Subgroups — C₂³×#13133138 #28247600

Order 13568.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133138	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	—	2	1	no	—
6	C₂³×#88322	—	2	8	no	—
7	C₂³×#88322	C₂²	1	14	yes	complemented
8	C₂²×C₄⋊Dic₅₃	—	2	7	no	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	14	yes	complemented
10	C₂×#13133138	C₂²	1	112	yes	direct product factor
11	C₂⁶×C₅₃	—	2	1	no	—
12	C₂³×D₄×C₅₃	—	2	7	no	—
13	C₂³×D₄×C₅₃	—	2	8	no	—
14	C₂³×D₄×C₅₃	C₄	1	16	yes	complemented
15	C₂³×D₄×C₅₃	C₂²	1	14	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×Dic₅₃	—	2	14	no	—
18	C₂³×Dic₅₃	—	4	8	no	—
19	C₂²×#88322	—	2	56	no	—
20	C₂²×#88322	C₂³	1	28	yes	complemented
21	C₂×C₄⋊Dic₅₃	—	2	42	no	—
22	C₂×C₄⋊Dic₅₃	C₂³	1	28	yes	complemented
23	#13133138	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₅₃	—	2	16	no	—
25	C₂⁵×C₅₃	—	2	14	no	—
26	C₂⁵×C₅₃	—	4	8	no	—
27	C₂⁵×C₅₃	—	4	8	no	—
28	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₅₃	—	2	112	no	—
30	C₂²×D₄×C₅₃	—	2	42	no	—
31	C₂²×D₄×C₅₃	—	2	56	no	—
32	C₂²×D₄×C₅₃	C₂×C₄	1	112	yes	complemented
33	C₂²×D₄×C₅₃	C₂³	1	28	yes	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₅₃	D₄	1	7	yes	—
36	C₂³×C₄×C₅₃	D₄	1	8	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
38	C₂⁴×C₅₃	—	2	112	no	—
39	C₂⁴×C₅₃	—	2	28	no	—
40	C₂⁴×C₅₃	—	4	128	no	—
41	C₂⁴×C₅₃	—	4	56	no	—
42	C₂⁴×C₅₃	—	4	56	no	—
43	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
44	C₂⁴×C₅₃	D₈	1	8	yes	—
45	C₂⁴×C₅₃	SD₁₆	1	8	yes	—
46	C₂⁴×C₅₃	C₂×D₄	1	7	yes	—
47	C₂×D₄×C₅₃	—	2	336	no	—
48	C₂×D₄×C₅₃	—	2	28	no	—
49	C₂×D₄×C₅₃	—	2	56	no	—
50	C₂×D₄×C₅₃	C₂²×C₄	1	112	yes	complemented
51	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	56	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
57	C₂²×Dic₅₃	—	2	28	no	—
58	C₂²×Dic₅₃	—	4	56	no	—
59	C₄⋊Dic₅₃	—	2	28	no	—
60	C₄⋊Dic₅₃	C₂⁴	1	8	yes	complemented
61	C₂×#88322	—	2	56	no	—
62	C₂×#88322	C₂⁴	1	8	yes	complemented
63	C₂³×C₅₃	—	2	112	no	—
64	C₂³×C₅₃	—	2	8	no	—
65	C₂³×C₅₃	—	4	56	no	—
66	C₂³×C₅₃	—	4	56	no	—
67	C₂³×C₅₃	—	4	448	no	—
68	C₂³×C₅₃	#102	1	64	yes	complemented
69	C₂³×C₅₃	C₂×C₂²⋊C₄	1	28	yes	—
70	C₂³×C₅₃	C₂×D₈	1	28	yes	—
71	C₂³×C₅₃	C₂×SD₁₆	1	28	yes	—
72	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
73	D₄×C₅₃	—	2	112	no	—
74	D₄×C₅₃	—	2	8	no	—
75	D₄×C₅₃	C₂³×C₄	1	16	yes	complemented
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	84	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
80	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	8	no	—
82	C₂×Dic₅₃	—	4	56	no	—
83	#88322	—	2	8	no	—
84	C₂³×#102	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	16	no	—
86	C₂²×C₅₃	—	4	224	no	—
87	C₂²×C₅₃	—	4	8	no	—
88	C₂²×C₅₃	—	4	8	no	—
89	C₂²×C₅₃	C₂×#102	1	112	yes	complemented
90	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
91	C₂²×C₅₃	C₂²×D₈	1	14	yes	—
92	C₂²×C₅₃	C₂²×SD₁₆	1	14	yes	—
93	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
94	C₄×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
95	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
97	Dic₅₃	—	4	8	no	—
98	C₂⁴×D₄	D₅₃	1	1	yes	characteristic
99	C₂⁴×C₈	—	53	1	no	—
100	C₂³×C₄⋊C₄	—	53	1	no	—
101	C₂²×#102	—	53	28	no	—
102	C₂×C₅₃	—	4	16	no	—
103	C₂×C₅₃	C₂²×#102	1	28	yes	complemented
104	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
106	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
107	C₂⁶	—	2	1	no	—
108	C₂³×D₄	—	2	8	no	—
109	C₂³×D₄	—	2	7	no	—
110	C₂³×D₄	Dic₅₃	1	16	yes	complemented
111	C₂³×D₄	C₂×D₅₃	1	14	yes	—
112	C₂⁴×C₄	—	106	1	no	—
113	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂³×C₈	—	53	14	no	—
115	C₂³×C₈	—	106	8	no	—
116	C₂²×C₄⋊C₄	—	53	14	no	—
117	C₂²×C₄⋊C₄	—	106	7	no	—
118	C₂×#102	—	53	112	no	—
119	C₅₃	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
120	C₂⁵	—	2	14	no	—
121	C₂⁵	—	2	16	no	—
122	C₂⁵	—	4	8	no	—
123	C₂⁵	—	4	8	no	—
124	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×D₄	—	2	42	no	—
126	C₂²×D₄	—	2	56	no	—
127	C₂²×D₄	—	2	112	no	—
128	C₂²×D₄	C₂×Dic₅₃	1	112	yes	complemented
129	C₂²×D₄	C₂²×D₅₃	1	28	yes	—
130	C₂³×C₄	—	106	14	no	—
131	C₂³×C₄	—	212	8	no	—
132	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
133	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
134	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
135	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
136	C₂²×C₈	—	53	28	no	—
137	C₂²×C₈	—	106	56	no	—
138	C₂×C₄⋊C₄	—	53	28	no	—
139	C₂×C₄⋊C₄	—	106	42	no	—
140	#102	—	53	64	no	—
141	C₂⁴	—	2	112	no	—
142	C₂⁴	—	2	28	no	—
143	C₂⁴	—	4	128	no	—
144	C₂⁴	—	4	56	no	—
145	C₂⁴	—	4	56	no	—
146	C₂⁴	#11719922	1	8	yes	—
147	C₂⁴	#11719923	1	8	yes	—
148	C₂⁴	C₂²⋊Dic₅₃	1	8	yes	—
149	C₂⁴	C₂×C₂²⋊D₅₃	1	7	yes	—
150	C₂×D₄	—	2	336	no	—
151	C₂×D₄	—	2	28	no	—
152	C₂×D₄	—	2	56	no	—
153	C₂×D₄	C₂²×Dic₅₃	1	112	yes	complemented
154	C₂×D₄	C₂³×D₅₃	1	8	yes	—
155	C₂²×C₄	—	106	28	no	—
156	C₂²×C₄	—	212	56	no	—
157	C₂²×C₄	C₂²⋊Dic₅₃	1	56	yes	—
158	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₅₃	1	28	yes	—
161	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
162	C₂×C₈	—	53	8	no	—
163	C₂×C₈	—	106	56	no	—
164	C₄⋊C₄	—	53	8	no	—
165	C₄⋊C₄	—	106	28	no	—
166	C₂³	—	2	112	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	448	no	—
169	C₂³	—	4	56	no	—
170	C₂³	—	4	56	no	—
171	C₂³	#13133138	1	64	yes	direct product factor
172	C₂³	C₂×C₂²⋊Dic₅₃	1	28	yes	—
173	C₂³	C₂×#11719922	1	28	yes	—
174	C₂³	C₂×#11719923	1	28	yes	—
175	C₂³	C₂²×C₂²⋊D₅₃	1	7	yes	—
176	D₄	—	2	112	no	—
177	D₄	—	2	8	no	—
178	D₄	C₂³×Dic₅₃	1	16	yes	complemented
179	C₂×C₄	—	106	8	no	—
180	C₂×C₄	—	212	56	no	—
181	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	84	yes	—
182	C₂×C₄	C₂³×Dic₅₃	1	14	yes	—
183	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₅₃	1	14	yes	—
185	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
186	C₈	—	106	8	no	—
187	C₂²	—	2	16	no	—
188	C₂²	—	4	224	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	4	8	no	—
191	C₂²	C₂×#13133138	1	112	yes	direct product factor
192	C₂²	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
193	C₂²	C₂²×#11719922	1	14	yes	—
194	C₂²	C₂²×#11719923	1	14	yes	—
195	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄	—	212	8	no	—
197	C₄	C₂²×C₂²⋊Dic₅₃	1	14	yes	—
198	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
199	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂	—	4	16	no	—
201	C₂	C₂²×#13133138	1	28	yes	direct product factor
202	C₂	C₂³×C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂	C₂³×#11719922	1	1	yes	characteristic
204	C₂	C₂³×#11719923	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting, prime radical	C₂⁴×D₄×C₅₃	C₂
socle	C₂⁵×C₅₃	D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×D₄	D₅₃
hypocenter	C₅₃	C₂³×#102
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#88322	C₂	semidirect
2. C₂³×C₄⋊Dic₅₃	C₂	semidirect
3. C₂²×#13133138	C₂	direct (N × H)
7. C₂³×#88322	C₂²	semidirect
9. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
10. C₂×#13133138	C₂²	direct (N × H)
14. C₂³×D₄×C₅₃	C₄	semidirect
20. C₂²×#88322	C₂³	semidirect
22. C₂×C₄⋊Dic₅₃	C₂³	semidirect
23. #13133138	C₂³	direct (N × H)
32. C₂²×D₄×C₅₃	C₂×C₄	semidirect
50. C₂×D₄×C₅₃	C₂²×C₄	semidirect
60. C₄⋊Dic₅₃	C₂⁴	semidirect
62. C₂×#88322	C₂⁴	semidirect
68. C₂³×C₅₃	#102	semidirect
75. D₄×C₅₃	C₂³×C₄	semidirect
89. C₂²×C₅₃	C₂×#102	semidirect
103. C₂×C₅₃	C₂²×#102	semidirect
110. C₂³×D₄	Dic₅₃	semidirect
119. C₅₃	C₂³×#102	semidirect
128. C₂²×D₄	C₂×Dic₅₃	semidirect
153. C₂×D₄	C₂²×Dic₅₃	semidirect
171. C₂³	#13133138	direct (N × H)
178. D₄	C₂³×Dic₅₃	semidirect
191. C₂²	C₂×#13133138	direct (N × H)
201. C₂	C₂²×#13133138	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×#88322	2
C₂³×C₄⋊Dic₅₃	2
C₂²×#13133138	2
C₂⁴×D₄×C₅₃	2
C₂³×#102	53

Subgroups — C23×#13133138 #28247600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133138 #28247600