Subgroups — #28300040

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290430	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₈×#88324	C₂	1	2	yes	maximal
5	#20274459	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88324	C₄	1	4	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	2	yes	—
10	#16811489	C₄	1	4	yes	—
11	#88324⋊C₈	C₄	1	8	yes	complemented
12	C₈×#88324	C₄	1	8	yes	complemented
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810378	C₄	1	1	yes	characteristic
16	#16810377	C₄	1	2	yes	—
17	C₅₃×#336	C₄	1	1	yes	characteristic
18	C₂²×#88324	C₂×C₄	1	4	yes	—
19	C₂²×#88324	C₂×C₄	1	2	yes	—
20	C₂²×#88324	C₂×C₄	1	2	yes	—
21	C₄×#88324	C₈	1	16	yes	complemented
22	C₄×#88324	C₂×C₄	1	4	yes	—
23	C₄×#88324	C₂×C₄	1	4	yes	—
24	C₄×#88324	C₂×C₄	1	4	yes	—
25	C₄×#88324	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
32	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₄²	1	2	yes	—
38	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#88324	—	2	8	no	—
41	C₂×#88324	C₄²	1	8	yes	—
42	C₂×#88324	C₄²	1	4	yes	—
43	C₂×#88324	C₄²	1	4	yes	—
44	C₂×#88324	C₂×C₈	1	8	yes	—
45	C₂×#88324	C₂×C₈	1	8	yes	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₄²	1	4	yes	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	4	no	—
53	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
54	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	C₄²	1	4	yes	—
59	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
64	C₂×C₄×C₅₃	#131	1	2	yes	—
65	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
70	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
73	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
74	C₂×Dic₅₃	#131	1	4	yes	—
75	#88324	—	2	8	no	—
76	#88324	C₄×C₈	1	8	yes	—
77	#88324	C₄×C₈	1	8	yes	—
78	C₈×C₅₃	—	2	4	no	—
79	#88322	—	2	4	no	—
80	#7841	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄³	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
83	C₂²×C₅₃	#405	1	1	yes	characteristic
84	C₂²×C₅₃	#405	1	1	yes	characteristic
85	C₂²×C₅₃	#405	1	2	yes	—
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	#405	1	2	yes	—
88	C₄×C₅₃	#405	1	2	yes	—
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
91	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
92	Dic₅₃	#405	1	4	yes	—
93	#1744	—	53	1	no	—
94	#1744	—	53	1	no	—
95	C₄²×C₈	—	53	1	no	—
96	C₂×#322	—	53	2	no	—
97	C₂×C₈²	—	53	2	no	—
98	C₂×C₅₃	#1450	1	2	yes	—
99	C₂×C₅₃	#1450	1	2	yes	—
100	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
101	C₂×C₅₃	#1726	1	1	yes	characteristic
102	C₂×C₅₃	#1726	1	1	yes	characteristic
103	C₂×C₄×C₈	—	53	4	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	2	no	—
108	C₄³	—	53	1	no	—
109	#336	—	53	4	no	—
110	#336	—	53	2	no	—
111	#336	—	53	1	no	—
112	#336	#6157	1	1	yes	characteristic
113	#322	—	53	8	no	—
114	C₈²	—	53	8	no	—
115	C₅₃	#7841	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	—	53	4	no	—
119	C₂²×C₈	—	53	2	no	—
120	C₂²×C₈	C₂×#6157	1	2	yes	—
121	C₂×C₄²	—	53	2	no	—
122	C₂×C₄²	—	53	1	no	—
123	C₂×C₄²	—	53	2	no	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
126	C₄×C₈	—	53	4	no	—
127	C₄×C₈	—	53	4	no	—
128	C₄×C₈	—	53	4	no	—
129	C₄×C₈	—	53	4	no	—
130	C₄×C₈	—	53	16	no	—
131	C₄×C₈	—	53	4	no	—
132	C₄×C₈	—	53	4	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	C₄×#6157	1	2	yes	—
137	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	53	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	53	8	no	—
143	C₂×C₈	—	53	8	no	—
144	C₂×C₈	—	53	8	no	—
145	C₂×C₈	—	53	4	no	—
146	C₂×C₈	—	106	8	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	—	106	2	no	—
149	C₂×C₈	C₄×#6157	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	53	8	no	—
152	C₄²	—	53	2	no	—
153	C₄²	—	53	2	no	—
154	C₄²	—	106	4	no	—
155	C₄²	—	106	1	no	—
156	C₄²	—	106	1	no	—
157	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	—	53	2	no	—
163	C₂×C₄	—	53	4	no	—
164	C₂×C₄	—	53	4	no	—
165	C₂×C₄	—	53	4	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	106	2	no	—
168	C₂×C₄	C₈×#6157	1	4	yes	—
169	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
170	C₂×C₄	#13133304	1	1	yes	characteristic
171	C₂×C₄	#13133304	1	2	yes	—
172	C₂×C₄	#13133305	1	1	yes	characteristic
173	C₈	—	2	4	no	—
174	C₈	—	53	8	no	—
175	C₈	—	53	8	no	—
176	C₈	—	106	8	no	—
177	C₈	—	106	4	no	—
178	C₂²	C₄²×#6157	1	1	yes	characteristic
179	C₂²	C₂×C₈×#6157	1	2	yes	—
180	C₂²	#16811587	1	1	yes	characteristic
181	C₂²	#16811588	1	1	yes	characteristic
182	C₂²	#16811589	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	53	2	no	—
185	C₄	—	53	2	no	—
186	C₄	—	53	4	no	—
187	C₄	—	106	4	no	—
188	C₄	#16811587	1	2	yes	—
189	C₄	#16811588	1	2	yes	—
190	C₂	#20289920	1	2	yes	—
191	C₂	#20289921	1	2	yes	—
192	C₂	C₄×C₈×#6157	1	1	yes	characteristic
193	C₂	#20290372	1	1	yes	characteristic
194	C₂	#20290373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₅₃×#336	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#336	#6157
hypocenter	C₅₃	#7841
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
11. #88324⋊C₈	C₄	semidirect
12. C₈×#88324	C₄	semidirect
21. C₄×#88324	C₈	semidirect
115. C₅₃	#7841	semidirect

Maximal subgroups

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

subgroup	index
#20290430	2
C₄²×#88324	2
C₂×#88324⋊C₈	2
C₂×C₈×#88324	2
#20274459	2
#7841	53