Subgroups — C₂²×#16810735 #28248137

Order 13568.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×#16810735	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133138	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	—	2	2	no	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×D₄×Dic₅₃	—	2	12	no	—
10	C₂×D₄×Dic₅₃	C₂²	1	24	yes	complemented
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₅₃	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810735	C₂²	1	64	yes	direct product factor
15	C₂×#13133149	C₂²	1	12	yes	complemented
16	C₂×#13133138	C₂²	1	24	yes	complemented
17	C₂×#13133127	C₂²	1	12	yes	complemented
18	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×D₈×C₅₃	C₄	1	16	yes	complemented
20	C₂²×D₈×C₅₃	C₂²	1	12	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₅₃	—	2	2	no	—
23	C₂³×Dic₅₃	—	2	24	no	—
24	C₂³×Dic₅₃	—	4	16	no	—
25	C₂³×Dic₅₃	D₄	1	2	yes	—
26	C₂²×#88322	—	2	4	no	—
27	C₂²×#88322	C₂³	1	6	yes	—
28	D₄×Dic₅₃	—	2	48	no	—
29	D₄×Dic₅₃	C₂³	1	32	yes	complemented
30	C₂×C₄⋊Dic₅₃	—	2	6	no	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
32	C₂×C₂²⋊Dic₅₃	—	2	24	no	—
33	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
35	#13133149	C₂³	1	16	yes	complemented
36	#13133138	C₂³	1	32	yes	complemented
37	#13133127	C₂³	1	16	yes	complemented
38	C₂⁵×C₅₃	—	2	2	no	—
39	C₂²×D₄×C₅₃	—	2	6	no	—
40	C₂²×D₄×C₅₃	—	2	8	no	—
41	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
42	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×D₈×C₅₃	C₂×C₄	1	96	yes	complemented
45	C₂×D₈×C₅₃	C₂³	1	16	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
47	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
48	C₂⁴×C₅₃	—	2	16	no	—
49	C₂⁴×C₅₃	—	2	12	no	—
50	C₂⁴×C₅₃	—	4	8	no	—
51	C₂⁴×C₅₃	—	4	8	no	—
52	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×D₄×C₅₃	—	2	48	no	—
54	C₂×D₄×C₅₃	—	2	12	no	—
55	C₂×D₄×C₅₃	—	2	24	no	—
56	C₂×D₄×C₅₃	C₂²×C₄	1	48	yes	complemented
57	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₅₃	#41	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
62	C₂²×Dic₅₃	—	2	8	no	—
63	C₂²×Dic₅₃	—	2	12	no	—
64	C₂²×Dic₅₃	—	2	32	no	—
65	C₂²×Dic₅₃	—	4	96	no	—
66	C₂²×Dic₅₃	—	4	8	no	—
67	C₂²×Dic₅₃	C₂×D₄	1	12	yes	—
68	D₈×C₅₃	C₂²×C₄	1	64	yes	complemented
69	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	complemented
70	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
71	C₂²⋊Dic₅₃	—	2	32	no	—
72	C₄⋊Dic₅₃	—	2	12	no	—
73	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
74	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
75	C₄×Dic₅₃	C₂⁴	1	4	yes	—
76	C₂×#88322	—	2	12	no	—
77	C₂×#88322	C₂⁴	1	4	yes	—
78	C₂³×C₅₃	—	2	48	no	—
79	C₂³×C₅₃	—	2	8	no	—
80	C₂³×C₅₃	—	4	24	no	—
81	C₂³×C₅₃	—	4	24	no	—
82	C₂³×C₅₃	—	4	64	no	—
83	C₂³×C₅₃	C₄×D₄	1	4	yes	—
84	C₂³×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
85	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
86	D₄×C₅₃	—	2	48	no	—
87	D₄×C₅₃	—	2	8	no	—
88	D₄×C₅₃	C₂³×C₄	1	16	yes	complemented
89	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
92	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
93	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₅₃	—	2	24	no	—
95	C₂×Dic₅₃	—	2	8	no	—
96	C₂×Dic₅₃	—	4	64	no	—
97	C₂×Dic₅₃	—	4	24	no	—
98	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
99	C₈×C₅₃	C₂³×C₄	1	8	yes	complemented
100	#88322	—	2	4	no	—
101	C₂²×#442	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	16	no	—
103	C₂²×C₅₃	—	4	96	no	—
104	C₂²×C₅₃	—	4	8	no	—
105	C₂²×C₅₃	—	4	8	no	—
106	C₂²×C₅₃	#442	1	16	yes	complemented
107	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
108	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
109	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
111	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
113	Dic₅₃	—	2	8	no	—
114	Dic₅₃	—	4	8	no	—
115	C₂³×D₈	D₅₃	1	1	yes	characteristic
116	C₂²×C₄×D₄	—	53	2	no	—
117	C₂×#442	—	53	24	no	—
118	C₂²×#106	—	53	1	no	—
119	C₂²×#102	—	53	2	no	—
120	C₂²×#97	—	53	1	no	—
121	C₂×C₅₃	—	4	16	no	—
122	C₂×C₅₃	C₂×#442	1	12	yes	complemented
123	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
125	C₂³×D₄	C₂×D₅₃	1	2	yes	—
126	C₂⁴×C₄	—	106	2	no	—
127	C₂²×D₈	Dic₅₃	1	16	yes	complemented
128	C₂²×D₈	C₂×D₅₃	1	12	yes	—
129	C₂³×C₈	—	53	1	no	—
130	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	53	24	no	—
132	C₂×C₄×D₄	—	106	12	no	—
133	C₂²×C₄⋊C₄	—	53	2	no	—
134	C₂²×C₂²⋊C₄	—	106	2	no	—
135	C₂²×C₄²	—	53	1	no	—
136	#442	—	53	64	no	—
137	C₂×#106	—	53	12	no	—
138	C₂×#102	—	53	24	no	—
139	C₂×#97	—	53	12	no	—
140	C₅₃	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	2	2	no	—
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	—	2	8	no	—
144	C₂²×D₄	C₂×Dic₅₃	1	16	yes	complemented
145	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
146	C₂³×C₄	—	53	2	no	—
147	C₂³×C₄	—	106	24	no	—
148	C₂³×C₄	—	106	2	no	—
149	C₂³×C₄	—	212	16	no	—
150	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	C₂×D₈	C₂×Dic₅₃	1	96	yes	complemented
152	C₂×D₈	C₂²×D₅₃	1	16	yes	—
153	C₂²×C₈	—	53	6	no	—
154	C₂²×C₈	—	106	4	no	—
155	C₂²×C₈	C₂×Dic₅₃	1	8	yes	complemented
156	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
157	C₄×D₄	—	53	32	no	—
158	C₄×D₄	—	106	48	no	—
159	C₂×C₄⋊C₄	—	53	12	no	—
160	C₂×C₄⋊C₄	—	106	6	no	—
161	C₂×C₂²⋊C₄	—	106	24	no	—
162	C₂×C₄²	—	53	6	no	—
163	C₂×C₄²	—	53	6	no	—
164	#106	—	53	16	no	—
165	#102	—	53	32	no	—
166	#97	—	53	16	no	—
167	C₂⁴	—	2	16	no	—
168	C₂⁴	—	2	12	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	—	4	8	no	—
171	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
172	C₂×D₄	—	2	48	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	2	24	no	—
175	C₂×D₄	C₂²×Dic₅₃	1	48	yes	complemented
176	C₂×D₄	C₂³×D₅₃	1	8	yes	—
177	C₂²×C₄	—	53	12	no	—
178	C₂²×C₄	—	106	32	no	—
179	C₂²×C₄	—	106	8	no	—
180	C₂²×C₄	—	106	12	no	—
181	C₂²×C₄	—	212	96	no	—
182	C₂²×C₄	—	212	8	no	—
183	C₂²×C₄	D₄×D₅₃	1	3	yes	—
184	C₂²×C₄	#11719947	1	4	yes	—
185	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
186	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
187	D₈	C₂²×Dic₅₃	1	64	yes	complemented
188	C₂×C₈	—	53	4	no	—
189	C₂×C₈	—	106	12	no	—
190	C₂×C₈	C₂²×Dic₅₃	1	24	yes	complemented
191	C₂×C₈	C₂³×D₅₃	1	4	yes	—
192	C₄⋊C₄	—	53	8	no	—
193	C₄⋊C₄	—	106	12	no	—
194	C₂²⋊C₄	—	106	32	no	—
195	C₄²	—	53	12	no	—
196	C₄²	—	53	4	no	—
197	C₂³	—	2	48	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	24	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	4	64	no	—
202	C₂³	D₄×Dic₅₃	1	4	yes	—
203	C₂³	#13133241	1	8	yes	—
204	C₂³	C₂×D₄×D₅₃	1	3	yes	—
205	D₄	—	2	48	no	—
206	D₄	—	2	8	no	—
207	D₄	C₂³×Dic₅₃	1	16	yes	complemented
208	C₂×C₄	—	53	8	no	—
209	C₂×C₄	—	106	24	no	—
210	C₂×C₄	—	106	8	no	—
211	C₂×C₄	—	212	64	no	—
212	C₂×C₄	—	212	24	no	—
213	C₂×C₄	D₄×Dic₅₃	1	12	yes	—
214	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
215	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
216	C₂×C₄	C₂×#11719947	1	6	yes	—
217	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
218	C₈	—	106	4	no	—
219	C₈	C₂³×Dic₅₃	1	8	yes	complemented
220	C₂²	—	2	16	no	—
221	C₂²	—	4	96	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	8	no	—
224	C₂²	#16810735	1	16	yes	direct product factor
225	C₂²	C₂×D₄×Dic₅₃	1	6	yes	—
226	C₂²	C₂×#13133241	1	12	yes	—
227	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
228	C₄	—	106	8	no	—
229	C₄	—	212	8	no	—
230	C₄	C₂×D₄×Dic₅₃	1	6	yes	—
231	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
232	C₄	C₂²×#11719947	1	1	yes	characteristic
233	C₂	—	4	16	no	—
234	C₂	C₂×#16810735	1	12	yes	direct product factor
235	C₂	C₂²×D₄×Dic₅₃	1	1	yes	characteristic
236	C₂	C₂²×#13133241	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting, prime radical	C₂³×D₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×D₈	D₅₃
hypocenter	C₅₃	C₂²×#442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₅₃	C₂	semidirect
2. C₂×#16810735	C₂	direct (N × H)
3. C₂²×#13133149	C₂	semidirect
4. C₂²×#13133138	C₂	semidirect
5. C₂²×#13133127	C₂	semidirect
10. C₂×D₄×Dic₅₃	C₂²	semidirect
14. #16810735	C₂²	direct (N × H)
15. C₂×#13133149	C₂²	semidirect
16. C₂×#13133138	C₂²	semidirect
17. C₂×#13133127	C₂²	semidirect
19. C₂²×D₈×C₅₃	C₄	semidirect
29. D₄×Dic₅₃	C₂³	semidirect
35. #13133149	C₂³	semidirect
36. #13133138	C₂³	semidirect
37. #13133127	C₂³	semidirect
41. C₂²×D₄×C₅₃	C₂×C₄	semidirect
44. C₂×D₈×C₅₃	C₂×C₄	semidirect
46. C₂²×C₈×C₅₃	C₂×C₄	semidirect
56. C₂×D₄×C₅₃	C₂²×C₄	semidirect
68. D₈×C₅₃	C₂²×C₄	semidirect
69. C₂×C₈×C₅₃	C₂²×C₄	semidirect
88. D₄×C₅₃	C₂³×C₄	semidirect
99. C₈×C₅₃	C₂³×C₄	semidirect
106. C₂²×C₅₃	#442	semidirect
122. C₂×C₅₃	C₂×#442	semidirect
127. C₂²×D₈	Dic₅₃	semidirect
140. C₅₃	C₂²×#442	semidirect
144. C₂²×D₄	C₂×Dic₅₃	semidirect
151. C₂×D₈	C₂×Dic₅₃	semidirect
155. C₂²×C₈	C₂×Dic₅₃	semidirect
175. C₂×D₄	C₂²×Dic₅₃	semidirect
187. D₈	C₂²×Dic₅₃	semidirect
190. C₂×C₈	C₂²×Dic₅₃	semidirect
207. D₄	C₂³×Dic₅₃	semidirect
219. C₈	C₂³×Dic₅₃	semidirect
224. C₂²	#16810735	direct (N × H)
234. C₂	C₂×#16810735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₄×Dic₅₃	2
C₂×#16810735	2
C₂²×#13133149	2
C₂²×#13133138	2
C₂²×#13133127	2
C₂³×D₈×C₅₃	2
C₂²×#442	53

Subgroups — C22×#16810735 #28248137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810735 #28248137