Subgroups — #15623415

Order 2816.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13082021	C₂	1	2	yes	maximal
2	#13079155	C₂	1	8	yes	maximal, complemented
3	#13079147	C₂	1	8	yes	maximal, complemented
4	#13079140	C₂	1	4	yes	maximal, complemented
5	#13079130	C₂	1	4	yes	maximal, complemented
6	#13077206	C₂	1	2	yes	maximal
7	C₂×#10622719	C₂	1	2	yes	maximal
8	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
12	C₂×#66979	C₂²	1	4	yes	—
13	#10623050	C₂²	1	16	yes	—
14	#10623048	—	2	8	no	—
15	#10623031	—	2	4	no	—
16	#10623014	—	2	4	no	—
17	#10623001	C₂²	1	8	yes	—
18	C₁₁×#405	C₂²	1	4	yes	—
19	C₁₁×#405	C₂²	1	4	yes	—
20	C₂×C₁₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
22	#10622734	C₂²	1	16	yes	—
23	#10622719	C₂²	1	4	yes	—
24	#10622719	C₂²	1	8	yes	—
25	#10622719	C₂²	1	4	yes	—
26	#10622719	C₂²	1	4	yes	—
27	#10622719	C₂²	1	4	yes	—
28	C₂×#66882	C₂²	1	4	yes	—
29	C₂×#66881	C₂²	1	4	yes	—
30	#10622315	C₂²	1	32	yes	complemented
31	#10622296	C₂²	1	8	yes	complemented
32	#10622296	C₂²	1	8	yes	complemented
33	#10622295	C₂²	1	8	yes	complemented
34	#10622293	C₂²	1	8	yes	complemented
35	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₁₁×M₁₆	Q₈	1	8	yes	complemented
37	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₁	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
44	C₁₁×#117	D₄	1	4	yes	—
45	C₁₁×#117	C₂³	1	4	yes	—
46	C₂×C₁₁×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
48	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊Dic₁₁	—	2	4	no	—
50	#66979	—	2	4	no	—
51	#66979	—	2	4	no	—
52	#66979	C₂³	1	8	yes	—
53	#66979	C₂³	1	8	yes	—
54	C₂×C₄⋊Dic₁₁	C₂³	1	8	yes	—
55	#66977	—	2	16	no	—
56	C₂×C₄×Dic₁₁	—	2	4	no	—
57	#66943	—	2	16	no	—
58	#66941	—	2	16	no	—
59	#66931	—	2	32	no	—
60	#66923	C₂³	1	16	yes	—
61	#66922	C₂³	1	16	yes	—
62	C₄×Dic₂₂	—	2	16	no	—
63	C₁₁×#97	C₂³	1	8	yes	—
64	C₄×C₈×C₁₁	C₂³	1	4	yes	—
65	C₄×C₈×C₁₁	C₂³	1	4	yes	—
66	#66882	C₂³	1	16	yes	—
67	#66882	C₂³	1	8	yes	—
68	#66882	C₂³	1	8	yes	—
69	#66881	C₂³	1	16	yes	—
70	#66881	C₂³	1	8	yes	—
71	#66881	C₂³	1	8	yes	—
72	#60660	—	11	1	no	maximal, Sylow
73	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₁₁	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₁	—	2	8	no	—
82	C₂×Dic₂₂	—	2	16	no	—
83	C₁₁×M₁₆	C₂×Q₈	1	16	yes	—
84	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₁₁	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₁₁	C₂×Q₈	1	8	yes	—
88	C₂×C₈×C₁₁	C₂×Q₈	1	8	yes	—
89	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
91	C₂×C₈×C₁₁	C₂⁴	1	2	yes	—
92	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₁₁×C₄⋊C₄	C₂×D₄	1	8	yes	—
94	C₁₁×C₂²⋊C₄	C₂×D₄	1	8	yes	—
95	C₄²×C₁₁	C₂×D₄	1	4	yes	—
96	C₄²×C₁₁	C₂⁴	1	4	yes	—
97	C₂²⋊Dic₁₁	—	2	8	no	—
98	C₂²⋊Dic₁₁	—	2	8	no	—
99	C₄⋊Dic₁₁	—	2	16	no	—
100	C₄⋊Dic₁₁	—	2	8	no	—
101	C₄⋊Dic₁₁	C₂⁴	1	16	yes	—
102	Dic₁₁⋊C₄	—	2	32	no	—
103	C₄×Dic₁₁	—	2	8	no	—
104	C₄×Dic₁₁	—	2	8	no	—
105	#3458	—	11	2	no	—
106	#3164	—	11	8	no	—
107	#3163	—	11	8	no	—
108	#3162	—	11	4	no	—
109	#3160	—	11	4	no	—
110	#2911	—	11	2	no	—
111	C₂×#427	—	11	2	no	—
112	C₂×#405	D₁₁	1	1	yes	characteristic
113	C₂³×C₁₁	—	4	2	no	—
114	C₂³×C₁₁	C₄⋊Q₈	1	4	yes	—
115	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
116	C₂³×C₁₁	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₁₁	—	4	4	no	—
118	C₂×C₄×C₁₁	C₄⋊Q₈	1	8	yes	—
119	C₂×C₄×C₁₁	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
121	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²×Q₈	1	4	yes	—
124	C₂×C₄×C₁₁	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₁₁	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₁₁	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
128	C₂×Dic₁₁	—	2	16	no	—
129	C₂×Dic₁₁	—	8	8	no	—
130	Dic₂₂	—	8	16	no	—
131	C₈×C₁₁	C₂²×Q₈	1	2	yes	—
132	C₈×C₁₁	C₂²×Q₈	1	4	yes	—
133	C₈×C₁₁	C₂²×Q₈	1	2	yes	—
134	C₈×C₁₁	C₂²×Q₈	1	8	yes	—
135	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
137	#544	—	11	16	no	—
138	#544	—	22	4	no	—
139	#543	—	22	4	no	—
140	#533	—	11	8	no	—
141	#533	—	22	8	no	—
142	C₂×#117	—	11	4	no	—
143	C₂×#117	C₂×D₁₁	1	1	yes	characteristic
144	#501	—	11	32	no	—
145	#500	—	11	8	no	—
146	#499	—	11	8	no	—
147	#499	—	11	8	no	—
148	#498	—	11	8	no	—
149	#428	—	11	16	no	—
150	#427	—	11	4	no	—
151	#427	—	11	8	no	—
152	#427	—	11	4	no	—
153	#427	—	11	4	no	—
154	#427	—	11	4	no	—
155	C₂×#107	—	11	4	no	—
156	C₂×#106	—	11	4	no	—
157	#405	C₂×D₁₁	1	4	yes	—
158	#405	C₂×D₁₁	1	4	yes	—
159	C₂×#97	C₂×D₁₁	1	2	yes	—
160	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
161	C₂²×C₁₁	—	2	2	no	—
162	C₂²×C₁₁	—	2	4	no	—
163	C₂²×C₁₁	—	4	4	no	—
164	C₂²×C₁₁	C₂×C₄⋊Q₈	1	2	yes	—
165	C₂²×C₁₁	C₂×C₄⋊Q₈	1	4	yes	—
166	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
167	C₄×C₁₁	—	2	8	no	—
168	C₄×C₁₁	C₂×C₄⋊Q₈	1	4	yes	—
169	C₄×C₁₁	C₂×C₄⋊Q₈	1	2	yes	—
170	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
171	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
172	Dic₁₁	—	4	16	no	—
173	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
174	C₂×M₁₆	Dic₂₂	1	8	yes	complemented
175	C₂×M₁₆	C₂²×D₁₁	1	4	yes	—
176	C₂²×C₈	Dic₂₂	1	8	yes	complemented
177	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
180	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
181	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
182	C₄⋊Q₈	—	11	16	no	—
183	C₄⋊Q₈	—	22	16	no	—
184	#125	—	11	16	no	—
185	#125	—	22	16	no	—
186	C₂²⋊Q₈	—	22	16	no	—
187	C₂²⋊Q₈	—	22	32	no	—
188	C₄×Q₈	—	22	16	no	—
189	#117	—	11	8	no	—
190	#117	—	11	8	no	—
191	#117	—	22	4	no	—
192	#117	—	22	4	no	—
193	#117	D₄₄	1	4	yes	—
194	#117	C₂²×D₁₁	1	4	yes	—
195	C₂×C₄⋊C₄	—	11	8	no	—
196	C₂×C₄⋊C₄	D₄₄	1	2	yes	—
197	C₂×C₂²⋊C₄	—	22	4	no	—
198	C₂×C₂²⋊C₄	D₄₄	1	2	yes	—
199	C₂×C₄²	—	22	4	no	—
200	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
201	#107	—	11	8	no	—
202	#107	—	11	8	no	—
203	#107	—	11	16	no	—
204	#106	—	11	8	no	—
205	#106	—	11	8	no	—
206	#106	—	11	16	no	—
207	#97	C₂²×D₁₁	1	8	yes	—
208	C₄×C₈	C₂²×D₁₁	1	4	yes	—
209	C₄×C₈	C₂²×D₁₁	1	4	yes	—
210	C₂×C₁₁	—	2	2	no	—
211	C₂×C₁₁	—	2	4	no	—
212	C₂×C₁₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
213	C₂×C₁₁	#3584	1	2	yes	—
214	C₂⁴	C₂×D₄₄	1	1	yes	characteristic
215	C₂×Q₈	—	22	16	no	—
216	C₂²×C₄	—	22	8	no	—
217	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
218	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
219	C₂²×C₄	C₂×D₄₄	1	4	yes	—
220	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄₄	1	2	yes	—
222	C₂²×C₄	C₂³×D₁₁	1	2	yes	—
223	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
224	M₁₆	C₂×Dic₂₂	1	16	yes	—
225	C₂×C₈	C₂×Dic₂₂	1	4	yes	—
226	C₂×C₈	C₂×Dic₂₂	1	4	yes	—
227	C₂×C₈	C₂×Dic₂₂	1	8	yes	—
228	C₂×C₈	C₂×Dic₂₂	1	8	yes	—
229	C₂×C₈	C₂×Dic₂₂	1	8	yes	—
230	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₁₁	1	4	yes	—
233	C₂×C₈	C₂³×D₁₁	1	2	yes	—
234	C₄⋊C₄	—	11	16	no	—
235	C₄⋊C₄	—	22	8	no	—
236	C₄⋊C₄	—	22	32	no	—
237	C₄⋊C₄	—	22	16	no	—
238	C₄⋊C₄	C₂×D₄₄	1	8	yes	—
239	C₂²⋊C₄	—	22	8	no	—
240	C₂²⋊C₄	—	22	8	no	—
241	C₂²⋊C₄	C₂×D₄₄	1	8	yes	—
242	C₄²	—	22	8	no	—
243	C₄²	—	22	8	no	—
244	C₄²	C₂×D₄₄	1	4	yes	—
245	C₄²	C₂³×D₁₁	1	4	yes	—
246	C₁₁	#60660	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	#66922	1	4	yes	—
249	C₂³	C₂²×D₄₄	1	1	yes	characteristic
250	C₂³	C₂²×D₄₄	1	2	yes	—
251	Q₈	—	88	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	22	16	no	—
254	C₂×C₄	—	88	8	no	—
255	C₂×C₄	#66922	1	4	yes	—
256	C₂×C₄	#66922	1	8	yes	—
257	C₂×C₄	C₂²×Dic₂₂	1	4	yes	—
258	C₂×C₄	C₂²×Dic₂₂	1	2	yes	—
259	C₂×C₄	C₂²×Dic₂₂	1	2	yes	—
260	C₂×C₄	C₂²×Dic₂₂	1	4	yes	—
261	C₂×C₄	C₂²×D₄₄	1	8	yes	—
262	C₂×C₄	C₂²×D₄₄	1	2	yes	—
263	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
264	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
265	C₈	C₂²×Dic₂₂	1	2	yes	—
266	C₈	C₂²×Dic₂₂	1	2	yes	—
267	C₈	C₂²×Dic₂₂	1	4	yes	—
268	C₈	C₂²×Dic₂₂	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#66922	1	2	yes	—
273	C₂²	C₂×#66922	1	4	yes	—
274	C₂²	C₂³×D₄₄	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	44	16	no	—
277	C₄	C₂×#66922	1	4	yes	—
278	C₄	C₂×#66922	1	2	yes	—
279	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
280	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#66922	1	1	yes	characteristic
284	C₂	#13083390	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂×C₁₁×#405	C₂
socle	C₂²×C₁₁	C₂³×D₄
prime radical	C₂⁴×C₁₁	C₂×D₄
hypercenter	C₂×#405	D₁₁
hypocenter	C₁₁	#60660
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13079155	C₂	semidirect
3. #13079147	C₂	semidirect
4. #13079140	C₂	semidirect
5. #13079130	C₂	semidirect
30. #10622315	C₂²	semidirect
31. #10622296	C₂²	semidirect
32. #10622296	C₂²	semidirect
33. #10622295	C₂²	semidirect
34. #10622293	C₂²	semidirect
36. C₂×C₁₁×M₁₆	Q₈	semidirect
38. C₂²×C₈×C₁₁	Q₈	semidirect
174. C₂×M₁₆	Dic₂₂	semidirect
176. C₂²×C₈	Dic₂₂	semidirect
246. C₁₁	#60660	semidirect

Maximal subgroups

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

subgroup	index
#13082021	2
#13079155	2
#13079147	2
#13079140	2
#13079130	2
#13077206	2
C₂×#10622719	2
C₂×C₁₁×#405	2
#60660	11