Subgroups — #27302234

Order 13568.

310 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×#16810446	C₂	1	2	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	1	yes	maximal, characteristic
4	#20272821	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	1	yes	characteristic
7	#16810661	C₂²	1	1	yes	characteristic
8	C₂×#13133149	—	2	1	no	—
9	C₂×#13133149	C₂²	1	2	yes	—
10	#16810578	—	2	1	no	—
11	#16810493	—	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	4	yes	complemented
17	C₂×#11719917	—	2	1	no	—
18	#13133176	—	2	1	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	4	2	no	—
21	#13133153	—	2	1	no	—
22	#13133153	D₄	1	2	yes	—
23	#13133151	Q₈	1	2	yes	—
24	#13133149	—	2	1	no	—
25	#13133149	—	2	2	no	—
26	#13133149	—	2	2	no	—
27	#13133149	—	4	2	no	—
28	#13133127	—	4	1	no	—
29	C₈×Dic₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	2	no	—
34	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²⋊Dic₅₃	—	4	1	no	—
54	C₄⋊Dic₅₃	—	2	2	no	—
55	C₄⋊Dic₅₃	—	4	2	no	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	8	4	no	—
58	C₄×Dic₅₃	—	4	1	no	—
59	#11719917	—	2	2	no	—
60	#11719917	—	4	1	no	—
61	C₂×#88322	—	4	1	no	—
62	C₁₆×C₅₃	—	2	4	no	—
63	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
72	C₂×Dic₅₃	—	4	2	no	—
73	C₂×Dic₅₃	—	8	4	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₈×C₅₃	#106	1	2	yes	—
76	C₈×C₅₃	#106	1	2	yes	—
77	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₅₃	#125	1	1	yes	characteristic
80	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	#88322	—	4	2	no	—
82	#13186	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	#428	1	1	yes	characteristic
85	C₂²×C₅₃	#460	1	1	yes	characteristic
86	C₂²×C₅₃	#476	1	1	yes	characteristic
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	#389	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₅₃	#461	1	1	yes	characteristic
91	C₄×C₅₃	#475	1	1	yes	characteristic
92	Dic₅₃	—	8	2	no	—
93	Dic₅₃	—	16	2	no	—
94	C₂×#365	—	53	2	no	—
95	C₂×#364	D₅₃	1	1	yes	characteristic
96	#1844	—	53	1	no	—
97	#1772	—	53	1	no	—
98	#1386	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1948	1	1	yes	characteristic
102	C₂×C₅₃	#2230	1	2	yes	—
103	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
104	C₂×#108	—	53	1	no	—
105	#427	—	53	1	no	—
106	C₂×#106	—	53	2	no	—
107	C₂×#106	—	106	1	no	—
108	#405	—	106	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#384	—	106	1	no	—
111	#365	—	106	4	no	—
112	#364	Dic₅₃	1	4	yes	complemented
113	#339	—	106	1	no	—
114	C₅₃	#13186	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	106	1	no	—
116	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	#117	—	106	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	212	2	no	—
121	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₂×Dic₅₃	1	4	yes	—
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	53	2	no	—
126	#108	—	106	1	no	—
127	#107	—	53	2	no	—
128	#106	—	106	1	no	—
129	#106	—	106	2	no	—
130	#106	—	106	2	no	—
131	#106	—	212	2	no	—
132	#97	—	212	1	no	—
133	C₄×C₈	—	212	1	no	—
134	C₄×C₈	Dic₁₀₆	1	2	yes	—
135	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	212	2	no	—
138	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	M₁₆	—	106	2	no	—
140	M₁₆	—	212	1	no	—
141	C₂×C₈	—	212	1	no	—
142	C₂×C₈	#11719913	1	1	yes	characteristic
143	C₂×C₈	#11719913	1	1	yes	characteristic
144	C₂×C₈	#11719913	1	2	yes	—
145	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
146	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
148	C₂×C₈	#11719947	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
150	C₂×C₈	#11719949	1	1	yes	characteristic
151	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	106	2	no	—
153	C₄⋊C₄	—	424	4	no	—
154	C₄⋊C₄	—	212	2	no	—
155	C₄⋊C₄	—	212	2	no	—
156	C₂²⋊C₄	—	212	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	212	1	no	—
159	C₄²	C₄⋊Dic₅₃	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	#13133245	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	424	4	no	—
165	C₂×C₄	—	424	2	no	—
166	C₂×C₄	—	212	2	no	—
167	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	#13133192	1	1	yes	characteristic
169	C₂×C₄	#13133197	1	1	yes	characteristic
170	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
172	C₂×C₄	#13133250	1	1	yes	characteristic
173	C₈	—	212	2	no	—
174	C₈	#13133149	1	2	yes	—
175	C₈	#13133149	1	2	yes	—
176	C₈	D₄×Dic₅₃	1	1	yes	characteristic
177	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
178	C₈	#13133208	1	1	yes	characteristic
179	C₈	#13133216	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#16810665	1	1	yes	characteristic
182	C₂²	#16810824	1	1	yes	characteristic
183	C₂²	#16810914	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	848	2	no	—
186	C₄	—	424	2	no	—
187	C₄	#16810522	1	1	yes	characteristic
188	C₄	C₂×#13133149	1	1	yes	characteristic
189	C₄	#16810829	1	1	yes	characteristic
190	C₄	#16810909	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20275479	1	1	yes	characteristic
194	C₂	#20276887	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810914
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#13186
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ 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
112. #364	Dic₅₃	semidirect
114. C₅₃	#13186	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810446	2
#20274872	2
#20274594	2
#20272821	2
C₂×C₅₃×#364	2
#13186	53