	no	—
217	#117	—	22	2	no	—
218	#117	C₂×Dic₁₁	1	4	yes	—
219	#117	C₂²×D₁₁	1	4	yes	—
220	C₂×C₄⋊C₄	—	22	1	no	—
221	C₂×C₄⋊C₄	C₂×Dic₁₁	1	4	yes	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	22	1	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	—	11	2	no	—
226	#108	—	22	4	no	—
227	#108	—	22	4	no	—
228	#107	C₂×Dic₁₁	1	4	yes	—
229	#107	C₂×Dic₁₁	1	4	yes	—
230	#106	C₂×Dic₁₁	1	4	yes	—
231	#106	C₂×Dic₁₁	1	4	yes	—
232	C₄⋊C₈	—	22	16	no	—
233	#97	—	22	4	no	—
234	C₄×C₈	—	22	4	no	—
235	C₂×C₁₁	—	2	1	no	—
236	C₂×C₁₁	—	2	1	no	—
237	C₂×C₁₁	—	2	2	no	—
238	C₂×C₁₁	—	2	2	no	—
239	C₂×C₁₁	C₂×#389	1	1	yes	characteristic
240	C₂×C₁₁	#2957	1	2	yes	—
241	C₂⁴	#4341	1	1	yes	characteristic
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	22	2	no	—
244	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
245	C₂²×C₄	C₂×D₄₄	1	2	yes	—
246	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
248	C₂²×C₄	#4341	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₁₁	1	2	yes	—
250	C₂²×C₄	#4343	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
254	M₁₆	—	22	4	no	—
255	M₁₆	—	22	4	no	—
256	M₁₆	—	22	4	no	—
257	M₁₆	—	22	4	no	—
258	M₁₆	—	44	4	no	—
259	C₂×C₈	—	11	4	no	—
260	C₂×C₈	—	22	2	no	—
261	C₂×C₈	—	22	2	no	—
262	C₂×C₈	—	22	4	no	—
263	C₂×C₈	—	44	4	no	—
264	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
265	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
266	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
267	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
268	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
269	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
270	C₂×C₈	C₂×D₄₄	1	2	yes	—
271	C₂×C₈	C₂×D₄₄	1	2	yes	—
272	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	22	4	no	—
278	C₄⋊C₄	C₂²×Dic₁₁	1	8	yes	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	22	4	no	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	11	4	no	—
285	C₄²	—	22	2	no	—
286	C₁₁	#21163	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	4	2	no	—
288	C₂³	#66943	1	2	yes	—
289	C₂³	#66950	1	2	yes	—
290	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
291	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
292	C₂³	C₂×#4341	1	1	yes	characteristic
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	22	4	no	—
296	C₂×C₄	—	44	4	no	—
297	C₂×C₄	#66941	1	2	yes	—
298	C₂×C₄	#66941	1	2	yes	—
299	C₂×C₄	#66943	1	2	yes	—
300	C₂×C₄	#66948	1	2	yes	—
301	C₂×C₄	#66948	1	2	yes	—
302	C₂×C₄	#66950	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
305	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
307	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
308	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
309	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
312	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
313	C₂×C₄	C₂×#4343	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
315	C₈	—	22	4	no	—
316	C₈	—	44	4	no	—
317	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
318	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
319	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
320	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#10622598	1	2	yes	—
327	C₂²	#10622598	1	2	yes	—
328	C₂²	C₂×#66943	1	1	yes	characteristic
329	C₂²	C₂×#66950	1	1	yes	characteristic
330	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
331	C₄	—	4	8	no	—
332	C₄	—	44	4	no	—
333	C₄	#10622598	1	2	yes	—
334	C₄	#10622598	1	2	yes	—
335	C₄	C₂×#66941	1	1	yes	characteristic
336	C₄	C₂×#66948	1	1	yes	characteristic
337	C₄	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
338	C₄	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	2	no	—
343	C₂	C₂×#10622598	1	1	yes	characteristic
344	C₂	#13077533	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×#427	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂⁴×C₁₁	#41
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×#427	D₁₁
hypocenter	C₁₁	#21163
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #13069370	C₂	semidirect
5. #13069365	C₂	semidirect
13. C₁₁×#427	C₄	semidirect
14. C₁₁×#427	C₄	semidirect
17. C₂×C₁₁×#107	C₄	semidirect
18. C₂×C₁₁×#106	C₄	semidirect
164. #427	Dic₁₁	semidirect
165. #427	Dic₁₁	semidirect
168. C₂×#107	Dic₁₁	semidirect
169. C₂×#106	Dic₁₁	semidirect
286. C₁₁	#21163	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66883	2
#13077180	2
C₂×#10622714	2
#13069370	2
#13069365	2
#13068325	2
C₂×C₁₁×#427	2
#21163	11