Subgroups — C₂³×#66948 #15660411

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×D₄₄	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#66948	C₂	1	56	yes	maximal, direct product factor
5	C₂³×(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₁	C₂²	1	2	yes	—
8	C₂⁵×D₁₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄₄	—	2	32	no	—
13	C₂³×D₄₄	—	2	16	no	—
14	C₂³×D₄₄	C₂²	1	56	yes	complemented
15	C₂³×D₄₄	C₂²	1	28	yes	complemented
16	C₂³×C₄×D₁₁	—	2	16	no	—
17	C₂³×C₄×D₁₁	C₂²	1	28	yes	complemented
18	C₂²×C₁₁×C₄⋊C₄	C₂²	1	28	yes	complemented
19	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	56	yes	complemented
20	C₂×#66948	C₂²	1	448	yes	direct product factor
21	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
22	C₂⁴×D₁₁	—	2	8	no	—
23	C₂⁴×D₁₁	—	2	8	no	—
24	C₂⁴×D₁₁	—	2	16	no	—
25	C₂⁴×D₁₁	—	2	16	no	—
26	C₂⁴×D₁₁	—	2	16	no	—
27	C₂⁴×D₁₁	D₄	1	16	yes	complemented
28	C₂⁴×D₁₁	C₂³	1	28	yes	—
29	C₂⁴×D₁₁	C₂³	1	14	yes	—
30	C₂³×C₄×C₁₁	—	2	16	no	—
31	C₂³×C₄×C₁₁	D₄	1	16	yes	complemented
32	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
33	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
34	C₂³×Dic₁₁	—	2	8	no	—
35	C₂³×Dic₁₁	C₂³	1	14	yes	—
36	C₂²×D₄₄	—	2	448	no	—
37	C₂²×D₄₄	—	2	224	no	—
38	C₂²×D₄₄	C₂³	1	224	yes	complemented
39	C₂²×D₄₄	C₂³	1	112	yes	complemented
40	C₂²×C₄×D₁₁	—	2	224	no	—
41	C₂²×C₄×D₁₁	C₂³	1	112	yes	complemented
42	C₂×C₁₁×C₄⋊C₄	C₂³	1	112	yes	complemented
43	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	224	yes	complemented
44	#66948	C₂³	1	512	yes	direct product factor
45	C₂³×#121	—	11	1	no	maximal, Sylow
46	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
47	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
48	C₂⁴×C₁₁	#41	1	8	yes	—
49	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
50	C₂³×D₁₁	—	2	128	no	—
51	C₂³×D₁₁	—	2	56	no	—
52	C₂³×D₁₁	—	2	56	no	—
53	C₂³×D₁₁	—	2	112	no	—
54	C₂³×D₁₁	—	2	112	no	—
55	C₂³×D₁₁	—	2	112	no	—
56	C₂³×D₁₁	—	4	128	no	—
57	C₂³×D₁₁	C₂×D₄	1	112	yes	complemented
58	C₂³×D₁₁	C₂⁴	1	56	yes	—
59	C₂³×D₁₁	C₂⁴	1	28	yes	—
60	C₂²×C₄×C₁₁	—	2	112	no	—
61	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	complemented
62	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
63	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
64	C₂²×Dic₁₁	—	2	56	no	—
65	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
66	C₂×D₄₄	—	2	896	no	—
67	C₂×D₄₄	—	2	448	no	—
68	C₂×D₄₄	C₂⁴	1	128	yes	complemented
69	C₂×D₄₄	C₂⁴	1	64	yes	complemented
70	C₂×C₄×D₁₁	—	2	448	no	—
71	C₂×C₄×D₁₁	C₂⁴	1	64	yes	complemented
72	C₁₁×C₄⋊C₄	C₂⁴	1	64	yes	complemented
73	(C₂×D₁₁)⋊C₄	C₂⁴	1	128	yes	complemented
74	C₂⁴×D₄	—	11	2	no	—
75	C₂⁴×D₄	—	11	1	no	—
76	C₂⁵×C₄	—	11	1	no	—
77	C₂²×#121	—	11	56	no	—
78	C₂³×C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
79	C₂³×C₂²⋊C₄	—	11	2	no	—
80	C₂³×C₁₁	#121	1	64	yes	complemented
81	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
82	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
83	C₂³×C₁₁	C₂×#41	1	28	yes	—
84	C₂³×C₁₁	C₂⁵	1	7	yes	—
85	C₂²×D₁₁	—	2	448	no	—
86	C₂²×D₁₁	—	2	56	no	—
87	C₂²×D₁₁	—	2	56	no	—
88	C₂²×D₁₁	—	2	112	no	—
89	C₂²×D₁₁	—	2	112	no	—
90	C₂²×D₁₁	—	2	112	no	—
91	C₂²×D₁₁	—	4	448	no	—
92	C₂²×D₁₁	C₂²×D₄	1	112	yes	complemented
93	C₂²×D₁₁	C₂⁵	1	16	yes	—
94	C₂²×D₁₁	C₂⁵	1	8	yes	—
95	C₂×C₄×C₁₁	—	2	112	no	—
96	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	complemented
97	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
98	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
99	C₂×Dic₁₁	—	2	56	no	—
100	C₂×Dic₁₁	C₂⁵	1	8	yes	—
101	D₄₄	—	2	256	no	—
102	D₄₄	—	2	128	no	—
103	C₄×D₁₁	—	2	128	no	—
104	C₂⁶	—	11	1	no	—
105	C₂⁶	—	11	2	no	—
106	C₂³×D₄	—	11	56	no	—
107	C₂³×D₄	—	11	28	no	—
108	C₂³×D₄	—	22	32	no	—
109	C₂³×D₄	—	22	16	no	—
110	C₂⁴×C₄	—	11	1	no	—
111	C₂⁴×C₄	—	11	28	no	—
112	C₂⁴×C₄	—	22	16	no	—
113	C₂⁴×C₄	C₂×D₁₁	1	2	yes	—
114	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
115	C₂×#121	—	11	448	no	—
116	C₂²×C₄⋊C₄	C₂×D₁₁	1	28	yes	complemented
117	C₂²×C₂²⋊C₄	—	11	56	no	—
118	C₂²×C₁₁	C₂×#121	1	112	yes	complemented
119	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
120	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
121	C₂²×C₁₁	C₂²×#41	1	14	yes	—
122	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
123	C₂×D₁₁	—	2	8	no	—
124	C₂×D₁₁	—	2	8	no	—
125	C₂×D₁₁	—	2	224	no	—
126	C₂×D₁₁	—	2	16	no	—
127	C₂×D₁₁	—	2	16	no	—
128	C₂×D₁₁	—	2	16	no	—
129	C₂×D₁₁	—	4	224	no	—
130	C₂×D₁₁	C₂³×D₄	1	16	yes	complemented
131	C₄×C₁₁	—	2	16	no	—
132	C₄×C₁₁	C₂³×D₄	1	16	yes	complemented
133	Dic₁₁	—	2	8	no	—
134	C₂⁵	—	11	16	no	—
135	C₂⁵	—	11	14	no	—
136	C₂⁵	—	11	28	no	—
137	C₂⁵	—	22	8	no	—
138	C₂⁵	—	22	8	no	—
139	C₂⁵	—	22	16	no	—
140	C₂⁵	—	22	16	no	—
141	C₂⁵	—	22	16	no	—
142	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
143	C₂²×D₄	—	11	224	no	—
144	C₂²×D₄	—	11	112	no	—
145	C₂²×D₄	—	22	448	no	—
146	C₂²×D₄	—	22	224	no	—
147	C₂³×C₄	—	2	16	no	—
148	C₂³×C₄	—	11	14	no	—
149	C₂³×C₄	—	11	112	no	—
150	C₂³×C₄	—	22	8	no	—
151	C₂³×C₄	—	22	224	no	—
152	C₂³×C₄	D₄₄	1	16	yes	complemented
153	C₂³×C₄	C₂²×D₁₁	1	28	yes	—
154	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
155	#121	—	11	512	no	—
156	C₂×C₄⋊C₄	C₂²×D₁₁	1	112	yes	complemented
157	C₂×C₂²⋊C₄	—	11	224	no	—
158	C₂×C₁₁	C₂²×#121	1	28	yes	complemented
159	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
160	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
161	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
162	D₁₁	—	2	16	no	—
163	D₁₁	—	4	16	no	—
164	C₂⁴	—	11	112	no	—
165	C₂⁴	—	11	28	no	—
166	C₂⁴	—	11	56	no	—
167	C₂⁴	—	22	128	no	—
168	C₂⁴	—	22	56	no	—
169	C₂⁴	—	22	56	no	—
170	C₂⁴	—	22	112	no	—
171	C₂⁴	—	22	112	no	—
172	C₂⁴	—	22	112	no	—
173	C₂⁴	—	44	128	no	—
174	C₂⁴	C₂×D₄₄	1	8	yes	—
175	C₂⁴	D₄×D₁₁	1	8	yes	—
176	C₂⁴	#4343	1	8	yes	—
177	C₂⁴	C₂³×D₁₁	1	7	yes	—
178	C₂×D₄	—	11	128	no	—
179	C₂×D₄	—	11	64	no	—
180	C₂×D₄	—	22	896	no	—
181	C₂×D₄	—	22	448	no	—
182	C₂²×C₄	—	2	112	no	—
183	C₂²×C₄	—	11	28	no	—
184	C₂²×C₄	—	11	64	no	—
185	C₂²×C₄	—	22	56	no	—
186	C₂²×C₄	—	22	448	no	—
187	C₂²×C₄	C₂×D₄₄	1	112	yes	complemented
188	C₂²×C₄	C₂³×D₁₁	1	56	yes	—
189	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
190	C₄⋊C₄	C₂³×D₁₁	1	64	yes	complemented
191	C₂²⋊C₄	—	11	128	no	—
192	C₁₁	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	11	112	no	—
194	C₂³	—	11	8	no	—
195	C₂³	—	11	16	no	—
196	C₂³	—	22	448	no	—
197	C₂³	—	22	56	no	—
198	C₂³	—	22	56	no	—
199	C₂³	—	22	112	no	—
200	C₂³	—	22	112	no	—
201	C₂³	—	22	112	no	—
202	C₂³	—	44	448	no	—
203	C₂³	#66948	1	64	yes	direct product factor
204	C₂³	C₂²×D₄₄	1	28	yes	—
205	C₂³	C₂×D₄×D₁₁	1	28	yes	—
206	C₂³	C₂×#4343	1	28	yes	—
207	C₂³	C₂⁴×D₁₁	1	7	yes	—
208	D₄	—	22	256	no	—
209	D₄	—	22	128	no	—
210	C₂×C₄	—	2	112	no	—
211	C₂×C₄	—	11	8	no	—
212	C₂×C₄	—	22	56	no	—
213	C₂×C₄	—	22	128	no	—
214	C₂×C₄	C₂²×D₄₄	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₁₁	1	16	yes	—
216	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
217	C₂²	—	11	16	no	—
218	C₂²	—	22	8	no	—
219	C₂²	—	22	8	no	—
220	C₂²	—	22	224	no	—
221	C₂²	—	22	16	no	—
222	C₂²	—	22	16	no	—
223	C₂²	—	22	16	no	—
224	C₂²	—	44	224	no	—
225	C₂²	C₂×#66948	1	112	yes	direct product factor
226	C₂²	C₂³×D₄₄	1	14	yes	—
227	C₂²	C₂²×D₄×D₁₁	1	14	yes	—
228	C₂²	C₂²×#4343	1	14	yes	—
229	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
230	C₄	—	2	16	no	—
231	C₄	—	22	8	no	—
232	C₄	C₂³×D₄₄	1	16	yes	complemented
233	C₂	—	22	16	no	—
234	C₂	—	44	16	no	—
235	C₂	C₂²×#66948	1	28	yes	direct product factor
236	C₂	C₂³×D₄×D₁₁	1	1	yes	characteristic
237	C₂	C₂⁴×D₄₄	1	1	yes	characteristic
238	C₂	C₂³×#4343	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂³×C₁₁×C₄⋊C₄	C₂
socle	C₂⁵×C₁₁	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₁
hypocenter	C₁₁	C₂³×#121
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄₄	C₂	semidirect
2. C₂⁴×D₄₄	C₂	semidirect
3. C₂⁴×C₄×D₁₁	C₂	semidirect
4. C₂²×#66948	C₂	direct (N × H)
5. C₂³×(C₂×D₁₁)⋊C₄	C₂	semidirect
6. C₂³×C₁₁×C₄⋊C₄	C₂	semidirect
14. C₂³×D₄₄	C₂²	semidirect
15. C₂³×D₄₄	C₂²	semidirect
17. C₂³×C₄×D₁₁	C₂²	semidirect
18. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
19. C₂²×(C₂×D₁₁)⋊C₄	C₂²	semidirect
20. C₂×#66948	C₂²	direct (N × H)
27. C₂⁴×D₁₁	D₄	semidirect
31. C₂³×C₄×C₁₁	D₄	semidirect
38. C₂²×D₄₄	C₂³	semidirect
39. C₂²×D₄₄	C₂³	semidirect
41. C₂²×C₄×D₁₁	C₂³	semidirect
42. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
43. C₂×(C₂×D₁₁)⋊C₄	C₂³	semidirect
44. #66948	C₂³	direct (N × H)
57. C₂³×D₁₁	C₂×D₄	semidirect
61. C₂²×C₄×C₁₁	C₂×D₄	semidirect
68. C₂×D₄₄	C₂⁴	semidirect
69. C₂×D₄₄	C₂⁴	semidirect
71. C₂×C₄×D₁₁	C₂⁴	semidirect
72. C₁₁×C₄⋊C₄	C₂⁴	semidirect
73. (C₂×D₁₁)⋊C₄	C₂⁴	semidirect
78. C₂³×C₄⋊C₄	D₁₁	semidirect
80. C₂³×C₁₁	#121	semidirect
92. C₂²×D₁₁	C₂²×D₄	semidirect
96. C₂×C₄×C₁₁	C₂²×D₄	semidirect
116. C₂²×C₄⋊C₄	C₂×D₁₁	semidirect
118. C₂²×C₁₁	C₂×#121	semidirect
130. C₂×D₁₁	C₂³×D₄	semidirect
132. C₄×C₁₁	C₂³×D₄	semidirect
152. C₂³×C₄	D₄₄	semidirect
156. C₂×C₄⋊C₄	C₂²×D₁₁	semidirect
158. C₂×C₁₁	C₂²×#121	semidirect
187. C₂²×C₄	C₂×D₄₄	semidirect
190. C₄⋊C₄	C₂³×D₁₁	semidirect
192. C₁₁	C₂³×#121	semidirect
203. C₂³	#66948	direct (N × H)
214. C₂×C₄	C₂²×D₄₄	semidirect
225. C₂²	C₂×#66948	direct (N × H)
232. C₄	C₂³×D₄₄	semidirect
235. C₂	C₂²×#66948	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×D₄₄	2
C₂⁴×C₄×D₁₁	2
C₂²×#66948	2
C₂³×(C₂×D₁₁)⋊C₄	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×#121	11

Subgroups — C23×#66948 #15660411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66948 #15660411