Subgroups — #27536657

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811020	C₂	1	4	yes	maximal
2	#20283112	C₂	1	2	yes	maximal
3	#20274872	C₂	1	1	yes	maximal, characteristic
4	#20274871	C₂	1	2	yes	maximal
5	#20274870	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133216	—	2	4	no	—
9	C₂×#13133216	C₂²	1	4	yes	—
10	#16811020	—	2	16	no	—
11	C₂×#13133149	C₂²	1	4	yes	—
12	C₂×#13133148	C₂²	1	8	yes	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	#16810493	C₂²	1	2	yes	—
17	#16810492	—	2	4	no	—
18	#16810492	C₂²	1	4	yes	—
19	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×#106×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₂²×Dic₁₀₆	—	2	4	no	—
24	C₂²×#88322	C₂³	1	2	yes	—
25	#13133216	—	2	16	no	—
26	#13133216	—	4	16	no	—
27	C₂×C₄⋊Dic₅₃	—	2	4	no	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
29	C₂×#11719919	—	2	8	no	—
30	C₂×#11719919	—	2	4	no	—
31	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
34	#13133149	—	2	8	no	—
35	#13133148	—	2	16	no	—
36	C₄×#88322	Q₈	1	4	yes	complemented
37	C₈×Dic₅₃	D₄	1	8	yes	—
38	C₈×Dic₅₃	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#106×C₅₃	—	2	8	no	—
44	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	4	no	—
49	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
50	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
51	C₂×Dic₁₀₆	—	4	8	no	—
52	C₂×Dic₁₀₆	—	4	16	no	—
53	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
56	C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₄⋊C₄×C₅₃	—	4	8	no	—
58	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
59	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
60	C₄⋊Dic₅₃	—	2	8	no	—
61	C₄⋊Dic₅₃	—	4	8	no	—
62	#11719919	—	4	8	no	—
63	#11719919	—	4	16	no	—
64	C₄×Dic₅₃	SD₁₆	1	16	yes	—
65	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
66	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
67	C₂×#88322	C₂×D₄	1	4	yes	—
68	C₂×#88322	C₂×Q₈	1	4	yes	—
69	C₂×#88322	C₂×Q₈	1	4	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×SD₁₆	1	8	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂×Dic₅₃	—	4	4	no	—
80	C₂×Dic₅₃	—	4	8	no	—
81	C₂×Dic₅₃	#127	1	4	yes	—
82	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
83	C₂×Dic₅₃	C₂×SD₁₆	1	8	yes	—
84	C₂×Dic₅₃	C₂×SD₁₆	1	8	yes	—
85	Dic₁₀₆	—	8	16	no	—
86	C₈×C₅₃	C₄⋊Q₈	1	8	yes	complemented
87	#88322	C₄⋊Q₈	1	8	yes	complemented
88	#33310	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₅₃	C₂²×SD₁₆	1	4	yes	—
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#498	1	8	yes	complemented
95	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
96	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
97	Dic₅₃	—	8	4	no	—
98	Dic₅₃	#492	1	8	yes	—
99	Dic₅₃	#498	1	8	yes	complemented
100	C₂×#498	—	53	4	no	—
101	#2868	—	53	2	no	—
102	#1844	—	53	4	no	—
103	#1844	—	53	1	no	—
104	#1844	—	53	2	no	—
105	#1844	D₅₃	1	1	yes	characteristic
106	C₄²×C₈	—	53	1	no	—
107	C₂×C₅₃	#2868	1	1	yes	characteristic
108	C₂×C₅₃	C₂×#492	1	2	yes	—
109	C₂×C₅₃	C₂×#498	1	2	yes	—
110	C₂×C₅₃	C₂×#498	1	2	yes	—
111	C₂×C₄⋊Q₈	—	53	4	no	—
112	C₂×C₄⋊Q₈	—	106	4	no	—
113	#498	—	106	16	no	—
114	C₂×#106	—	53	4	no	—
115	C₂×#106	—	53	8	no	—
116	C₂×#106	C₂×D₅₃	1	4	yes	—
117	C₂×C₄×C₈	—	53	2	no	—
118	C₂×C₄×C₈	—	53	2	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	#384	—	53	4	no	—
122	#384	—	53	2	no	—
123	#384	—	106	4	no	—
124	#384	C₂×D₅₃	1	2	yes	—
125	C₄³	—	53	1	no	—
126	C₅₃	#33310	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	106	4	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
130	C₄⋊Q₈	—	106	16	no	—
131	C₄⋊Q₈	—	212	16	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	53	4	no	—
134	C₂×C₄⋊C₄	—	106	4	no	—
135	C₂×C₄⋊C₄	—	106	8	no	—
136	C₂×C₄⋊C₄	—	106	4	no	—
137	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
138	C₂×C₄²	—	53	4	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	#106	—	2	8	no	—
143	#106	—	106	16	no	—
144	#106	—	106	8	no	—
145	C₄×C₈	—	53	8	no	—
146	C₄×C₈	—	53	8	no	—
147	C₄×C₈	—	53	4	no	—
148	C₄×C₈	Dic₁₀₆	1	4	yes	complemented
149	C₂×Q₈	—	212	8	no	—
150	C₂×Q₈	—	212	16	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	53	2	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	—	106	4	no	—
155	C₂²×C₄	D₄×D₅₃	1	2	yes	—
156	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	53	4	no	—
160	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
161	C₂×C₈	D₄×D₅₃	1	4	yes	—
162	C₂×C₈	Q₈×D₅₃	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄⋊C₄	—	106	8	no	—
166	C₄⋊C₄	—	212	8	no	—
167	C₄⋊C₄	—	212	16	no	—
168	C₄⋊C₄	—	212	8	no	—
169	C₄²	—	53	16	no	—
170	C₄²	—	53	4	no	—
171	C₄²	—	53	4	no	—
172	C₄²	C₂×Dic₁₀₆	1	2	yes	—
173	C₄²	Q₈×D₅₃	1	2	yes	—
174	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
175	Q₈	—	424	16	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	53	8	no	—
179	C₂×C₄	—	53	8	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	53	4	no	—
182	C₂×C₄	—	212	8	no	—
183	C₂×C₄	—	212	4	no	—
184	C₂×C₄	#13133216	1	4	yes	—
185	C₂×C₄	SD₁₆×D₅₃	1	8	yes	—
186	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
187	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
189	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
190	C₈	—	53	8	no	—
191	C₈	#13133216	1	8	yes	complemented
192	C₂²	#127×D₅₃	1	1	yes	characteristic
193	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
194	C₂²	C₂×#13133216	1	1	yes	characteristic
195	C₂²	C₂×SD₁₆×D₅₃	1	4	yes	—
196	C₄	—	8	4	no	—
197	C₄	—	53	8	no	—
198	C₄	—	53	8	no	—
199	C₄	—	424	4	no	—
200	C₄	#16811020	1	8	yes	complemented
201	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
202	C₄	C₂×#13133216	1	2	yes	—
203	C₂	#20283112	1	1	yes	characteristic
204	C₂	#492×D₅₃	1	2	yes	—
205	C₂	#498×D₅₃	1	2	yes	—
206	C₂	C₂×#16811020	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#33310
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
36. C₄×#88322	Q₈	semidirect
38. C₈×Dic₅₃	Q₈	semidirect
44. C₄×C₈×C₅₃	Q₈	semidirect
86. C₈×C₅₃	C₄⋊Q₈	semidirect
87. #88322	C₄⋊Q₈	semidirect
94. C₄×C₅₃	#498	semidirect
99. Dic₅₃	#498	semidirect
126. C₅₃	#33310	semidirect
148. C₄×C₈	Dic₁₀₆	semidirect
191. C₈	#13133216	semidirect
200. C₄	#16811020	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811020	2
#20283112	2
#20274872	2
#20274871	2
#20274870	2
C₄×C₈×Dic₅₃	2
C₅₃×#1844	2
#33310	53