Subgroups — C₂³×#66991 #15660407

Order 2816.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	C₂⁴×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#66991	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂²⋊Dic₁₁	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₁	C₂²	1	2	yes	—
8	C₂⁵×D₁₁	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₁	—	2	16	no	—
10	C₂³×D₄×C₁₁	C₂²	1	28	yes	complemented
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₂²⋊D₁₁	—	2	32	no	—
13	C₂³×C₂²⋊D₁₁	C₂²	1	56	yes	complemented
14	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₁₁	C₂²	1	2	yes	—
16	C₂³×C₄×D₁₁	—	2	16	no	—
17	C₂³×C₄×D₁₁	C₂²	1	28	yes	complemented
18	C₂²×C₂²⋊Dic₁₁	C₂²	1	56	yes	complemented
19	C₂×#66991	C₂²	1	448	yes	direct product factor
20	C₂²×C₄⋊Dic₁₁	C₂²	1	28	yes	complemented
21	C₂⁵×C₁₁	—	2	16	no	—
22	C₂⁵×C₁₁	—	2	16	no	—
23	C₂⁵×C₁₁	—	2	16	no	—
24	C₂⁵×C₁₁	C₂³	1	28	yes	—
25	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
26	C₂⁴×D₁₁	—	2	8	no	—
27	C₂⁴×D₁₁	—	2	8	no	—
28	C₂⁴×D₁₁	D₄	1	16	yes	complemented
29	C₂⁴×D₁₁	C₂³	1	14	yes	—
30	C₂²×D₄×C₁₁	—	2	224	no	—
31	C₂²×D₄×C₁₁	C₂³	1	112	yes	complemented
32	C₂³×C₄×C₁₁	D₄	1	16	yes	complemented
33	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
34	C₂²×C₂²⋊D₁₁	—	2	448	no	—
35	C₂²×C₂²⋊D₁₁	C₂³	1	224	yes	complemented
36	C₂³×Dic₁₁	—	2	8	no	—
37	C₂³×Dic₁₁	—	2	16	no	—
38	C₂³×Dic₁₁	C₂³	1	28	yes	—
39	C₂³×Dic₁₁	C₂³	1	14	yes	—
40	C₂²×C₄×D₁₁	—	2	224	no	—
41	C₂²×C₄×D₁₁	C₂³	1	112	yes	complemented
42	C₂×C₂²⋊Dic₁₁	C₂³	1	224	yes	complemented
43	#66991	C₂³	1	512	yes	direct product factor
44	C₂×C₄⋊Dic₁₁	C₂³	1	112	yes	complemented
45	C₂³×#121	—	11	1	no	maximal, Sylow
46	C₂⁴×C₁₁	—	2	112	no	—
47	C₂⁴×C₁₁	—	2	112	no	—
48	C₂⁴×C₁₁	—	2	112	no	—
49	C₂⁴×C₁₁	—	4	128	no	—
50	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
51	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
52	C₂⁴×C₁₁	#41	1	8	yes	—
53	C₂⁴×C₁₁	C₂⁴	1	56	yes	—
54	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
55	C₂³×D₁₁	—	2	128	no	—
56	C₂³×D₁₁	—	2	56	no	—
57	C₂³×D₁₁	—	2	56	no	—
58	C₂³×D₁₁	C₂×D₄	1	112	yes	complemented
59	C₂³×D₁₁	C₂⁴	1	28	yes	—
60	C₂×D₄×C₁₁	—	2	448	no	—
61	C₂×D₄×C₁₁	C₂⁴	1	64	yes	complemented
62	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	complemented
63	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
64	C₂×C₂²⋊D₁₁	—	2	896	no	—
65	C₂×C₂²⋊D₁₁	C₂⁴	1	128	yes	complemented
66	C₂²×Dic₁₁	—	2	56	no	—
67	C₂²×Dic₁₁	—	2	112	no	—
68	C₂²×Dic₁₁	C₂⁴	1	56	yes	—
69	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
70	C₂×C₄×D₁₁	—	2	448	no	—
71	C₂×C₄×D₁₁	C₂⁴	1	64	yes	complemented
72	C₂²⋊Dic₁₁	C₂⁴	1	128	yes	complemented
73	C₄⋊Dic₁₁	C₂⁴	1	64	yes	complemented
74	C₂⁴×D₄	—	11	2	no	—
75	C₂⁴×D₄	D₁₁	1	1	yes	characteristic, complemented
76	C₂⁵×C₄	—	11	1	no	—
77	C₂²×#121	—	11	56	no	—
78	C₂³×C₄⋊C₄	—	11	1	no	—
79	C₂³×C₂²⋊C₄	—	11	2	no	—
80	C₂³×C₁₁	—	2	112	no	—
81	C₂³×C₁₁	—	2	112	no	—
82	C₂³×C₁₁	—	2	112	no	—
83	C₂³×C₁₁	—	4	448	no	—
84	C₂³×C₁₁	#121	1	64	yes	complemented
85	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
86	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
87	C₂³×C₁₁	C₂×#41	1	28	yes	—
88	C₂³×C₁₁	C₂⁵	1	16	yes	—
89	C₂³×C₁₁	C₂⁵	1	7	yes	—
90	C₂²×D₁₁	—	2	448	no	—
91	C₂²×D₁₁	—	2	56	no	—
92	C₂²×D₁₁	—	2	56	no	—
93	C₂²×D₁₁	C₂²×D₄	1	112	yes	complemented
94	C₂²×D₁₁	C₂⁵	1	8	yes	—
95	D₄×C₁₁	—	2	128	no	—
96	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	complemented
97	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
98	C₂²⋊D₁₁	—	2	256	no	—
99	C₂×Dic₁₁	—	2	56	no	—
100	C₂×Dic₁₁	—	2	112	no	—
101	C₂×Dic₁₁	C₂⁵	1	16	yes	—
102	C₂×Dic₁₁	C₂⁵	1	8	yes	—
103	C₄×D₁₁	—	2	128	no	—
104	C₂⁶	—	11	1	no	—
105	C₂⁶	C₂×D₁₁	1	2	yes	—
106	C₂³×D₄	—	2	16	no	—
107	C₂³×D₄	—	11	56	no	—
108	C₂³×D₄	—	22	32	no	—
109	C₂³×D₄	C₂×D₁₁	1	28	yes	complemented
110	C₂⁴×C₄	—	11	1	no	—
111	C₂⁴×C₄	—	11	2	no	—
112	C₂⁴×C₄	—	11	28	no	—
113	C₂⁴×C₄	—	22	16	no	—
114	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
115	C₂×#121	—	11	448	no	—
116	C₂²×C₄⋊C₄	—	11	28	no	—
117	C₂²×C₂²⋊C₄	—	11	56	no	—
118	C₂²×C₁₁	—	2	16	no	—
119	C₂²×C₁₁	—	2	16	no	—
120	C₂²×C₁₁	—	2	16	no	—
121	C₂²×C₁₁	—	4	224	no	—
122	C₂²×C₁₁	C₂×#121	1	112	yes	complemented
123	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
124	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
125	C₂²×C₁₁	C₂²×#41	1	14	yes	—
126	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
127	C₂×D₁₁	—	2	8	no	—
128	C₂×D₁₁	—	2	8	no	—
129	C₂×D₁₁	—	2	224	no	—
130	C₂×D₁₁	C₂³×D₄	1	16	yes	complemented
131	C₄×C₁₁	C₂³×D₄	1	16	yes	complemented
132	Dic₁₁	—	2	8	no	—
133	Dic₁₁	—	2	16	no	—
134	C₂⁵	—	2	16	no	—
135	C₂⁵	—	2	16	no	—
136	C₂⁵	—	2	16	no	—
137	C₂⁵	—	11	16	no	—
138	C₂⁵	—	11	14	no	—
139	C₂⁵	—	22	8	no	—
140	C₂⁵	—	22	8	no	—
141	C₂⁵	C₂²×D₁₁	1	28	yes	—
142	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
143	C₂²×D₄	—	2	224	no	—
144	C₂²×D₄	—	11	224	no	—
145	C₂²×D₄	—	22	448	no	—
146	C₂²×D₄	C₂²×D₁₁	1	112	yes	complemented
147	C₂³×C₄	—	11	14	no	—
148	C₂³×C₄	—	11	28	no	—
149	C₂³×C₄	—	11	112	no	—
150	C₂³×C₄	—	22	16	no	—
151	C₂³×C₄	—	22	8	no	—
152	C₂³×C₄	—	22	224	no	—
153	C₂³×C₄	C₂²⋊D₁₁	1	16	yes	complemented
154	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
155	#121	—	11	512	no	—
156	C₂×C₄⋊C₄	—	11	112	no	—
157	C₂×C₂²⋊C₄	—	11	224	no	—
158	C₂×C₁₁	—	4	16	no	—
159	C₂×C₁₁	C₂²×#121	1	28	yes	complemented
160	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
161	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
162	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
163	D₁₁	—	2	16	no	—
164	C₂⁴	—	2	112	no	—
165	C₂⁴	—	2	112	no	—
166	C₂⁴	—	2	112	no	—
167	C₂⁴	—	4	128	no	—
168	C₂⁴	—	11	112	no	—
169	C₂⁴	—	11	28	no	—
170	C₂⁴	—	22	128	no	—
171	C₂⁴	—	22	56	no	—
172	C₂⁴	—	22	56	no	—
173	C₂⁴	D₄×D₁₁	1	8	yes	—
174	C₂⁴	#4341	1	8	yes	—
175	C₂⁴	C₂×C₂²⋊D₁₁	1	8	yes	—
176	C₂⁴	C₂³×D₁₁	1	56	yes	—
177	C₂⁴	C₂³×D₁₁	1	7	yes	—
178	C₂×D₄	—	2	448	no	—
179	C₂×D₄	—	11	128	no	—
180	C₂×D₄	—	22	896	no	—
181	C₂×D₄	C₂³×D₁₁	1	64	yes	complemented
182	C₂²×C₄	—	11	28	no	—
183	C₂²×C₄	—	11	56	no	—
184	C₂²×C₄	—	11	64	no	—
185	C₂²×C₄	—	22	112	no	—
186	C₂²×C₄	—	22	56	no	—
187	C₂²×C₄	—	22	448	no	—
188	C₂²×C₄	C₂×C₂²⋊D₁₁	1	112	yes	complemented
189	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
190	C₄⋊C₄	—	11	64	no	—
191	C₂²⋊C₄	—	11	128	no	—
192	C₁₁	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	112	no	—
194	C₂³	—	2	112	no	—
195	C₂³	—	2	112	no	—
196	C₂³	—	4	448	no	—
197	C₂³	—	11	112	no	—
198	C₂³	—	11	8	no	—
199	C₂³	—	22	448	no	—
200	C₂³	—	22	56	no	—
201	C₂³	—	22	56	no	—
202	C₂³	#66991	1	64	yes	direct product factor
203	C₂³	C₂×D₄×D₁₁	1	28	yes	—
204	C₂³	C₂×#4341	1	28	yes	—
205	C₂³	C₂²×C₂²⋊D₁₁	1	28	yes	—
206	C₂³	C₂⁴×D₁₁	1	16	yes	—
207	C₂³	C₂⁴×D₁₁	1	7	yes	—
208	D₄	—	2	128	no	—
209	D₄	—	22	256	no	—
210	C₂×C₄	—	11	8	no	—
211	C₂×C₄	—	11	16	no	—
212	C₂×C₄	—	22	112	no	—
213	C₂×C₄	—	22	56	no	—
214	C₂×C₄	—	22	128	no	—
215	C₂×C₄	C₂²×C₂²⋊D₁₁	1	112	yes	complemented
216	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
217	C₂²	—	2	16	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	4	224	no	—
221	C₂²	—	11	16	no	—
222	C₂²	—	22	8	no	—
223	C₂²	—	22	8	no	—
224	C₂²	—	22	224	no	—
225	C₂²	C₂×#66991	1	112	yes	direct product factor
226	C₂²	C₂²×D₄×D₁₁	1	14	yes	—
227	C₂²	C₂²×#4341	1	14	yes	—
228	C₂²	C₂³×C₂²⋊D₁₁	1	14	yes	—
229	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
230	C₄	—	22	16	no	—
231	C₄	—	22	8	no	—
232	C₄	C₂³×C₂²⋊D₁₁	1	16	yes	complemented
233	C₂	—	4	16	no	—
234	C₂	—	22	16	no	—
235	C₂	C₂²×#66991	1	28	yes	direct product factor
236	C₂	C₂³×D₄×D₁₁	1	1	yes	characteristic
237	C₂	C₂⁴×C₂²⋊D₁₁	1	1	yes	characteristic
238	C₂	C₂³×#4341	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂⁴×D₄×C₁₁	C₂
socle	C₂⁵×C₁₁	C₂³
hypercenter	C₂⁴×D₄	D₁₁
hypocenter	C₁₁	C₂³×#121
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₁₁	C₂	semidirect
2. C₂⁴×C₄×D₁₁	C₂	semidirect
3. C₂²×#66991	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₁	C₂	semidirect
5. C₂³×C₂²⋊Dic₁₁	C₂	semidirect
6. C₂⁴×D₄×C₁₁	C₂	semidirect
10. C₂³×D₄×C₁₁	C₂²	semidirect
13. C₂³×C₂²⋊D₁₁	C₂²	semidirect
17. C₂³×C₄×D₁₁	C₂²	semidirect
18. C₂²×C₂²⋊Dic₁₁	C₂²	semidirect
19. C₂×#66991	C₂²	direct (N × H)
20. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
28. C₂⁴×D₁₁	D₄	semidirect
31. C₂²×D₄×C₁₁	C₂³	semidirect
32. C₂³×C₄×C₁₁	D₄	semidirect
35. C₂²×C₂²⋊D₁₁	C₂³	semidirect
41. C₂²×C₄×D₁₁	C₂³	semidirect
42. C₂×C₂²⋊Dic₁₁	C₂³	semidirect
43. #66991	C₂³	direct (N × H)
44. C₂×C₄⋊Dic₁₁	C₂³	semidirect
58. C₂³×D₁₁	C₂×D₄	semidirect
61. C₂×D₄×C₁₁	C₂⁴	semidirect
62. C₂²×C₄×C₁₁	C₂×D₄	semidirect
65. C₂×C₂²⋊D₁₁	C₂⁴	semidirect
71. C₂×C₄×D₁₁	C₂⁴	semidirect
72. C₂²⋊Dic₁₁	C₂⁴	semidirect
73. C₄⋊Dic₁₁	C₂⁴	semidirect
75. C₂⁴×D₄	D₁₁	semidirect
84. C₂³×C₁₁	#121	semidirect
93. C₂²×D₁₁	C₂²×D₄	semidirect
96. C₂×C₄×C₁₁	C₂²×D₄	semidirect
109. C₂³×D₄	C₂×D₁₁	semidirect
122. C₂²×C₁₁	C₂×#121	semidirect
130. C₂×D₁₁	C₂³×D₄	semidirect
131. C₄×C₁₁	C₂³×D₄	semidirect
146. C₂²×D₄	C₂²×D₁₁	semidirect
153. C₂³×C₄	C₂²⋊D₁₁	semidirect
159. C₂×C₁₁	C₂²×#121	semidirect
181. C₂×D₄	C₂³×D₁₁	semidirect
188. C₂²×C₄	C₂×C₂²⋊D₁₁	semidirect
192. C₁₁	C₂³×#121	semidirect
202. C₂³	#66991	direct (N × H)
215. C₂×C₄	C₂²×C₂²⋊D₁₁	semidirect
225. C₂²	C₂×#66991	direct (N × H)
232. C₄	C₂³×C₂²⋊D₁₁	semidirect
235. C₂	C₂²×#66991	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×#66991	2
C₂³×C₄⋊Dic₁₁	2
C₂³×C₂²⋊Dic₁₁	2
C₂⁴×D₄×C₁₁	2
C₂³×#121	11

Subgroups — C23×#66991 #15660407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66991 #15660407