Subgroups — #28310424

Order 13568.

390 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	C₂×#16811512	C₂	1	2	yes	maximal
2	#20289754	C₂	1	2	yes	maximal
3	#20289753	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	#16811512	C₄	1	8	yes	complemented
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	4	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	C₂×C₄	1	8	yes	—
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#88322	—	2	2	no	—
24	C₈×Dic₅₃	—	2	2	no	—
25	C₈×Dic₅₃	—	2	2	no	—
26	C₈×Dic₅₃	C₈	1	4	yes	—
27	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₄²×C₅₃	—	2	1	no	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₄×Dic₅₃	—	2	4	no	—
49	C₄×Dic₅₃	C₄²	1	2	yes	—
50	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	C₂×C₈	1	2	yes	—
54	C₂×#88322	C₂²×C₄	1	2	yes	—
55	C₂×#88322	#41	1	2	yes	—
56	C₂×#88322	#41	1	2	yes	—
57	#11719910	—	2	8	no	—
58	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
73	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
74	C₂×Dic₅₃	M₃₂	1	4	yes	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	#97	1	4	yes	—
77	#88322	—	2	2	no	—
78	#88322	#117	1	4	yes	—
79	#12591	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
81	C₂²×C₅₃	#431	1	1	yes	characteristic
82	C₂²×C₅₃	#431	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
84	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	#376	1	1	yes	characteristic
87	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
88	C₄×C₅₃	#431	1	1	yes	characteristic
89	C₄×C₅₃	#431	1	1	yes	characteristic
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	—	2	2	no	—
92	Dic₅₃	—	2	4	no	—
93	C₂×#346	—	53	2	no	—
94	C₄²×C₈	—	53	1	no	—
95	#1374	—	53	2	no	—
96	#1374	—	53	2	no	—
97	C₂×C₅₃	#1744	1	1	yes	characteristic
98	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
99	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
100	C₂×C₅₃	#2163	1	2	yes	—
101	C₂×C₅₃	#2163	1	2	yes	—
102	C₂²×C₁₆	—	53	4	no	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	#6157	1	1	yes	characteristic
109	C₄³	—	53	1	no	—
110	#346	—	53	8	no	—
111	C₅₃	#12591	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	53	1	no	—
113	C₂²×C₈	—	53	1	no	—
114	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
115	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
116	C₂×C₄²	—	53	1	no	—
117	C₂×C₄²	—	53	1	no	—
118	C₂×C₄²	—	53	2	no	—
119	C₂×C₄²	—	53	2	no	—
120	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
121	C₂×C₁₆	—	53	8	no	—
122	C₂×C₁₆	—	106	4	no	—
123	C₂×C₁₆	—	106	4	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	53	4	no	—
126	C₄×C₈	—	53	4	no	—
127	C₄×C₈	—	106	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₂²×C₄	—	53	1	no	—
131	C₂²×C₄	—	53	1	no	—
132	C₂²×C₄	—	53	2	no	—
133	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
134	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	53	2	no	—
138	C₂×C₈	—	53	2	no	—
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	106	1	no	—
142	C₂×C₈	—	106	1	no	—
143	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
144	C₂×C₈	#11719936	1	2	yes	—
145	C₂×C₈	C₄×#6157	1	2	yes	—
146	C₂×C₈	#11719940	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	53	2	no	—
150	C₄²	—	53	2	no	—
151	C₄²	—	106	4	no	—
152	C₄²	—	106	4	no	—
153	C₄²	—	106	1	no	—
154	C₄²	—	106	1	no	—
155	C₁₆	—	106	8	no	—
156	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	53	2	no	—
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	53	4	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	2	no	—
164	C₂×C₄	—	106	2	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	#13133292	1	4	yes	—
167	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
168	C₂×C₄	#13133296	1	1	yes	characteristic
169	C₂×C₄	#13133296	1	1	yes	characteristic
170	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	53	4	no	—
175	C₈	—	106	2	no	—
176	C₈	#13133272	1	4	yes	—
177	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
178	C₂²	#16811640	1	1	yes	characteristic
179	C₂²	#16811642	1	1	yes	characteristic
180	C₂²	D₅₃⋊M₃₂	1	2	yes	—
181	C₂²	C₂×#13133292	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	106	4	no	—
184	C₄	—	106	2	no	—
185	C₄	—	106	2	no	—
186	C₄	#16811559	1	1	yes	characteristic
187	C₄	C₂×#13133272	1	1	yes	characteristic
188	C₄	#16811640	1	1	yes	characteristic
189	C₄	#16811642	1	1	yes	characteristic
190	C₂	#20290430	1	1	yes	characteristic
191	C₂	C₄×#13133292	1	1	yes	characteristic
192	C₂	#20290922	1	1	yes	characteristic
193	C₂	#20291077	1	2	yes	—
194	C₂	#20291079	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12591
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #16811512	C₄	semidirect
111. C₅₃	#12591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811512	2
#20289754	2
#20289753	2
C₄×C₈×Dic₅₃	2
#12591	53