Subgroups — #14767651

Order 2816.

846 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	#13079044	C₂	1	1	yes	maximal, characteristic
2	#13077329	C₂	1	2	yes	maximal, complemented
3	#13069444	C₂	1	2	yes	maximal, complemented
4	#13068942	C₂	1	8	yes	maximal, complemented
5	#13068754	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#66957	C₂²	1	4	yes	—
8	C₂×#66954	C₂²	1	4	yes	—
9	#10623008	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₄³×C₁₁	C₂²	1	1	yes	characteristic
13	#10622723	C₂²	1	4	yes	—
14	C₂×#66869	C₂²	1	8	yes	—
15	#10622537	C₂²	1	2	yes	—
16	#10622312	C₄	1	1	yes	characteristic, complemented
17	#10622309	C₄	1	4	yes	complemented
18	#10622308	C₄	1	1	yes	characteristic, complemented
19	#10622306	C₄	1	2	yes	complemented
20	#10622301	—	2	2	no	—
21	#10622301	—	2	2	no	—
22	C₈×D₄₄	—	2	2	no	—
23	C₈×D₄₄	—	2	2	no	—
24	C₁₁×#336	C₂²	1	4	yes	—
25	#10622190	C₂²	1	4	yes	—
26	C₂×#4339	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
29	C₂×C₄²×C₁₁	—	2	1	no	—
30	C₂×C₄²×C₁₁	—	2	1	no	—
31	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	#66983	—	2	2	no	—
34	#66983	—	2	2	no	—
35	C₂×#4318	C₂³	1	4	yes	—
36	C₂×Dic₄₄	C₂×C₄	1	4	yes	—
37	#66957	—	4	8	no	—
38	C₂×D₈₈	C₂×C₄	1	4	yes	—
39	C₂×#4314	C₂×C₄	1	8	yes	—
40	#66954	—	4	8	no	—
41	C₂×#4313	—	2	4	no	—
42	C₂×C₈×D₁₁	—	2	4	no	—
43	#66928	C₂×C₄	1	4	yes	—
44	#66927	C₂×C₄	1	2	yes	—
45	C₄×D₄₄	—	2	2	no	—
46	C₄×D₄₄	—	2	2	no	—
47	#66922	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁	D₄	1	2	yes	—
53	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
54	(C₂×D₁₁)⋊C₈	—	2	4	no	—
55	(C₂×D₁₁)⋊C₈	—	2	4	no	—
56	#66869	—	4	8	no	—
57	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
59	#22280	—	11	1	no	maximal, Sylow
60	C₂²×C₄×C₁₁	—	2	2	no	—
61	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₁	—	2	4	no	—
65	#4339	—	2	8	no	—
66	#4339	—	2	8	no	—
67	C₂×D₄₄	—	2	2	no	—
68	C₂×D₄₄	—	2	2	no	—
69	C₂×D₄₄	C₂²×C₄	1	4	yes	—
70	C₂×C₄×D₁₁	—	2	4	no	—
71	C₂×Dic₂₂	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₁	—	2	4	no	—
73	C₂×C₈×C₁₁	—	2	4	no	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
78	C₄²×C₁₁	—	2	4	no	—
79	C₄²×C₁₁	—	2	4	no	—
80	C₄²×C₁₁	—	4	2	no	—
81	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
85	(C₂×D₁₁)⋊C₄	—	2	4	no	—
86	(C₂×D₁₁)⋊C₄	—	2	4	no	—
87	C₄⋊Dic₁₁	—	2	2	no	—
88	C₄⋊Dic₁₁	—	2	2	no	—
89	#4318	—	2	8	no	—
90	C₂×C₁₁⋊C₈	—	2	4	no	—
91	Dic₄₄	—	4	4	no	—
92	D₈₈	—	4	4	no	—
93	#4314	—	4	8	no	—
94	#4313	—	4	8	no	—
95	C₈×D₁₁	—	4	8	no	—
96	#3147	—	11	1	no	—
97	#2931	—	11	2	no	—
98	#1962	—	11	2	no	—
99	#1876	—	11	8	no	—
100	#1837	—	11	1	no	—
101	#1744	D₁₁	1	1	yes	characteristic, complemented
102	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
103	C₂²×D₁₁	—	2	4	no	—
104	C₂×C₄×C₁₁	—	2	2	no	—
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	4	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	—	4	2	no	—
109	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	H(5,𝔽₂)	1	4	yes	—
116	C₂²⋊D₁₁	—	4	8	no	—
117	C₂×Dic₁₁	—	2	4	no	—
118	D₄₄	—	2	8	no	—
119	D₄₄	—	4	4	no	—
120	D₄₄	—	4	4	no	—
121	C₄×D₁₁	—	4	4	no	—
122	C₄×D₁₁	—	4	4	no	—
123	Dic₂₂	—	2	8	no	—
124	C₈×C₁₁	—	2	2	no	—
125	C₈×C₁₁	—	2	2	no	—
126	C₈×C₁₁	—	2	2	no	—
127	C₈×C₁₁	—	2	2	no	—
128	C₁₁⋊C₈	—	4	4	no	—
129	C₂×#135	—	11	4	no	—
130	C₂×#131	—	11	4	no	—
131	#532	—	11	2	no	—
132	#495	—	11	4	no	—
133	#494	—	11	1	no	—
134	#493	—	11	1	no	—
135	#492	—	11	2	no	—
136	#436	—	22	2	no	—
137	#436	—	22	2	no	—
138	C₈×D₄	—	22	2	no	—
139	C₈×D₄	—	22	2	no	—
140	#423	—	11	2	no	—
141	C₂×#104	—	11	8	no	—
142	#408	—	11	4	no	—
143	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
145	C₄³	C₂×D₁₁	1	1	yes	characteristic
146	#341	—	11	4	no	—
147	#336	C₂×D₁₁	1	4	yes	—
148	C₂²×C₁₁	—	2	1	no	—
149	C₂²×C₁₁	—	2	1	no	—
150	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₁₁	#540	1	2	yes	—
152	C₂×D₁₁	—	4	4	no	—
153	C₂×D₁₁	—	4	4	no	—
154	C₂×D₁₁	—	4	4	no	—
155	C₄×C₁₁	—	2	1	no	—
156	C₄×C₁₁	—	2	1	no	—
157	C₄×C₁₁	—	4	4	no	—
158	C₄×C₁₁	#443	1	2	yes	—
159	C₄×C₁₁	#444	1	2	yes	—
160	C₄×C₁₁	#518	1	2	yes	—
161	C₄×C₁₁	#535	1	2	yes	—
162	Dic₁₁	—	4	4	no	—
163	C₂×#41	—	11	4	no	—
164	#135	—	44	8	no	—
165	C₂×Q₁₆	—	11	4	no	—
166	C₂×SD₁₆	—	11	8	no	—
167	C₂×D₈	—	11	4	no	—
168	#131	—	44	8	no	—
169	C₂×M₁₆	—	11	4	no	—
170	C₂×M₁₆	—	22	4	no	—
171	C₂²×C₈	—	22	4	no	—
172	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
173	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
174	C₄⋊Q₈	—	11	2	no	—
175	#127	—	11	2	no	—
176	#124	—	11	4	no	—
177	#121	—	22	2	no	—
178	#121	—	22	2	no	—
179	C₄×D₄	—	22	2	no	—
180	C₄×D₄	—	22	2	no	—
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
184	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
185	C₄⋊C₈	—	22	2	no	—
186	C₄⋊C₈	—	22	2	no	—
187	#104	—	44	8	no	—
188	C₂²⋊C₈	—	22	4	no	—
189	C₂²⋊C₈	—	22	4	no	—
190	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
191	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
192	C₄×C₈	C₄×D₁₁	1	2	yes	—
193	C₄×C₈	D₄₄	1	1	yes	characteristic
194	C₄×C₈	D₄₄	1	2	yes	—
195	C₄×C₈	D₄₄	1	1	yes	characteristic
196	C₂×C₁₁	—	2	2	no	—
197	C₂×C₁₁	#2328	1	1	yes	characteristic
198	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
199	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
200	D₁₁	—	8	4	no	—
201	#41	—	22	8	no	—
202	#41	—	22	8	no	—
203	C₂×Q₈	—	11	4	no	—
204	C₂×D₄	—	11	4	no	—
205	C₂×D₄	—	22	4	no	—
206	C₂×D₄	—	22	2	no	—
207	C₂×D₄	—	22	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	22	4	no	—
210	C₂²×C₄	#4339	1	1	yes	characteristic
211	C₂²×C₄	#4339	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
213	Q₁₆	—	44	4	no	—
214	SD₁₆	—	44	8	no	—
215	D₈	—	44	4	no	—
216	M₁₆	—	22	8	no	—
217	M₁₆	—	44	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	22	4	no	—
221	C₂×C₈	—	44	8	no	—
222	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
224	C₂×C₈	C₂×D₄₄	1	2	yes	—
225	C₂×C₈	C₂×D₄₄	1	2	yes	—
226	C₄⋊C₄	—	22	2	no	—
227	C₄⋊C₄	—	22	2	no	—
228	C₂²⋊C₄	—	22	4	no	—
229	C₂²⋊C₄	—	22	4	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	4	2	no	—
233	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
234	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
235	C₄²	C₂×D₄₄	1	1	yes	characteristic
236	C₄²	C₂×D₄₄	1	1	yes	characteristic
237	C₁₁	#22280	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	22	4	no	—
239	C₂³	C₂×#4339	1	1	yes	characteristic
240	Q₈	—	22	8	no	—
241	D₄	—	22	8	no	—
242	D₄	—	44	8	no	—
243	D₄	—	44	4	no	—
244	D₄	—	44	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	22	4	no	—
251	C₂×C₄	—	44	4	no	—
252	C₂×C₄	—	44	4	no	—
253	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
259	C₂×C₄	#67042	1	4	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	44	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	44	4	no	—
268	C₂²	—	44	4	no	—
269	C₂²	—	44	4	no	—
270	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
271	C₂²	#10623066	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	44	4	no	—
276	C₄	#10622313	1	2	yes	—
277	C₄	#10622330	1	2	yes	—
278	C₄	#10623052	1	2	yes	—
279	C₄	#10623056	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	88	4	no	—
282	C₂	#13071341	1	1	yes	characteristic
283	C₂	C₂×#10622313	1	1	yes	characteristic
284	C₂	C₂×#10622330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#1744	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#13079044	C₂
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#1744	D₁₁
hypocenter	C₁₁	#22280
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
2. #13077329	C₂	semidirect
3. #13069444	C₂	semidirect
4. #13068942	C₂	semidirect
5. #13068754	C₂	semidirect
6. C₁₁×#1744	C₂	semidirect
16. #10622312	C₄	semidirect
17. #10622309	C₄	semidirect
18. #10622308	C₄	semidirect
19. #10622306	C₄	semidirect
101. #1744	D₁₁	semidirect
237. C₁₁	#22280	semidirect

Maximal subgroups

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

subgroup	index
#13079044	2
#13077329	2
#13069444	2
#13068942	2
#13068754	2
C₁₁×#1744	2
#22280	11