Subgroups — #27275283

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274570	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272549	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133127	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
11	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	—	2	2	no	—
13	#16810374	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#334	C₄	1	4	yes	complemented
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	C₂×C₄	1	4	yes	—
25	#13133127	—	4	4	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₈×Dic₅₃	—	2	4	no	—
29	C₈×Dic₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	C₄²	1	2	yes	—
48	C₂×C₈×C₅₃	C₄²	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	#41	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	2	no	—
57	C₄×Dic₅₃	—	2	4	no	—
58	C₄×Dic₅₃	SD₁₆	1	4	yes	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	M₁₆	1	4	yes	—
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#135	1	2	yes	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	#106	1	4	yes	—
82	C₂×Dic₅₃	#108	1	4	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	4	4	no	—
85	C₈×C₅₃	#97	1	4	yes	complemented
86	#88322	—	2	4	no	—
87	#88322	—	4	4	no	—
88	#8273	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	#423	1	2	yes	—
91	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
92	C₂²×C₅₃	#427	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×#108	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
97	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
98	C₄×C₅₃	#432	1	2	yes	—
99	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
100	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
101	Dic₅₃	—	2	4	no	—
102	Dic₅₃	—	2	4	no	—
103	#1767	—	53	2	no	—
104	C₄²×C₈	—	53	1	no	—
105	C₂×#334	—	53	1	no	—
106	C₂×#334	D₅₃	1	1	yes	characteristic
107	#1277	—	53	2	no	—
108	C₂×C₅₃	#1566	1	2	yes	—
109	C₂×C₅₃	#1568	1	2	yes	—
110	C₂×C₅₃	#1767	1	1	yes	characteristic
111	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
112	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	53	2	no	—
114	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
115	C₂×#97	—	106	2	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₄³	—	53	1	no	—
121	#336	—	106	2	no	—
122	#334	—	53	4	no	—
123	#334	Dic₅₃	1	4	yes	complemented
124	C₅₃	#8273	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	106	2	no	—
128	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄²	—	53	1	no	—
131	C₂×C₄²	—	53	2	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	2	no	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	53	4	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
139	#97	—	212	4	no	—
140	C₄×C₈	—	106	4	no	—
141	C₄×C₈	—	106	4	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	C₄×D₅₃	1	2	yes	—
145	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	53	4	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	#11719912	1	4	yes	—
162	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
163	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
164	C₄²	—	53	4	no	—
165	C₄²	—	106	4	no	—
166	C₄²	—	106	2	no	—
167	C₄²	—	106	2	no	—
168	C₄²	—	106	2	no	—
169	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₄²	#11719947	1	1	yes	characteristic
171	C₄²	#11719949	1	1	yes	characteristic
172	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	53	4	no	—
176	C₂×C₄	—	53	4	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	—	106	2	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	#13133175	1	1	yes	characteristic
183	C₂×C₄	#13133179	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133182	1	1	yes	characteristic
186	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
188	C₂×C₄	C₂×#11719912	1	2	yes	—
189	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
190	C₂×C₄	#13133238	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	—	106	4	no	—
194	C₈	—	212	4	no	—
195	C₈	#13133127	1	4	yes	complemented
196	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
197	C₂²	#16810645	1	2	yes	—
198	C₂²	#106×D₅₃	1	1	yes	characteristic
199	C₂²	#16810658	1	1	yes	characteristic
200	C₂²	#108×D₅₃	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	106	4	no	—
204	C₄	—	106	4	no	—
205	C₄	C₂×#13133127	1	1	yes	characteristic
206	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
207	C₄	#16810679	1	2	yes	—
208	C₄	#16810710	1	2	yes	—
209	C₄	SD₁₆×Dic₅₃	1	2	yes	—
210	C₂	#20273469	1	2	yes	—
211	C₂	#20273475	1	2	yes	—
212	C₂	#20274570	1	1	yes	characteristic
213	C₂	#106×Dic₅₃	1	1	yes	characteristic
214	C₂	#108×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#8273
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #16810374	C₄	semidirect
16. C₅₃×#334	C₄	semidirect
85. C₈×C₅₃	#97	semidirect
123. #334	Dic₅₃	semidirect
124. C₅₃	#8273	semidirect
195. C₈	#13133127	semidirect

Maximal subgroups

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

subgroup	index
#20274570	2
C₄×C₈×Dic₅₃	2
C₂×#16810374	2
#20272549	2
C₂×C₅₃×#334	2
#8273	53