Subgroups — #28300053

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290430	C₂	1	2	yes	maximal
2	C₂×#88324⋊C₈	C₂	1	2	yes	maximal
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88324	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	#16811489	C₄	1	8	yes	—
9	#88324⋊C₈	C₄	1	8	yes	complemented
10	#16811464	C₄	1	4	yes	complemented
11	C₈×#88324	C₄	1	4	yes	complemented
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×#13133127	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	C₂×C₄	1	4	yes	—
20	C₂²×#88324	C₂×C₄	1	4	yes	—
21	C₄×#88324	C₂×C₄	1	4	yes	—
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	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
30	#13133128	—	2	2	no	—
31	#13133127	—	2	2	no	—
32	#13133127	—	2	2	no	—
33	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
34	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#97×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₄²	1	2	yes	—
43	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
44	C₂×#88324	—	2	8	no	—
45	C₂×#88324	—	2	8	no	—
46	C₂×#88324	C₄²	1	8	yes	—
47	C₂×#88324	C₄²	1	8	yes	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	C₄²	1	2	yes	—
51	C₂×C₈×C₅₃	C₄²	1	2	yes	—
52	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	C₄×Dic₅₃	—	2	1	no	—
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	—	2	8	no	—
59	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	C₄²	1	2	yes	—
65	C₂×#88322	C₄²	1	2	yes	—
66	C₂×#88322	M₁₆	1	2	yes	—
67	C₂×#88322	M₁₆	1	2	yes	—
68	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	2	4	no	—
79	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
80	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
81	C₂×Dic₅₃	#131	1	8	yes	—
82	#88324	—	2	8	no	—
83	#88324	—	2	8	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	#97	1	4	yes	—
86	#88322	—	2	2	no	—
87	#88322	#97	1	4	yes	—
88	#7844	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₄³	1	1	yes	characteristic
90	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₅₃	#405	1	4	yes	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
98	Dic₅₃	—	4	2	no	—
99	Dic₅₃	#405	1	4	yes	—
100	Dic₅₃	#405	1	4	yes	—
101	#1744	—	53	2	no	—
102	C₄×#97	—	53	1	no	—
103	C₂×#322	—	53	1	no	—
104	C₂×#322	—	53	2	no	—
105	C₂×C₈²	—	53	1	no	—
106	C₂×C₅₃	#1453	1	2	yes	—
107	C₂×C₅₃	#1453	1	2	yes	—
108	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
109	C₂×C₅₃	#1726	1	2	yes	—
110	C₂×#97	—	53	1	no	—
111	C₂×#97	—	53	1	no	—
112	C₂×#97	—	53	1	no	—
113	C₂×#97	#6157	1	1	yes	characteristic
114	C₂×C₄×C₈	—	53	2	no	—
115	C₂×C₄×C₈	—	53	2	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₄³	—	53	1	no	—
119	#336	—	53	8	no	—
120	#322	—	53	4	no	—
121	#322	—	53	8	no	—
122	C₈²	—	53	4	no	—
123	C₅₃	#7844	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	4	no	—
126	C₂²×C₈	—	53	4	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
129	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
130	C₂×C₄²	—	53	1	no	—
131	C₂×C₄²	—	53	4	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
134	#97	—	2	2	no	—
135	#97	—	106	2	no	—
136	#97	—	106	2	no	—
137	#97	—	106	2	no	—
138	C₄×C₈	—	53	4	no	—
139	C₄×C₈	—	53	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	—	53	2	no	—
143	C₄×C₈	—	53	2	no	—
144	C₄×C₈	—	53	4	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	C₄×#6157	1	2	yes	—
148	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	8	no	—
155	C₂×C₈	—	53	8	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	—	106	8	no	—
159	C₂×C₈	—	106	1	no	—
160	C₂×C₈	—	106	1	no	—
161	C₂×C₈	#11719936	1	2	yes	—
162	C₂×C₈	C₄×#6157	1	2	yes	—
163	C₂×C₈	C₄×#6157	1	2	yes	—
164	C₂×C₈	#11719940	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	53	1	no	—
168	C₄²	—	53	1	no	—
169	C₄²	—	53	2	no	—
170	C₄²	—	106	8	no	—
171	C₄²	—	106	1	no	—
172	C₄²	—	106	1	no	—
173	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	2	no	—
178	C₂×C₄	—	53	8	no	—
179	C₂×C₄	—	106	2	no	—
180	C₂×C₄	—	106	4	no	—
181	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
184	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
185	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	53	4	no	—
189	C₈	—	106	8	no	—
190	C₈	—	106	8	no	—
191	C₈	—	106	2	no	—
192	C₈	#13133272	1	4	yes	—
193	C₂²	C₄²×#6157	1	1	yes	characteristic
194	C₂²	C₄×#11719940	1	1	yes	characteristic
195	C₂²	C₄×#11719936	1	1	yes	characteristic
196	C₂²	#16811589	1	4	yes	—
197	C₄	—	4	2	no	—
198	C₄	—	53	4	no	—
199	C₄	—	53	4	no	—
200	C₄	—	212	2	no	—
201	C₄	C₂×#13133272	1	1	yes	characteristic
202	C₄	#16811582	1	1	yes	characteristic
203	C₄	C₄×#11719940	1	1	yes	characteristic
204	C₄	C₄×#11719936	1	1	yes	characteristic
205	C₂	#20289934	1	2	yes	—
206	C₂	#20289935	1	2	yes	—
207	C₂	C₄×#13133272	1	1	yes	characteristic
208	C₂	#20290373	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#7844
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
9. #88324⋊C₈	C₄	semidirect
10. #16811464	C₄	semidirect
11. C₈×#88324	C₄	semidirect
123. C₅₃	#7844	semidirect

Maximal subgroups

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

subgroup	index
#20290430	2
C₂×#88324⋊C₈	2
C₂×#16811464	2
C₂×C₈×#88324	2
#97×Dic₅₃	2
#7844	53