Subgroups — #27299409

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810443	C₂	1	2	yes	maximal
2	M₃₂×Dic₅₃	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	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	#16810443	C₄	1	8	yes	complemented
14	#16810398	—	2	2	no	—
15	C₁₆×Dic₅₃	—	2	2	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	—	2	2	no	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	#13133157	—	2	2	no	—
24	#13133157	C₂×C₄	1	4	yes	—
25	C₂×#11719908	—	2	2	no	—
26	C₄×#88322	C₈	1	4	yes	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	C₈	1	4	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
30	C₈×Dic₅₃	D₄	1	1	yes	characteristic
31	C₈×Dic₅₃	D₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₅₃	—	2	2	no	—
38	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	1	no	—
44	C₂²×Dic₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	2	yes	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄×Dic₅₃	—	2	8	no	—
55	C₄×Dic₅₃	—	2	1	no	—
56	C₄×Dic₅₃	C₄²	1	2	yes	—
57	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
60	C₂×#88322	C₄²	1	2	yes	—
61	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₂×#88322	C₂×C₈	1	2	yes	—
65	C₂×#88322	C₂×C₈	1	2	yes	—
66	C₂×#88322	M₁₆	1	2	yes	—
67	C₁₆×C₅₃	—	2	4	no	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	4	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	2	4	no	—
81	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
82	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₄×D₄	1	2	yes	—
85	C₈×C₅₃	C₄×Q₈	1	2	yes	—
86	#88322	C₄×C₈	1	4	yes	—
87	#88322	C₄⋊C₈	1	2	yes	—
88	#88322	C₄⋊C₈	1	2	yes	—
89	C₄×#364	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₃	#431	1	1	yes	characteristic
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₅₃	#431	1	1	yes	characteristic
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	—	2	2	no	—
101	Dic₅₃	#364	1	4	yes	complemented
102	C₂×#364	—	53	2	no	—
103	C₄×M₃₂	—	53	2	no	—
104	C₄²×C₈	—	53	1	no	—
105	#1376	—	53	1	no	—
106	#1376	D₅₃	1	1	yes	characteristic
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#364	1	2	yes	—
111	C₂×M₃₂	—	53	2	no	—
112	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄³	—	53	1	no	—
120	#364	—	53	8	no	—
121	#346	—	106	2	no	—
122	C₄×C₁₆	—	106	2	no	—
123	C₅₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄²	—	53	1	no	—
129	C₂×C₄²	—	53	1	no	—
130	C₂×C₄²	—	106	2	no	—
131	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	53	4	no	—
134	M₃₂	—	106	2	no	—
135	M₃₂	C₄×D₅₃	1	4	yes	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	53	4	no	—
140	C₄×C₈	—	53	1	no	—
141	C₄×C₈	—	53	1	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	53	4	no	—
145	C₄×C₈	—	53	4	no	—
146	C₄×C₈	—	106	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	—	106	1	no	—
150	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	53	2	no	—
160	C₂×C₈	C₈×D₅₃	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
162	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
163	C₂×C₈	#11719947	1	2	yes	—
164	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
165	C₄²	—	2	1	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	1	no	—
168	C₄²	—	53	2	no	—
169	C₄²	—	53	2	no	—
170	C₄²	—	106	8	no	—
171	C₄²	—	106	1	no	—
172	C₄²	C₈×D₅₃	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	106	4	no	—
175	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	106	4	no	—
181	C₂×C₄	—	106	4	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
184	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133173	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	53	4	no	—
191	C₈	—	53	2	no	—
192	C₈	—	53	2	no	—
193	C₈	#13133177	1	2	yes	—
194	C₈	#13133182	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	#16810575	1	1	yes	characteristic
197	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
198	C₂²	#16810673	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	53	4	no	—
201	C₄	—	106	4	no	—
202	C₄	—	106	2	no	—
203	C₄	#16810474	1	1	yes	characteristic
204	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
205	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
206	C₄	#16810673	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#20274563	1	1	yes	characteristic
210	C₂	#364×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	C₄×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
13. #16810443	C₄	semidirect
101. Dic₅₃	#364	semidirect
123. C₅₃	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810443	2
M₃₂×Dic₅₃	2
C₄×C₈×Dic₅₃	2
#20272797	2
C₅₃×#1376	2
C₄×#364	53