Subgroups — #28310418

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#6157	C₂	1	1	yes	maximal, characteristic
3	(C₂×#6157)⋊C₁₆	C₂	1	4	yes	maximal
4	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
6	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×#6157	C₄	1	2	yes	—
8	C₂×C₈×#6157	C₂²	1	2	yes	—
9	C₄²×#6157	C₄	1	1	yes	characteristic
10	#16811509	C₄	1	2	yes	complemented
11	C₄×#11719910	C₄	1	2	yes	complemented
12	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
13	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₄×#11719908	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×#6157	C₈	1	2	yes	—
19	C₂×C₄×#6157	C₈	1	2	yes	—
20	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
21	D₅₃⋊C₁₆	—	2	4	no	—
22	C₂×#11719910	C₂×C₄	1	4	yes	—
23	C₄×#88324	C₈	1	2	yes	—
24	C₄×#88324	C₈	1	2	yes	—
25	C₈×#6157	—	2	4	no	—
26	C₈×#6157	—	2	4	no	—
27	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
29	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	C₂×C₄	1	2	yes	—
31	C₁₆×D₅₃	—	2	4	no	—
32	C₄×#88322	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×#6157	C₂×C₈	1	4	yes	—
38	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×#88324	C₂×C₈	1	4	yes	—
42	C₄×#6157	—	2	4	no	—
43	C₄×#6157	—	2	2	no	—
44	C₄×#6157	—	2	2	no	—
45	C₄×#6157	C₁₆	1	8	yes	complemented
46	D₅₃⋊C₈	—	2	2	no	—
47	D₅₃⋊C₈	—	2	2	no	—
48	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₄²	1	1	yes	characteristic
51	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
52	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂×#88322	C₄²	1	1	yes	characteristic
55	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
56	C₈×D₅₃	M₁₆	1	4	yes	—
57	C₈×D₅₃	#41	1	4	yes	—
58	#11719910	—	2	4	no	—
59	C₁₆×C₅₃	—	2	2	no	—
60	#11719908	—	2	2	no	—
61	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×#6157	—	2	4	no	—
63	C₂×#6157	—	2	4	no	—
64	C₂×#6157	C₂×C₁₆	1	8	yes	—
65	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
69	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
70	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
72	C₄×D₅₃	M₃₂	1	4	yes	—
73	C₄×D₅₃	#117	1	2	yes	—
74	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
75	#88324	—	2	4	no	—
76	C₈×C₅₃	#97	1	2	yes	—
77	C₈×C₅₃	#117	1	2	yes	—
78	#88322	#97	1	2	yes	—
79	#88322	#117	1	2	yes	—
80	#12590	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂×D₅₃	#431	1	2	yes	—
83	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
84	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
85	#6157	—	2	8	no	—
86	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
87	C₄×C₅₃	#346	1	2	yes	—
88	C₄×C₅₃	#376	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
90	Dic₅₃	C₄×C₁₆	1	2	yes	—
91	Dic₅₃	#346	1	2	yes	—
92	Dic₅₃	#431	1	2	yes	—
93	C₂×C₄×C₁₆	—	53	1	no	—
94	C₂×C₄×C₁₆	—	53	1	no	—
95	C₄²×C₈	—	53	1	no	—
96	#1374	—	53	4	no	—
97	C₂×C₅₃	#1744	1	1	yes	characteristic
98	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
100	D₅₃	#2162	1	4	yes	—
101	C₂²×C₁₆	—	53	2	no	—
102	C₂²×C₁₆	—	53	2	no	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₄³	—	53	1	no	—
108	C₄×C₁₆	—	53	2	no	—
109	C₄×C₁₆	—	53	2	no	—
110	C₄×C₁₆	—	53	2	no	—
111	C₄×C₁₆	—	53	1	no	—
112	C₄×C₁₆	#6157	1	1	yes	characteristic, complemented
113	C₅₃	#12590	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂×C₄²	—	53	2	no	—
118	C₂×C₄²	—	53	1	no	—
119	C₂×C₄²	—	53	2	no	—
120	C₂×C₄²	—	53	2	no	—
121	C₂×C₁₆	—	53	4	no	—
122	C₂×C₁₆	—	53	2	no	—
123	C₂×C₁₆	—	106	4	no	—
124	C₂×C₁₆	—	106	4	no	—
125	C₂×C₁₆	C₂×#6157	1	2	yes	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	1	no	—
129	C₄×C₈	—	53	1	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	106	4	no	—
132	C₄×C₈	—	106	4	no	—
133	C₄×C₈	C₂×#6157	1	1	yes	characteristic
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	53	4	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	1	no	—
140	C₂×C₈	—	53	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	53	4	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	C₄×#6157	1	1	yes	characteristic
146	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
147	C₄²	—	53	8	no	—
148	C₄²	—	53	1	no	—
149	C₄²	—	53	1	no	—
150	C₄²	—	53	1	no	—
151	C₄²	—	106	2	no	—
152	C₄²	—	106	2	no	—
153	C₄²	—	106	4	no	—
154	C₄²	C₄×#6157	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	—	106	4	no	—
157	C₁₆	—	106	2	no	—
158	C₂³	—	53	1	no	—
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	—	53	2	no	—
163	C₂×C₄	—	53	2	no	—
164	C₂×C₄	—	53	4	no	—
165	C₂×C₄	—	53	4	no	—
166	C₂×C₄	—	53	8	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	106	4	no	—
169	C₂×C₄	C₈×#6157	1	1	yes	characteristic
170	C₂×C₄	C₈×#6157	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
172	C₈	—	53	2	no	—
173	C₈	—	53	2	no	—
174	C₈	—	106	4	no	—
175	C₈	#13133271	1	2	yes	—
176	C₈	#13133296	1	2	yes	—
177	C₂²	—	53	2	no	—
178	C₂²	—	53	2	no	—
179	C₂²	—	53	2	no	—
180	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
181	C₄	—	53	2	no	—
182	C₄	—	53	2	no	—
183	C₄	—	53	2	no	—
184	C₄	—	106	8	no	—
185	C₄	C₁₆×#6157	1	2	yes	—
186	C₄	#16811510	1	2	yes	—
187	C₄	#16811559	1	1	yes	characteristic
188	C₄	C₂×#13133271	1	1	yes	characteristic
189	C₂	—	53	4	no	—
190	C₂	#20290428	1	1	yes	characteristic
191	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic
192	C₂	C₂×#16811510	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#12590
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16811509	C₄	semidirect
11. C₄×#11719910	C₄	semidirect
14. C₄×#11719908	C₄	semidirect
15. C₁₆×Dic₅₃	C₄	semidirect
16. C₄×C₁₆×C₅₃	C₄	semidirect
45. C₄×#6157	C₁₆	semidirect
112. C₄×C₁₆	#6157	semidirect
113. C₅₃	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₅₃⋊C₁₆	2
C₄×C₈×#6157	2
(C₂×#6157)⋊C₁₆	2
C₄×C₁₆×D₅₃	2
#12590	53