Subgroups — #27275341

Order 13568.

430 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	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₃	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	C₄	1	2	yes	—
11	#16810377	—	2	2	no	—
12	#16810376	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	C₄	1	4	yes	complemented
17	C₂²×#88322	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₃	—	2	2	no	—
20	#13133149	C₈	1	8	yes	complemented
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	C₈×Dic₅₃	—	4	4	no	—
24	#13133124	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₄²	1	2	yes	—
46	C₂×C₈×C₅₃	C₄²	1	2	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₅₃	#41	1	1	yes	characteristic
53	C₄⋊Dic₅₃	—	2	4	no	—
54	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
55	C₄×Dic₅₃	—	4	2	no	—
56	C₄×Dic₅₃	—	4	2	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	#131	1	2	yes	—
70	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₅₃	#137	1	2	yes	—
72	C₂×Dic₅₃	—	2	8	no	—
73	C₂×Dic₅₃	—	4	4	no	—
74	C₂×Dic₅₃	—	4	4	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	—	4	2	no	—
78	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
79	#88322	—	4	4	no	—
80	#8285	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×D₄	1	2	yes	—
83	C₂²×C₅₃	#440	1	2	yes	—
84	C₂²×C₅₃	#444	1	2	yes	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₅₃	#405	1	1	yes	characteristic
89	C₄×C₅₃	#428	1	2	yes	—
90	C₄×C₅₃	#441	1	1	yes	characteristic
91	C₄×C₅₃	#442	1	1	yes	characteristic
92	C₄×C₅₃	C₈×Q₈	1	2	yes	—
93	Dic₅₃	—	4	8	no	—
94	C₄×#106	—	53	1	no	—
95	C₈×C₄⋊C₄	—	53	2	no	—
96	C₂×#335	—	53	1	no	—
97	C₂×#322	D₅₃	1	1	yes	characteristic
98	#1279	—	53	2	no	—
99	C₂×C₅₃	SD₁₆⋊C₈	1	4	yes	—
100	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₃	#1773	1	1	yes	characteristic
102	C₂×C₅₃	#1776	1	1	yes	characteristic
103	C₂×#106	—	53	2	no	—
104	C₂×C₄⋊C₈	—	53	2	no	—
105	C₂×C₄×C₈	—	106	2	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	53	2	no	—
110	#340	—	53	2	no	—
111	#336	—	106	2	no	—
112	#335	—	53	4	no	—
113	#322	Dic₅₃	1	4	yes	complemented
114	C₅₃	#8285	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	106	2	no	—
117	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	D₂₁₂	1	2	yes	—
120	C₂×C₄⋊C₄	—	53	4	no	—
121	C₂×C₄²	—	106	2	no	—
122	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	#106	—	53	8	no	—
124	C₄⋊C₈	—	53	4	no	—
125	C₄⋊C₈	—	106	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	212	4	no	—
128	C₄×C₈	Dic₁₀₆	1	2	yes	—
129	C₄×C₈	C₄×D₅₃	1	2	yes	—
130	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
131	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
133	#95	—	106	2	no	—
134	C₂²×C₄	—	106	4	no	—
135	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂²×C₄	#11719945	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	C₈×D₅₃	1	4	yes	—
147	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
148	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
150	C₄⋊C₄	—	53	8	no	—
151	C₄⋊C₄	—	106	4	no	—
152	C₄²	—	212	2	no	—
153	C₄²	—	212	2	no	—
154	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
156	C₄²	#11719945	1	1	yes	characteristic
157	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	106	8	no	—
161	C₂×C₄	—	212	4	no	—
162	C₂×C₄	—	212	4	no	—
163	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133176	1	1	yes	characteristic
166	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
167	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
170	C₂×C₄	#13133223	1	2	yes	—
171	C₂×C₄	#13133245	1	2	yes	—
172	C₂×C₄	#13133250	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	212	4	no	—
177	C₈	C₈×Dic₅₃	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂²	C₈×D₂₁₂	1	2	yes	—
180	C₂²	#16810725	1	2	yes	—
181	C₂²	#16810746	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	212	8	no	—
185	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
186	C₄	#16810578	1	1	yes	characteristic
187	C₄	#16810665	1	2	yes	—
188	C₄	#16810731	1	1	yes	characteristic
189	C₄	#16810736	1	1	yes	characteristic
190	C₄	C₈×Dic₁₀₆	1	2	yes	—
191	C₂	#20273529	1	4	yes	—
192	C₂	C₈×C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂	#20274599	1	1	yes	characteristic
194	C₂	#20274612	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8285
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #16810376	C₄	semidirect
16. C₅₃×#322	C₄	semidirect
20. #13133149	C₈	semidirect
78. C₈×C₅₃	C₄×C₈	semidirect
113. #322	Dic₅₃	semidirect
114. C₅₃	#8285	semidirect
177. C₈	C₈×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133149	2
C₈×C₄⋊Dic₅₃	2
C₂×#16810376	2
#20272555	2
C₂×C₅₃×#322	2
#8285	53