Subgroups — C₂³×#13133179 #28290656

Order 13568.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133179	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11719921	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	2	yes	—
8	C₂³×D₂₁₂	C₄	1	32	yes	complemented
9	C₂³×D₂₁₂	C₂²	1	28	yes	—
10	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₅₃	—	2	32	no	—
13	C₂³×C₄×D₅₃	C₂²	1	56	yes	complemented
14	C₂×#13133179	C₂²	1	448	yes	direct product factor
15	C₂²×#11719921	C₂²	1	56	yes	complemented
16	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	complemented
17	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	complemented
20	C₂⁴×D₅₃	—	2	16	no	—
21	C₂⁴×D₅₃	—	2	16	no	—
22	C₂⁴×D₅₃	C₂×C₄	1	32	yes	complemented
23	C₂⁴×D₅₃	C₂³	1	28	yes	—
24	C₂²×D₂₁₂	C₂×C₄	1	448	yes	complemented
25	C₂²×D₂₁₂	C₂³	1	112	yes	—
26	C₂³×Dic₅₃	—	2	8	no	—
27	C₂³×Dic₅₃	D₄	1	16	yes	complemented
28	C₂³×Dic₅₃	C₂³	1	14	yes	—
29	C₂³×Dic₅₃	C₂³	1	14	yes	—
30	C₂²×C₄×D₅₃	—	2	448	no	—
31	C₂²×C₄×D₅₃	C₂³	1	224	yes	complemented
32	#13133179	C₂³	1	512	yes	direct product factor
33	C₂×#11719921	C₂³	1	224	yes	complemented
34	C₂×C₄×Dic₅₃	C₂³	1	112	yes	complemented
35	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	—	2	16	no	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
38	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	complemented
41	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
43	C₂⁴×C₅₃	#41	1	8	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
45	C₂³×D₅₃	—	2	256	no	—
46	C₂³×D₅₃	—	2	112	no	—
47	C₂³×D₅₃	—	2	112	no	—
48	C₂³×D₅₃	C₂²×C₄	1	224	yes	complemented
49	C₂³×D₅₃	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₅₃	—	2	112	no	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	112	yes	complemented
52	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
54	C₂²×Dic₅₃	—	2	56	no	—
55	C₂²×Dic₅₃	C₂×D₄	1	112	yes	complemented
56	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
57	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
58	C₂×D₂₁₂	C₂²×C₄	1	896	yes	complemented
59	C₂×D₂₁₂	C₂⁴	1	64	yes	—
60	C₂×C₄×D₅₃	—	2	896	no	—
61	C₂×C₄×D₅₃	C₂⁴	1	128	yes	complemented
62	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
63	#11719921	C₂⁴	1	128	yes	complemented
64	C₄×Dic₅₃	C₂⁴	1	64	yes	complemented
65	C₂³×C₅₃	C₄×D₄	1	64	yes	complemented
66	C₂³×C₅₃	C₂³×C₄	1	28	yes	—
67	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
68	C₂³×C₅₃	C₂×#41	1	28	yes	—
69	C₂³×C₅₃	C₂⁵	1	7	yes	—
70	C₂²×D₅₃	—	2	896	no	—
71	C₂²×D₅₃	—	2	112	no	—
72	C₂²×D₅₃	—	2	112	no	—
73	C₂²×D₅₃	C₂³×C₄	1	224	yes	complemented
74	C₂²×D₅₃	C₂⁵	1	16	yes	—
75	C₂×C₄×C₅₃	—	2	112	no	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	112	yes	complemented
77	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
78	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
79	C₂×Dic₅₃	—	2	56	no	—
80	C₂×Dic₅₃	C₂²×D₄	1	112	yes	complemented
81	C₂×Dic₅₃	C₂⁵	1	8	yes	—
82	C₂×Dic₅₃	C₂⁵	1	8	yes	—
83	D₂₁₂	C₂³×C₄	1	256	yes	complemented
84	C₄×D₅₃	—	2	256	no	—
85	C₂³×C₄×D₄	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×C₄×D₄	1	112	yes	complemented
87	C₂²×C₅₃	C₂⁴×C₄	1	14	yes	—
88	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
89	C₂²×C₅₃	C₂²×#41	1	14	yes	—
90	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
91	C₂×D₅₃	—	2	16	no	—
92	C₂×D₅₃	—	2	16	no	—
93	C₂×D₅₃	—	2	448	no	—
94	C₂×D₅₃	C₂⁴×C₄	1	32	yes	complemented
95	C₄×C₅₃	—	2	16	no	—
96	C₄×C₅₃	C₂⁴×C₄	1	16	yes	complemented
97	Dic₅₃	—	2	8	no	—
98	Dic₅₃	C₂³×D₄	1	16	yes	complemented
99	C₂⁴×D₄	—	53	1	no	—
100	C₂⁵×C₄	—	53	2	no	—
101	C₂²×C₄×D₄	—	53	56	no	—
102	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
103	C₂³×C₂²⋊C₄	—	53	2	no	—
104	C₂³×C₄²	—	53	1	no	—
105	C₂×C₅₃	C₂²×C₄×D₄	1	28	yes	complemented
106	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
107	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
108	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
109	D₅₃	—	2	32	no	—
110	C₂⁶	—	53	2	no	—
111	C₂³×D₄	—	53	32	no	—
112	C₂³×D₄	—	53	28	no	—
113	C₂⁴×C₄	—	53	1	no	—
114	C₂⁴×C₄	—	53	1	no	—
115	C₂⁴×C₄	—	53	56	no	—
116	C₂⁴×C₄	—	106	32	no	—
117	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
118	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄×D₄	—	53	448	no	—
120	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	complemented
121	C₂²×C₂²⋊C₄	—	53	56	no	—
122	C₂²×C₄²	—	53	28	no	—
123	C₅₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	53	32	no	—
125	C₂⁵	—	53	28	no	—
126	C₂⁵	—	106	16	no	—
127	C₂⁵	—	106	16	no	—
128	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
129	C₂²×D₄	—	53	448	no	—
130	C₂²×D₄	—	53	112	no	—
131	C₂³×C₄	—	2	16	no	—
132	C₂³×C₄	—	53	16	no	—
133	C₂³×C₄	—	53	14	no	—
134	C₂³×C₄	—	53	14	no	—
135	C₂³×C₄	—	53	224	no	—
136	C₂³×C₄	—	106	448	no	—
137	C₂³×C₄	—	106	8	no	—
138	C₂³×C₄	C₄×D₅₃	1	16	yes	complemented
139	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
140	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
141	C₄×D₄	—	53	512	no	—
142	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	complemented
143	C₂×C₂²⋊C₄	—	53	224	no	—
144	C₂×C₄²	—	53	112	no	—
145	C₂⁴	—	53	224	no	—
146	C₂⁴	—	53	56	no	—
147	C₂⁴	—	106	256	no	—
148	C₂⁴	—	106	112	no	—
149	C₂⁴	—	106	112	no	—
150	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
151	C₂⁴	D₄×D₅₃	1	8	yes	—
152	C₂⁴	#11719949	1	8	yes	—
153	C₂⁴	C₂³×D₅₃	1	7	yes	—
154	C₂×D₄	—	53	896	no	—
155	C₂×D₄	—	53	64	no	—
156	C₂²×C₄	—	2	112	no	—
157	C₂²×C₄	—	53	112	no	—
158	C₂²×C₄	—	53	28	no	—
159	C₂²×C₄	—	53	28	no	—
160	C₂²×C₄	—	53	128	no	—
161	C₂²×C₄	—	106	896	no	—
162	C₂²×C₄	—	106	56	no	—
163	C₂²×C₄	C₂×C₄×D₅₃	1	112	yes	complemented
164	C₂²×C₄	C₂³×D₅₃	1	56	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
166	C₄⋊C₄	C₂³×D₅₃	1	64	yes	complemented
167	C₂²⋊C₄	—	53	128	no	—
168	C₄²	—	53	64	no	—
169	C₂³	—	53	224	no	—
170	C₂³	—	53	16	no	—
171	C₂³	—	106	896	no	—
172	C₂³	—	106	112	no	—
173	C₂³	—	106	112	no	—
174	C₂³	#13133179	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₅₃	1	28	yes	—
176	C₂³	C₂×D₄×D₅₃	1	28	yes	—
177	C₂³	C₂×#11719949	1	28	yes	—
178	C₂³	C₂⁴×D₅₃	1	7	yes	—
179	D₄	—	53	256	no	—
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	53	112	no	—
182	C₂×C₄	—	53	8	no	—
183	C₂×C₄	—	53	8	no	—
184	C₂×C₄	—	106	256	no	—
185	C₂×C₄	—	106	56	no	—
186	C₂×C₄	C₂²×C₄×D₅₃	1	112	yes	complemented
187	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
188	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
189	C₂²	—	53	32	no	—
190	C₂²	—	106	448	no	—
191	C₂²	—	106	16	no	—
192	C₂²	—	106	16	no	—
193	C₂²	C₂×#13133179	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₅₃	1	14	yes	—
195	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
196	C₂²	C₂²×#11719949	1	14	yes	—
197	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	53	16	no	—
200	C₄	—	106	8	no	—
201	C₄	C₂³×C₄×D₅₃	1	16	yes	complemented
202	C₂	—	106	32	no	—
203	C₂	C₂²×#13133179	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
205	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
206	C₂	C₂³×#11719949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂²×#13133179	C₂	direct (N × H)
4. C₂³×#11719921	C₂	semidirect
5. C₂³×C₄×Dic₅₃	C₂	semidirect
6. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
8. C₂³×D₂₁₂	C₄	semidirect
13. C₂³×C₄×D₅₃	C₂²	semidirect
14. C₂×#13133179	C₂²	direct (N × H)
15. C₂²×#11719921	C₂²	semidirect
16. C₂²×C₄×Dic₅₃	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
22. C₂⁴×D₅₃	C₂×C₄	semidirect
24. C₂²×D₂₁₂	C₂×C₄	semidirect
27. C₂³×Dic₅₃	D₄	semidirect
31. C₂²×C₄×D₅₃	C₂³	semidirect
32. #13133179	C₂³	direct (N × H)
33. C₂×#11719921	C₂³	semidirect
34. C₂×C₄×Dic₅₃	C₂³	semidirect
37. C₂³×C₄×C₅₃	C₂×C₄	semidirect
40. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
48. C₂³×D₅₃	C₂²×C₄	semidirect
51. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
55. C₂²×Dic₅₃	C₂×D₄	semidirect
58. C₂×D₂₁₂	C₂²×C₄	semidirect
61. C₂×C₄×D₅₃	C₂⁴	semidirect
62. C₄⋊C₄×C₅₃	C₂⁴	semidirect
63. #11719921	C₂⁴	semidirect
64. C₄×Dic₅₃	C₂⁴	semidirect
65. C₂³×C₅₃	C₄×D₄	semidirect
73. C₂²×D₅₃	C₂³×C₄	semidirect
76. C₂×C₄×C₅₃	C₂³×C₄	semidirect
80. C₂×Dic₅₃	C₂²×D₄	semidirect
83. D₂₁₂	C₂³×C₄	semidirect
86. C₂²×C₅₃	C₂×C₄×D₄	semidirect
94. C₂×D₅₃	C₂⁴×C₄	semidirect
96. C₄×C₅₃	C₂⁴×C₄	semidirect
98. Dic₅₃	C₂³×D₄	semidirect
102. C₂³×C₄⋊C₄	D₅₃	semidirect
105. C₂×C₅₃	C₂²×C₄×D₄	semidirect
120. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
123. C₅₃	C₂³×C₄×D₄	semidirect
138. C₂³×C₄	C₄×D₅₃	semidirect
142. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
163. C₂²×C₄	C₂×C₄×D₅₃	semidirect
166. C₄⋊C₄	C₂³×D₅₃	semidirect
174. C₂³	#13133179	direct (N × H)
186. C₂×C₄	C₂²×C₄×D₅₃	semidirect
193. C₂²	C₂×#13133179	direct (N × H)
201. C₄	C₂³×C₄×D₅₃	semidirect
203. C₂	C₂²×#13133179	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×D₂₁₂	2
C₂⁴×C₄×D₅₃	2
C₂²×#13133179	2
C₂³×#11719921	2
C₂³×C₄×Dic₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×C₄×D₄	53

Subgroups — C23×#13133179 #28290656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133179 #28290656