Subgroups — #27302244

Order 13568.

280 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	#20276488	C₂	1	2	yes	maximal
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	1	yes	maximal, characteristic
4	#20272827	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	C₂×#13133153	C₂²	1	1	yes	characteristic
9	#16810661	C₂²	1	1	yes	characteristic
10	#16810647	—	2	1	no	—
11	#16810578	—	2	1	no	—
12	#16810446	—	2	4	no	—
13	#16810383	—	2	1	no	—
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#364	C₄	1	2	yes	complemented
17	C₅₃×#364	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	1	no	—
19	C₂×#11719917	—	2	2	no	—
20	#13133176	—	2	1	no	—
21	#13133153	—	2	1	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133153	—	4	1	no	—
25	#13133153	—	4	1	no	—
26	#13133153	Q₈	1	2	yes	—
27	#13133151	—	2	1	no	—
28	#13133149	D₄	1	2	yes	—
29	C₄⋊#88322	—	4	2	no	—
30	#13133127	—	4	1	no	—
31	C₈×Dic₅₃	—	4	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²⋊Dic₅₃	—	4	1	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	4	1	no	—
62	#11719917	—	2	2	no	—
63	#11719917	—	4	1	no	—
64	#11719917	—	4	2	no	—
65	#11719917	—	4	2	no	—
66	C₂×#88322	—	4	1	no	—
67	C₂×#88322	—	4	2	no	—
68	C₁₆×C₅₃	—	2	4	no	—
69	C₂³×C₅₃	#137	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₅₃	—	4	2	no	—
80	C₈×C₅₃	#106	1	1	yes	characteristic
81	C₈×C₅₃	#106	1	1	yes	characteristic
82	C₈×C₅₃	#106	1	1	yes	characteristic
83	C₈×C₅₃	#106	1	1	yes	characteristic
84	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₃	#125	1	1	yes	characteristic
87	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
88	#88322	—	4	2	no	—
89	#88322	—	8	2	no	—
90	#13188	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	4	1	no	—
92	C₂²×C₅₃	#428	1	1	yes	characteristic
93	C₂²×C₅₃	#461	1	1	yes	characteristic
94	C₂²×C₅₃	#475	1	1	yes	characteristic
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	1	no	—
97	C₄×C₅₃	#389	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
99	C₄×C₅₃	#460	1	1	yes	characteristic
100	C₄×C₅₃	#476	1	1	yes	characteristic
101	Dic₅₃	—	8	2	no	—
102	#2155	—	53	2	no	—
103	C₂×#364	D₅₃	1	1	yes	characteristic
104	#1848	—	53	1	no	—
105	#1772	—	53	1	no	—
106	#1388	—	53	2	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#1948	1	1	yes	characteristic
111	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
112	C₂×#108	—	53	1	no	—
113	C₂×#108	—	53	2	no	—
114	C₂×#108	—	106	1	no	—
115	#427	—	53	1	no	—
116	#423	—	106	1	no	—
117	#405	—	106	1	no	—
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	#365	—	106	4	no	—
120	#364	Dic₅₃	1	2	yes	complemented
121	#364	Dic₅₃	1	2	yes	complemented
122	#339	—	106	1	no	—
123	C₅₃	#13188	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	106	1	no	—
125	C₂×M₁₆	—	106	2	no	—
126	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	#117	—	106	1	no	—
129	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×Dic₅₃	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	53	2	no	—
134	#108	—	106	2	no	—
135	#108	—	106	1	no	—
136	#108	—	106	2	no	—
137	#108	—	212	1	no	—
138	#108	—	212	1	no	—
139	#107	—	106	1	no	—
140	#106	—	53	2	no	—
141	C₄⋊C₈	—	212	2	no	—
142	#97	—	212	1	no	—
143	C₄×C₈	—	212	1	no	—
144	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
150	M₁₆	—	106	2	no	—
151	M₁₆	—	212	2	no	—
152	M₁₆	—	212	2	no	—
153	M₁₆	—	212	1	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	—	212	1	no	—
156	C₂×C₈	#11719913	1	1	yes	characteristic
157	C₂×C₈	#11719913	1	1	yes	characteristic
158	C₂×C₈	#11719913	1	1	yes	characteristic
159	C₂×C₈	#11719913	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
163	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
164	C₂×C₈	#11719947	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
166	C₂×C₈	#11719949	1	1	yes	characteristic
167	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	2	no	—
169	C₂²⋊C₄	—	212	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	212	1	no	—
172	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#13133250	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133192	1	1	yes	characteristic
182	C₂×C₄	#13133197	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
185	C₂×C₄	#13133245	1	1	yes	characteristic
186	C₈	—	424	2	no	—
187	C₈	—	212	2	no	—
188	C₈	#13133149	1	1	yes	characteristic
189	C₈	#13133149	1	1	yes	characteristic
190	C₈	#13133149	1	1	yes	characteristic
191	C₈	#13133149	1	1	yes	characteristic
192	C₈	D₄×Dic₅₃	1	1	yes	characteristic
193	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
194	C₈	#13133208	1	1	yes	characteristic
195	C₈	#13133216	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#16810665	1	1	yes	characteristic
198	C₂²	#16810829	1	1	yes	characteristic
199	C₂²	#16810909	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	424	2	no	—
203	C₄	#16810522	1	1	yes	characteristic
204	C₄	C₂×#13133149	1	1	yes	characteristic
205	C₄	#16810824	1	1	yes	characteristic
206	C₄	#16810914	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#20275479	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810914
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1948
prime radical	C₂³×C₅₃	#137
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#13188
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
16. C₅₃×#364	C₄	semidirect
17. C₅₃×#364	C₄	semidirect
120. #364	Dic₅₃	semidirect
121. #364	Dic₅₃	semidirect
123. C₅₃	#13188	semidirect

Maximal subgroups

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

subgroup	index
#20276488	2
#20274887	2
#20274594	2
#20272827	2
C₂×C₅₃×#364	2
#13188	53