Subgroups — C₂³×#13133210 #28290762

Order 13568.

9486 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	C₂²×#13133210	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×#13133210	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	complemented
15	C₂²×#11719919	C₂²	1	28	yes	complemented
16	C₂²×#11719919	C₂²	1	28	yes	complemented
17	C₂²×C₂²⋊Dic₅₃	C₂²	1	28	yes	—
18	C₂²×C₂²⋊Dic₅₃	C₂²	1	28	yes	—
19	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	complemented
20	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	28	yes	—
24	C₂³×Dic₅₃	—	2	8	no	—
25	C₂³×Dic₅₃	—	2	8	no	—
26	C₂³×Dic₅₃	—	2	8	no	—
27	C₂³×Dic₅₃	—	2	8	no	—
28	C₂³×Dic₅₃	C₂³	1	14	yes	—
29	C₂³×Dic₅₃	C₂³	1	14	yes	—
30	C₂³×Dic₅₃	C₂³	1	14	yes	—
31	C₂³×Dic₅₃	C₂³	1	14	yes	—
32	#13133210	C₂³	1	512	yes	direct product factor
33	C₂×C₄⋊Dic₅₃	C₂³	1	112	yes	complemented
34	C₂×#11719919	C₂³	1	112	yes	complemented
35	C₂×#11719919	C₂³	1	112	yes	complemented
36	C₂×C₂²⋊Dic₅₃	C₂³	1	112	yes	—
37	C₂×C₂²⋊Dic₅₃	C₂³	1	112	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	112	yes	complemented
39	C₂⁵×C₅₃	—	2	8	no	—
40	C₂⁵×C₅₃	—	2	8	no	—
41	C₂⁵×C₅₃	—	2	8	no	—
42	C₂⁵×C₅₃	C₂³	1	14	yes	—
43	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	8	no	—
45	C₂³×C₄×C₅₃	—	2	8	no	—
46	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂³	1	112	yes	—
49	C₂⁴×C₅₃	—	2	56	no	—
50	C₂⁴×C₅₃	—	2	56	no	—
51	C₂⁴×C₅₃	—	2	56	no	—
52	C₂⁴×C₅₃	—	4	64	no	—
53	C₂⁴×C₅₃	#41	1	8	yes	—
54	C₂⁴×C₅₃	#41	1	8	yes	—
55	C₂⁴×C₅₃	#41	1	8	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	28	yes	—
57	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
58	C₂²×C₄×C₅₃	—	2	56	no	—
59	C₂²×C₄×C₅₃	—	2	56	no	—
60	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
62	C₂²×Dic₅₃	—	2	56	no	—
63	C₂²×Dic₅₃	—	2	56	no	—
64	C₂²×Dic₅₃	—	2	56	no	—
65	C₂²×Dic₅₃	—	2	56	no	—
66	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
67	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
68	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
69	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
70	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	—
71	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
72	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
73	C₄⋊Dic₅₃	C₂⁴	1	64	yes	complemented
74	#11719919	C₂⁴	1	64	yes	complemented
75	#11719919	C₂⁴	1	64	yes	complemented
76	C₄×Dic₅₃	C₂⁴	1	64	yes	complemented
77	C₂³×C₅₃	—	2	56	no	—
78	C₂³×C₅₃	—	2	56	no	—
79	C₂³×C₅₃	—	2	56	no	—
80	C₂³×C₅₃	—	4	224	no	—
81	C₂³×C₅₃	#126	1	64	yes	complemented
82	C₂³×C₅₃	C₂×#41	1	28	yes	—
83	C₂³×C₅₃	C₂×#41	1	28	yes	—
84	C₂³×C₅₃	C₂×#41	1	28	yes	—
85	C₂³×C₅₃	C₂⁵	1	8	yes	—
86	C₂³×C₅₃	C₂⁵	1	7	yes	—
87	C₂×C₄×C₅₃	—	2	56	no	—
88	C₂×C₄×C₅₃	—	2	56	no	—
89	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
90	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
91	C₂×Dic₅₃	—	2	56	no	—
92	C₂×Dic₅₃	—	2	56	no	—
93	C₂×Dic₅₃	—	2	56	no	—
94	C₂×Dic₅₃	—	2	56	no	—
95	C₂×Dic₅₃	C₂⁵	1	8	yes	—
96	C₂×Dic₅₃	C₂⁵	1	8	yes	—
97	C₂×Dic₅₃	C₂⁵	1	8	yes	—
98	C₂×Dic₅₃	C₂⁵	1	8	yes	—
99	C₂³×#126	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	8	no	—
101	C₂²×C₅₃	—	2	8	no	—
102	C₂²×C₅₃	—	2	8	no	—
103	C₂²×C₅₃	—	4	112	no	—
104	C₂²×C₅₃	C₂×#126	1	112	yes	complemented
105	C₂²×C₅₃	C₂²×#41	1	14	yes	—
106	C₂²×C₅₃	C₂²×#41	1	14	yes	—
107	C₂²×C₅₃	C₂²×#41	1	14	yes	—
108	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
109	C₄×C₅₃	—	2	8	no	—
110	C₄×C₅₃	—	2	8	no	—
111	Dic₅₃	—	2	8	no	—
112	Dic₅₃	—	2	8	no	—
113	Dic₅₃	—	2	8	no	—
114	Dic₅₃	—	2	8	no	—
115	C₂²×#126	—	53	56	no	—
116	C₂³×C₄⋊C₄	—	53	1	no	—
117	C₂³×C₄⋊C₄	—	53	1	no	—
118	C₂³×C₄⋊C₄	—	53	1	no	—
119	C₂³×C₂²⋊C₄	—	53	1	no	—
120	C₂³×C₂²⋊C₄	—	53	1	no	—
121	C₂³×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
122	C₂³×C₄²	—	53	1	no	—
123	C₂×C₅₃	—	4	8	no	—
124	C₂×C₅₃	C₂²×#126	1	28	yes	complemented
125	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
126	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
127	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
128	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
129	C₂⁴×C₄	—	53	1	no	—
130	C₂⁴×C₄	—	53	1	no	—
131	C₂⁴×C₄	—	53	1	no	—
132	C₂⁴×C₄	—	53	1	no	—
133	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
135	C₂×#126	—	53	448	no	—
136	C₂²×C₄⋊C₄	—	53	28	no	—
137	C₂²×C₄⋊C₄	—	53	28	no	—
138	C₂²×C₄⋊C₄	—	53	28	no	—
139	C₂²×C₂²⋊C₄	—	53	28	no	—
140	C₂²×C₂²⋊C₄	—	53	28	no	—
141	C₂²×C₂²⋊C₄	C₂×D₅₃	1	28	yes	—
142	C₂²×C₄²	—	53	28	no	—
143	C₅₃	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
144	C₂⁵	—	2	8	no	—
145	C₂⁵	—	2	8	no	—
146	C₂⁵	—	2	8	no	—
147	C₂⁵	C₂²×D₅₃	1	14	yes	—
148	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	53	14	no	—
152	C₂³×C₄	—	53	14	no	—
153	C₂³×C₄	—	53	14	no	—
154	C₂³×C₄	—	53	14	no	—
155	C₂³×C₄	—	106	8	no	—
156	C₂³×C₄	—	106	8	no	—
157	C₂³×C₄	—	106	8	no	—
158	C₂³×C₄	—	106	8	no	—
159	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
160	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
161	#126	—	53	512	no	—
162	C₂×C₄⋊C₄	—	53	112	no	—
163	C₂×C₄⋊C₄	—	53	112	no	—
164	C₂×C₄⋊C₄	—	53	112	no	—
165	C₂×C₂²⋊C₄	—	53	112	no	—
166	C₂×C₂²⋊C₄	—	53	112	no	—
167	C₂×C₂²⋊C₄	C₂²×D₅₃	1	112	yes	—
168	C₂×C₄²	—	53	112	no	—
169	C₂⁴	—	2	56	no	—
170	C₂⁴	—	2	56	no	—
171	C₂⁴	—	2	56	no	—
172	C₂⁴	—	4	64	no	—
173	C₂⁴	#11719945	1	8	yes	—
174	C₂⁴	#11719947	1	8	yes	—
175	C₂⁴	#11719947	1	8	yes	—
176	C₂⁴	C₂³×D₅₃	1	28	yes	—
177	C₂⁴	C₂³×D₅₃	1	7	yes	—
178	C₂²×C₄	—	2	56	no	—
179	C₂²×C₄	—	2	56	no	—
180	C₂²×C₄	—	53	28	no	—
181	C₂²×C₄	—	53	28	no	—
182	C₂²×C₄	—	53	28	no	—
183	C₂²×C₄	—	53	28	no	—
184	C₂²×C₄	—	106	56	no	—
185	C₂²×C₄	—	106	56	no	—
186	C₂²×C₄	—	106	56	no	—
187	C₂²×C₄	—	106	56	no	—
188	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
189	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
190	C₄⋊C₄	—	53	64	no	—
191	C₄⋊C₄	—	53	64	no	—
192	C₄⋊C₄	—	53	64	no	—
193	C₂²⋊C₄	—	53	64	no	—
194	C₂²⋊C₄	—	53	64	no	—
195	C₂²⋊C₄	C₂³×D₅₃	1	64	yes	—
196	C₄²	—	53	64	no	—
197	C₂³	—	2	56	no	—
198	C₂³	—	2	56	no	—
199	C₂³	—	2	56	no	—
200	C₂³	—	4	224	no	—
201	C₂³	#13133210	1	64	yes	direct product factor
202	C₂³	C₂×#11719947	1	28	yes	—
203	C₂³	C₂×#11719947	1	28	yes	—
204	C₂³	C₂×#11719945	1	28	yes	—
205	C₂³	C₂⁴×D₅₃	1	8	yes	—
206	C₂³	C₂⁴×D₅₃	1	7	yes	—
207	C₂×C₄	—	2	56	no	—
208	C₂×C₄	—	2	56	no	—
209	C₂×C₄	—	53	8	no	—
210	C₂×C₄	—	53	8	no	—
211	C₂×C₄	—	53	8	no	—
212	C₂×C₄	—	53	8	no	—
213	C₂×C₄	—	106	56	no	—
214	C₂×C₄	—	106	56	no	—
215	C₂×C₄	—	106	56	no	—
216	C₂×C₄	—	106	56	no	—
217	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
218	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	8	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	4	112	no	—
223	C₂²	C₂×#13133210	1	112	yes	direct product factor
224	C₂²	C₂²×#11719947	1	14	yes	—
225	C₂²	C₂²×#11719947	1	14	yes	—
226	C₂²	C₂²×#11719945	1	14	yes	—
227	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	2	8	no	—
230	C₄	—	106	8	no	—
231	C₄	—	106	8	no	—
232	C₄	—	106	8	no	—
233	C₄	—	106	8	no	—
234	C₂	—	4	8	no	—
235	C₂	C₂²×#13133210	1	28	yes	direct product factor
236	C₂	C₂³×#11719947	1	1	yes	characteristic
237	C₂	C₂³×#11719947	1	1	yes	characteristic
238	C₂	C₂³×#11719945	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₂²⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#126
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133210	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₅₃	C₂	semidirect
3. C₂³×#11719919	C₂	semidirect
4. C₂³×#11719919	C₂	semidirect
7. C₂³×C₄×Dic₅₃	C₂	semidirect
13. C₂×#13133210	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
15. C₂²×#11719919	C₂²	semidirect
16. C₂²×#11719919	C₂²	semidirect
19. C₂²×C₄×Dic₅₃	C₂²	semidirect
32. #13133210	C₂³	direct (N × H)
33. C₂×C₄⋊Dic₅₃	C₂³	semidirect
34. C₂×#11719919	C₂³	semidirect
35. C₂×#11719919	C₂³	semidirect
38. C₂×C₄×Dic₅₃	C₂³	semidirect
73. C₄⋊Dic₅₃	C₂⁴	semidirect
74. #11719919	C₂⁴	semidirect
75. #11719919	C₂⁴	semidirect
76. C₄×Dic₅₃	C₂⁴	semidirect
81. C₂³×C₅₃	#126	semidirect
104. C₂²×C₅₃	C₂×#126	semidirect
124. C₂×C₅₃	C₂²×#126	semidirect
143. C₅₃	C₂³×#126	semidirect
201. C₂³	#13133210	direct (N × H)
223. C₂²	C₂×#13133210	direct (N × H)
235. C₂	C₂²×#13133210	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×#13133210	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×C₂²⋊Dic₅₃	2
C₂³×C₄×Dic₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂³×#126	53

Subgroups — C23×#13133210 #28290762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133210 #28290762