Subgroups — C₂³×#13133129 #28247518

Order 13568.

7646 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×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133129	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₅₃	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×D₅₃	—	2	8	no	—
8	C₂³×C₄×D₅₃	—	2	8	no	—
9	C₂³×C₄×D₅₃	C₄	1	14	yes	—
10	C₂³×C₄×D₅₃	C₂²	1	14	yes	—
11	C₂³×#88322	—	2	8	no	—
12	C₂³×#88322	C₂²	1	14	yes	complemented
13	C₂×#13133129	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	—	2	8	no	—
16	C₂³×C₈×C₅₃	C₂²	1	14	yes	complemented
17	C₂⁴×D₅₃	—	2	8	no	—
18	C₂⁴×D₅₃	—	2	8	no	—
19	C₂⁴×D₅₃	C₈	1	16	yes	complemented
20	C₂⁴×D₅₃	C₂×C₄	1	14	yes	—
21	C₂³×Dic₅₃	—	2	8	no	—
22	C₂³×Dic₅₃	C₂×C₄	1	14	yes	—
23	C₂²×C₄×D₅₃	—	2	112	no	—
24	C₂²×C₄×D₅₃	—	2	56	no	—
25	C₂²×C₄×D₅₃	—	2	56	no	—
26	C₂²×C₄×D₅₃	C₂×C₄	1	84	yes	—
27	C₂²×C₄×D₅₃	C₂³	1	28	yes	—
28	C₂²×#88322	—	2	56	no	—
29	C₂²×#88322	C₂³	1	28	yes	complemented
30	#13133129	C₂³	1	64	yes	direct product factor
31	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
33	C₂³×C₄×C₅₃	D₄	1	8	yes	—
34	C₂³×C₄×C₅₃	D₄	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
36	C₂²×C₈×C₅₃	—	2	56	no	—
37	C₂²×C₈×C₅₃	C₂³	1	28	yes	complemented
38	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
39	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
40	C₂⁴×C₅₃	M₁₆	1	8	yes	—
41	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
42	C₂³×D₅₃	—	2	128	no	—
43	C₂³×D₅₃	—	2	56	no	—
44	C₂³×D₅₃	—	2	56	no	—
45	C₂³×D₅₃	C₂×C₈	1	112	yes	complemented
46	C₂³×D₅₃	C₂²×C₄	1	28	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	56	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
52	C₂²×Dic₅₃	—	2	56	no	—
53	C₂²×Dic₅₃	C₂²×C₄	1	28	yes	—
54	C₂×C₄×D₅₃	—	2	336	no	—
55	C₂×C₄×D₅₃	—	2	56	no	—
56	C₂×C₄×D₅₃	—	2	56	no	—
57	C₂×C₄×D₅₃	C₂²×C₄	1	56	yes	—
58	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
59	C₂×C₈×C₅₃	—	2	56	no	—
60	C₂×C₈×C₅₃	C₂⁴	1	8	yes	complemented
61	C₂×#88322	—	2	56	no	—
62	C₂×#88322	C₂⁴	1	8	yes	complemented
63	C₂³×C₅₃	C₂²⋊C₈	1	64	yes	complemented
64	C₂³×C₅₃	C₂×C₂²⋊C₄	1	28	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	28	yes	—
66	C₂³×C₅₃	C₂×M₁₆	1	28	yes	—
67	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
68	C₂²×D₅₃	—	2	56	no	—
69	C₂²×D₅₃	—	2	56	no	—
70	C₂²×D₅₃	—	2	448	no	—
71	C₂²×D₅₃	C₂²×C₈	1	112	yes	complemented
72	C₂²×D₅₃	C₂³×C₄	1	8	yes	—
73	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	84	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
76	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
77	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	56	no	—
79	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
80	C₄×D₅₃	—	2	112	no	—
81	C₄×D₅₃	—	2	8	no	—
82	C₄×D₅₃	—	2	8	no	—
83	C₈×C₅₃	—	2	8	no	—
84	#88322	—	2	8	no	—
85	C₂³×C₂²⋊C₈	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×C₂²⋊C₈	1	112	yes	complemented
87	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
88	C₂²×C₅₃	C₂³×C₈	1	14	yes	—
89	C₂²×C₅₃	C₂²×M₁₆	1	14	yes	—
90	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
91	C₂×D₅₃	—	2	224	no	—
92	C₂×D₅₃	—	2	8	no	—
93	C₂×D₅₃	—	2	8	no	—
94	C₂×D₅₃	C₂³×C₈	1	16	yes	complemented
95	C₄×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
96	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
97	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
98	Dic₅₃	—	2	8	no	—
99	C₂⁵×C₄	—	53	1	no	—
100	C₂⁴×C₈	—	53	1	no	—
101	C₂⁴×C₈	D₅₃	1	1	yes	characteristic, complemented
102	C₂²×C₂²⋊C₈	—	53	28	no	—
103	C₂×C₅₃	C₂²×C₂²⋊C₈	1	28	yes	complemented
104	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
106	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
107	D₅₃	—	2	16	no	—
108	C₂⁶	—	53	1	no	—
109	C₂⁴×C₄	—	53	1	no	—
110	C₂⁴×C₄	—	53	14	no	—
111	C₂⁴×C₄	—	53	14	no	—
112	C₂⁴×C₄	—	106	8	no	—
113	C₂⁴×C₄	—	106	8	no	—
114	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
115	C₂³×C₈	—	2	8	no	—
116	C₂³×C₈	—	53	14	no	—
117	C₂³×C₈	—	106	8	no	—
118	C₂³×C₈	C₂×D₅₃	1	14	yes	complemented
119	C₂×C₂²⋊C₈	—	53	112	no	—
120	C₅₃	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
121	C₂⁵	—	53	14	no	—
122	C₂⁵	—	53	16	no	—
123	C₂⁵	—	106	8	no	—
124	C₂⁵	—	106	8	no	—
125	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
126	C₂³×C₄	—	53	14	no	—
127	C₂³×C₄	—	53	84	no	—
128	C₂³×C₄	—	53	28	no	—
129	C₂³×C₄	—	106	8	no	—
130	C₂³×C₄	—	106	56	no	—
131	C₂³×C₄	—	106	56	no	—
132	C₂³×C₄	—	106	112	no	—
133	C₂³×C₄	C₄×D₅₃	1	7	yes	—
134	C₂³×C₄	D₂₁₂	1	8	yes	—
135	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
136	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
137	C₂²×C₈	—	2	56	no	—
138	C₂²×C₈	—	53	28	no	—
139	C₂²×C₈	—	106	56	no	—
140	C₂²×C₈	C₂²×D₅₃	1	28	yes	complemented
141	C₂²⋊C₈	—	53	64	no	—
142	C₂⁴	—	53	112	no	—
143	C₂⁴	—	53	28	no	—
144	C₂⁴	—	106	128	no	—
145	C₂⁴	—	106	56	no	—
146	C₂⁴	—	106	56	no	—
147	C₂⁴	C₈×D₅₃	1	8	yes	—
148	C₂⁴	#11719912	1	8	yes	—
149	C₂⁴	#11719921	1	8	yes	—
150	C₂⁴	C₂×C₄×D₅₃	1	7	yes	—
151	C₂²×C₄	—	53	28	no	—
152	C₂²×C₄	—	53	56	no	—
153	C₂²×C₄	—	53	8	no	—
154	C₂²×C₄	—	106	56	no	—
155	C₂²×C₄	—	106	336	no	—
156	C₂²×C₄	—	106	56	no	—
157	C₂²×C₄	—	106	56	no	—
158	C₂²×C₄	#11719921	1	56	yes	—
159	C₂²×C₄	C₂×C₄×D₅₃	1	21	yes	—
160	C₂²×C₄	C₂×D₂₁₂	1	28	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₅₃	1	28	yes	—
162	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
163	C₂×C₈	—	2	56	no	—
164	C₂×C₈	—	53	8	no	—
165	C₂×C₈	—	106	56	no	—
166	C₂×C₈	C₂³×D₅₃	1	8	yes	complemented
167	C₂³	—	53	112	no	—
168	C₂³	—	53	8	no	—
169	C₂³	—	106	56	no	—
170	C₂³	—	106	448	no	—
171	C₂³	—	106	56	no	—
172	C₂³	#13133129	1	64	yes	direct product factor
173	C₂³	C₂×#11719921	1	28	yes	—
174	C₂³	C₂×C₈×D₅₃	1	28	yes	—
175	C₂³	C₂×#11719912	1	28	yes	—
176	C₂³	C₂²×C₄×D₅₃	1	7	yes	—
177	C₂×C₄	—	53	8	no	—
178	C₂×C₄	—	106	56	no	—
179	C₂×C₄	—	106	112	no	—
180	C₂×C₄	—	106	8	no	—
181	C₂×C₄	—	106	8	no	—
182	C₂×C₄	C₂×#11719921	1	84	yes	—
183	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₅₃	1	14	yes	—
185	C₂×C₄	C₂²×D₂₁₂	1	14	yes	—
186	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
187	C₈	—	2	8	no	—
188	C₈	—	106	8	no	—
189	C₂²	—	53	16	no	—
190	C₂²	—	106	224	no	—
191	C₂²	—	106	8	no	—
192	C₂²	—	106	8	no	—
193	C₂²	C₂×#13133129	1	112	yes	direct product factor
194	C₂²	C₂²×#11719921	1	14	yes	—
195	C₂²	C₂²×C₈×D₅₃	1	14	yes	—
196	C₂²	C₂²×#11719912	1	14	yes	—
197	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
198	C₄	—	106	8	no	—
199	C₄	C₂²×#11719921	1	14	yes	—
200	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
202	C₂	—	106	16	no	—
203	C₂	C₂²×#13133129	1	28	yes	direct product factor
204	C₂	C₂³×#11719921	1	1	yes	characteristic
205	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
206	C₂	C₂³×#11719912	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂×C₄
prime radical	C₂⁵×D₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂³×C₂²⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂⁴×#88322	C₂	semidirect
3. C₂²×#13133129	C₂	direct (N × H)
4. C₂⁴×C₈×C₅₃	C₂	semidirect
12. C₂³×#88322	C₂²	semidirect
13. C₂×#13133129	C₂²	direct (N × H)
16. C₂³×C₈×C₅₃	C₂²	semidirect
19. C₂⁴×D₅₃	C₈	semidirect
29. C₂²×#88322	C₂³	semidirect
30. #13133129	C₂³	direct (N × H)
37. C₂²×C₈×C₅₃	C₂³	semidirect
45. C₂³×D₅₃	C₂×C₈	semidirect
60. C₂×C₈×C₅₃	C₂⁴	semidirect
62. C₂×#88322	C₂⁴	semidirect
63. C₂³×C₅₃	C₂²⋊C₈	semidirect
71. C₂²×D₅₃	C₂²×C₈	semidirect
86. C₂²×C₅₃	C₂×C₂²⋊C₈	semidirect
94. C₂×D₅₃	C₂³×C₈	semidirect
101. C₂⁴×C₈	D₅₃	semidirect
103. C₂×C₅₃	C₂²×C₂²⋊C₈	semidirect
118. C₂³×C₈	C₂×D₅₃	semidirect
120. C₅₃	C₂³×C₂²⋊C₈	semidirect
140. C₂²×C₈	C₂²×D₅₃	semidirect
166. C₂×C₈	C₂³×D₅₃	semidirect
172. C₂³	#13133129	direct (N × H)
193. C₂²	C₂×#13133129	direct (N × H)
203. C₂	C₂²×#13133129	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×C₄×D₅₃	2
C₂⁴×#88322	2
C₂²×#13133129	2
C₂⁴×C₈×C₅₃	2
C₂³×C₂²⋊C₈	53

Subgroups — C23×#13133129 #28247518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133129 #28247518