Subgroups — C₂³×#66936 #15660420

Order 2816.

15150 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#66936	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×(C₂×D₁₁)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₁₁	—	2	16	no	—
14	C₂³×C₂²⋊D₁₁	C₂²	1	28	yes	—
15	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₁₁	—	2	16	no	—
19	C₂³×C₄×D₁₁	C₂²	1	28	yes	complemented
20	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	28	yes	complemented
21	C₂²×C₂²⋊Dic₁₁	C₂²	1	28	yes	complemented
22	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	28	yes	complemented
23	C₂²×C₄⋊Dic₁₁	C₂²	1	28	yes	complemented
24	C₂²×Dic₁₁⋊C₄	C₂²	1	28	yes	complemented
25	C₂×#66936	C₂²	1	448	yes	direct product factor
26	C₂⁵×C₁₁	—	2	8	no	—
27	C₂⁵×C₁₁	—	2	8	no	—
28	C₂⁵×C₁₁	—	2	8	no	—
29	C₂⁵×C₁₁	C₂³	1	14	yes	—
30	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₁	—	2	8	no	—
32	C₂⁴×D₁₁	—	2	8	no	—
33	C₂⁴×D₁₁	D₄	1	16	yes	—
34	C₂⁴×D₁₁	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₁	—	2	8	no	—
36	C₂³×C₄×C₁₁	—	2	8	no	—
37	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
38	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
39	C₂²×C₂²⋊D₁₁	—	2	224	no	—
40	C₂²×C₂²⋊D₁₁	C₂³	1	112	yes	—
41	C₂³×Dic₁₁	—	2	8	no	—
42	C₂³×Dic₁₁	—	2	8	no	—
43	C₂³×Dic₁₁	—	2	8	no	—
44	C₂³×Dic₁₁	C₂³	1	14	yes	—
45	C₂³×Dic₁₁	C₂³	1	14	yes	—
46	C₂³×Dic₁₁	C₂³	1	14	yes	—
47	C₂²×C₄×D₁₁	—	2	224	no	—
48	C₂²×C₄×D₁₁	C₂³	1	112	yes	complemented
49	C₂×C₁₁×C₂²⋊C₄	C₂³	1	112	yes	complemented
50	C₂×C₂²⋊Dic₁₁	C₂³	1	112	yes	complemented
51	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	112	yes	complemented
52	C₂×C₄⋊Dic₁₁	C₂³	1	112	yes	complemented
53	C₂×Dic₁₁⋊C₄	C₂³	1	112	yes	complemented
54	#66936	C₂³	1	512	yes	direct product factor
55	C₂³×#123	—	11	1	no	maximal, Sylow
56	C₂⁴×C₁₁	—	2	56	no	—
57	C₂⁴×C₁₁	—	2	56	no	—
58	C₂⁴×C₁₁	—	2	56	no	—
59	C₂⁴×C₁₁	—	4	64	no	—
60	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
61	C₂⁴×C₁₁	#41	1	8	yes	—
62	C₂⁴×C₁₁	#41	1	8	yes	—
63	C₂⁴×C₁₁	C₂⁴	1	28	yes	—
64	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
65	C₂³×D₁₁	—	2	128	no	—
66	C₂³×D₁₁	—	2	56	no	—
67	C₂³×D₁₁	—	2	56	no	—
68	C₂³×D₁₁	C₂×D₄	1	112	yes	—
69	C₂³×D₁₁	C₂⁴	1	28	yes	—
70	C₂²×C₄×C₁₁	—	2	56	no	—
71	C₂²×C₄×C₁₁	—	2	56	no	—
72	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
73	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
74	C₂×C₂²⋊D₁₁	—	2	448	no	—
75	C₂×C₂²⋊D₁₁	C₂⁴	1	64	yes	—
76	C₂²×Dic₁₁	—	2	56	no	—
77	C₂²×Dic₁₁	—	2	56	no	—
78	C₂²×Dic₁₁	—	2	56	no	—
79	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
80	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
81	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
82	C₂×C₄×D₁₁	—	2	448	no	—
83	C₂×C₄×D₁₁	C₂⁴	1	64	yes	complemented
84	C₁₁×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
85	C₂²⋊Dic₁₁	C₂⁴	1	64	yes	complemented
86	(C₂×D₁₁)⋊C₄	C₂⁴	1	64	yes	complemented
87	C₄⋊Dic₁₁	C₂⁴	1	64	yes	complemented
88	Dic₁₁⋊C₄	C₂⁴	1	64	yes	complemented
89	C₂⁴×D₄	—	11	1	no	—
90	C₂⁵×C₄	—	11	1	no	—
91	C₂²×#123	—	11	56	no	—
92	C₂³×C₄⋊C₄	—	11	1	no	—
93	C₂³×C₄⋊C₄	—	11	1	no	—
94	C₂³×C₂²⋊C₄	—	11	1	no	—
95	C₂³×C₂²⋊C₄	—	11	1	no	—
96	C₂³×C₂²⋊C₄	D₁₁	1	1	yes	characteristic, complemented
97	C₂³×C₁₁	—	2	56	no	—
98	C₂³×C₁₁	—	2	56	no	—
99	C₂³×C₁₁	—	2	56	no	—
100	C₂³×C₁₁	—	4	224	no	—
101	C₂³×C₁₁	#123	1	64	yes	complemented
102	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
103	C₂³×C₁₁	C₂×#41	1	28	yes	—
104	C₂³×C₁₁	C₂×#41	1	28	yes	—
105	C₂³×C₁₁	C₂⁵	1	8	yes	—
106	C₂³×C₁₁	C₂⁵	1	7	yes	—
107	C₂²×D₁₁	—	2	448	no	—
108	C₂²×D₁₁	—	2	56	no	—
109	C₂²×D₁₁	—	2	56	no	—
110	C₂²×D₁₁	C₂²×D₄	1	112	yes	—
111	C₂²×D₁₁	C₂⁵	1	8	yes	—
112	C₂×C₄×C₁₁	—	2	56	no	—
113	C₂×C₄×C₁₁	—	2	56	no	—
114	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
115	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
116	C₂²⋊D₁₁	—	2	128	no	—
117	C₂×Dic₁₁	—	2	56	no	—
118	C₂×Dic₁₁	—	2	56	no	—
119	C₂×Dic₁₁	—	2	56	no	—
120	C₂×Dic₁₁	C₂⁵	1	8	yes	—
121	C₂×Dic₁₁	C₂⁵	1	8	yes	—
122	C₂×Dic₁₁	C₂⁵	1	8	yes	—
123	C₄×D₁₁	—	2	128	no	—
124	C₂⁶	—	11	1	no	—
125	C₂⁶	C₂×D₁₁	1	1	yes	characteristic
126	C₂³×D₄	—	11	28	no	—
127	C₂³×D₄	—	22	16	no	—
128	C₂⁴×C₄	—	11	1	no	—
129	C₂⁴×C₄	—	11	1	no	—
130	C₂⁴×C₄	—	11	1	no	—
131	C₂⁴×C₄	—	11	28	no	—
132	C₂⁴×C₄	—	22	16	no	—
133	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
135	C₂×#123	—	11	448	no	—
136	C₂²×C₄⋊C₄	—	11	28	no	—
137	C₂²×C₄⋊C₄	—	11	28	no	—
138	C₂²×C₂²⋊C₄	—	11	28	no	—
139	C₂²×C₂²⋊C₄	—	11	28	no	—
140	C₂²×C₂²⋊C₄	C₂×D₁₁	1	28	yes	complemented
141	C₂²×C₁₁	—	2	8	no	—
142	C₂²×C₁₁	—	2	8	no	—
143	C₂²×C₁₁	—	2	8	no	—
144	C₂²×C₁₁	—	4	112	no	—
145	C₂²×C₁₁	C₂×#123	1	112	yes	complemented
146	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
147	C₂²×C₁₁	C₂²×#41	1	14	yes	—
148	C₂²×C₁₁	C₂²×#41	1	14	yes	—
149	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
150	C₂×D₁₁	—	2	8	no	—
151	C₂×D₁₁	—	2	8	no	—
152	C₂×D₁₁	—	2	224	no	—
153	C₂×D₁₁	C₂³×D₄	1	16	yes	—
154	C₄×C₁₁	—	2	8	no	—
155	C₄×C₁₁	—	2	8	no	—
156	Dic₁₁	—	2	8	no	—
157	Dic₁₁	—	2	8	no	—
158	Dic₁₁	—	2	8	no	—
159	C₂⁵	—	2	8	no	—
160	C₂⁵	—	2	8	no	—
161	C₂⁵	—	2	8	no	—
162	C₂⁵	—	11	16	no	—
163	C₂⁵	—	11	14	no	—
164	C₂⁵	—	22	8	no	—
165	C₂⁵	—	22	8	no	—
166	C₂⁵	C₂²×D₁₁	1	14	yes	—
167	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
168	C₂²×D₄	—	11	112	no	—
169	C₂²×D₄	—	22	224	no	—
170	C₂³×C₄	—	2	8	no	—
171	C₂³×C₄	—	2	8	no	—
172	C₂³×C₄	—	11	14	no	—
173	C₂³×C₄	—	11	14	no	—
174	C₂³×C₄	—	11	14	no	—
175	C₂³×C₄	—	11	112	no	—
176	C₂³×C₄	—	22	224	no	—
177	C₂³×C₄	—	22	8	no	—
178	C₂³×C₄	—	22	8	no	—
179	C₂³×C₄	—	22	8	no	—
180	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
181	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
182	#123	—	11	512	no	—
183	C₂×C₄⋊C₄	—	11	112	no	—
184	C₂×C₄⋊C₄	—	11	112	no	—
185	C₂×C₂²⋊C₄	—	11	112	no	—
186	C₂×C₂²⋊C₄	—	11	112	no	—
187	C₂×C₂²⋊C₄	C₂²×D₁₁	1	112	yes	complemented
188	C₂×C₁₁	—	4	8	no	—
189	C₂×C₁₁	C₂²×#123	1	28	yes	complemented
190	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
191	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
192	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
193	D₁₁	—	2	16	no	—
194	C₂⁴	—	2	56	no	—
195	C₂⁴	—	2	56	no	—
196	C₂⁴	—	2	56	no	—
197	C₂⁴	—	4	64	no	—
198	C₂⁴	—	11	112	no	—
199	C₂⁴	—	11	28	no	—
200	C₂⁴	—	22	128	no	—
201	C₂⁴	—	22	56	no	—
202	C₂⁴	—	22	56	no	—
203	C₂⁴	#4339	1	8	yes	—
204	C₂⁴	D₄×D₁₁	1	8	yes	—
205	C₂⁴	#4341	1	8	yes	—
206	C₂⁴	C₂³×D₁₁	1	28	yes	—
207	C₂⁴	C₂³×D₁₁	1	7	yes	—
208	C₂×D₄	—	11	64	no	—
209	C₂×D₄	—	22	448	no	—
210	C₂²×C₄	—	2	56	no	—
211	C₂²×C₄	—	2	56	no	—
212	C₂²×C₄	—	11	28	no	—
213	C₂²×C₄	—	11	28	no	—
214	C₂²×C₄	—	11	28	no	—
215	C₂²×C₄	—	11	64	no	—
216	C₂²×C₄	—	22	448	no	—
217	C₂²×C₄	—	22	56	no	—
218	C₂²×C₄	—	22	56	no	—
219	C₂²×C₄	—	22	56	no	—
220	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
221	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
222	C₄⋊C₄	—	11	64	no	—
223	C₄⋊C₄	—	11	64	no	—
224	C₂²⋊C₄	—	11	64	no	—
225	C₂²⋊C₄	—	11	64	no	—
226	C₂²⋊C₄	C₂³×D₁₁	1	64	yes	complemented
227	C₁₁	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	56	no	—
229	C₂³	—	2	56	no	—
230	C₂³	—	2	56	no	—
231	C₂³	—	4	224	no	—
232	C₂³	—	11	112	no	—
233	C₂³	—	11	8	no	—
234	C₂³	—	22	448	no	—
235	C₂³	—	22	56	no	—
236	C₂³	—	22	56	no	—
237	C₂³	#66936	1	64	yes	direct product factor
238	C₂³	C₂×#4339	1	28	yes	—
239	C₂³	C₂×D₄×D₁₁	1	28	yes	—
240	C₂³	C₂×#4341	1	28	yes	—
241	C₂³	C₂⁴×D₁₁	1	8	yes	—
242	C₂³	C₂⁴×D₁₁	1	7	yes	—
243	D₄	—	22	128	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	11	8	no	—
247	C₂×C₄	—	11	8	no	—
248	C₂×C₄	—	11	8	no	—
249	C₂×C₄	—	22	128	no	—
250	C₂×C₄	—	22	56	no	—
251	C₂×C₄	—	22	56	no	—
252	C₂×C₄	—	22	56	no	—
253	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
254	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	112	no	—
259	C₂²	—	11	16	no	—
260	C₂²	—	22	8	no	—
261	C₂²	—	22	8	no	—
262	C₂²	—	22	224	no	—
263	C₂²	C₂×#66936	1	112	yes	direct product factor
264	C₂²	C₂²×#4339	1	14	yes	—
265	C₂²	C₂²×D₄×D₁₁	1	14	yes	—
266	C₂²	C₂²×#4341	1	14	yes	—
267	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	22	8	no	—
271	C₄	—	22	8	no	—
272	C₄	—	22	8	no	—
273	C₂	—	4	8	no	—
274	C₂	—	22	16	no	—
275	C₂	C₂²×#66936	1	28	yes	direct product factor
276	C₂	C₂³×D₄×D₁₁	1	1	yes	characteristic
277	C₂	C₂³×#4341	1	1	yes	characteristic
278	C₂	C₂³×#4339	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂³×C₁₁×C₂²⋊C₄	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁴×C₂²⋊D₁₁	C₂
hypercenter	C₂³×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	C₂³×#123
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₁₁	C₂	semidirect
3. C₂²×#66936	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₁	C₂	semidirect
5. C₂³×Dic₁₁⋊C₄	C₂	semidirect
6. C₂³×C₂²⋊Dic₁₁	C₂	semidirect
7. C₂³×(C₂×D₁₁)⋊C₄	C₂	semidirect
8. C₂³×C₁₁×C₂²⋊C₄	C₂	semidirect
19. C₂³×C₄×D₁₁	C₂²	semidirect
20. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
21. C₂²×C₂²⋊Dic₁₁	C₂²	semidirect
22. C₂²×(C₂×D₁₁)⋊C₄	C₂²	semidirect
23. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
24. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
25. C₂×#66936	C₂²	direct (N × H)
48. C₂²×C₄×D₁₁	C₂³	semidirect
49. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
50. C₂×C₂²⋊Dic₁₁	C₂³	semidirect
51. C₂×(C₂×D₁₁)⋊C₄	C₂³	semidirect
52. C₂×C₄⋊Dic₁₁	C₂³	semidirect
53. C₂×Dic₁₁⋊C₄	C₂³	semidirect
54. #66936	C₂³	direct (N × H)
83. C₂×C₄×D₁₁	C₂⁴	semidirect
84. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
85. C₂²⋊Dic₁₁	C₂⁴	semidirect
86. (C₂×D₁₁)⋊C₄	C₂⁴	semidirect
87. C₄⋊Dic₁₁	C₂⁴	semidirect
88. Dic₁₁⋊C₄	C₂⁴	semidirect
96. C₂³×C₂²⋊C₄	D₁₁	semidirect
101. C₂³×C₁₁	#123	semidirect
140. C₂²×C₂²⋊C₄	C₂×D₁₁	semidirect
145. C₂²×C₁₁	C₂×#123	semidirect
187. C₂×C₂²⋊C₄	C₂²×D₁₁	semidirect
189. C₂×C₁₁	C₂²×#123	semidirect
226. C₂²⋊C₄	C₂³×D₁₁	semidirect
227. C₁₁	C₂³×#123	semidirect
237. C₂³	#66936	direct (N × H)
263. C₂²	C₂×#66936	direct (N × H)
275. C₂	C₂²×#66936	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₁₁	2
C₂⁴×C₄×D₁₁	2
C₂²×#66936	2
C₂³×C₄⋊Dic₁₁	2
C₂³×Dic₁₁⋊C₄	2
C₂³×C₂²⋊Dic₁₁	2
C₂³×(C₂×D₁₁)⋊C₄	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂³×#123	11

Subgroups — C23×#66936 #15660420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66936 #15660420