Subgroups — #27536661

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811023	C₂	1	4	yes	maximal
2	#20283076	C₂	1	2	yes	maximal
3	#20274876	C₂	1	4	yes	maximal
4	#20274874	C₂	1	2	yes	maximal
5	#20274872	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133208	C₂²	1	4	yes	—
9	#16811023	—	2	16	no	—
10	C₂×#13133150	C₂²	1	8	yes	—
11	C₂×#13133149	C₂²	1	4	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	#16810522	—	2	4	no	—
16	#16810519	—	2	4	no	—
17	#16810493	C₂²	1	2	yes	—
18	#16810488	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×#88322	C₂³	1	2	yes	—
24	#13133208	—	2	16	no	—
25	C₂×C₄⋊Dic₅₃	—	2	4	no	—
26	C₂×C₄⋊Dic₅₃	—	2	4	no	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
28	C₂×#11719919	—	2	4	no	—
29	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133150	—	2	16	no	—
33	#13133149	—	2	8	no	—
34	C₄×#88322	Q₈	1	4	yes	—
35	C₈×Dic₅₃	D₄	1	8	yes	—
36	C₈×Dic₅₃	Q₈	1	8	yes	—
37	#13133124	—	2	8	no	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#106×C₅₃	—	2	8	no	—
43	C₄×C₈×C₅₃	Q₈	1	4	yes	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	4	no	—
48	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
49	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	8	no	—
54	C₄⋊C₄×C₅₃	—	4	8	no	—
55	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
57	C₄⋊Dic₅₃	—	2	8	no	—
58	C₄⋊Dic₅₃	—	4	8	no	—
59	C₄⋊Dic₅₃	—	4	8	no	—
60	#11719919	—	4	8	no	—
61	C₄×Dic₅₃	—	2	8	no	—
62	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
63	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
64	C₂×#88322	C₂×D₄	1	4	yes	—
65	C₂×#88322	C₂×Q₈	1	4	yes	—
66	C₂×#88322	C₂×Q₈	1	4	yes	—
67	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	8	no	—
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
71	C₂×C₄×C₅₃	#135	1	8	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	4	8	no	—
80	C₂×Dic₅₃	#127	1	4	yes	—
81	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
82	C₈×C₅₃	C₄⋊Q₈	1	8	yes	—
83	#88322	C₄⋊Q₈	1	8	yes	—
84	#33311	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#135	1	4	yes	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#499	1	8	yes	complemented
91	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
92	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
93	Dic₅₃	—	2	4	no	—
94	Dic₅₃	—	2	4	no	—
95	Dic₅₃	—	8	4	no	—
96	C₂×#499	—	53	4	no	—
97	#2859	—	53	2	no	—
98	#1846	—	53	4	no	—
99	#1845	—	53	2	no	—
100	#1844	—	53	1	no	—
101	#1844	D₅₃	1	1	yes	characteristic
102	C₄²×C₈	—	53	1	no	—
103	C₂×C₅₃	#2868	1	1	yes	characteristic
104	C₂×C₅₃	#3147	1	2	yes	—
105	C₂×C₅₃	C₂×#499	1	2	yes	—
106	C₂×C₅₃	#3160	1	2	yes	—
107	C₂×#125	—	53	4	no	—
108	#499	—	106	16	no	—
109	C₂×#107	—	53	8	no	—
110	C₂×#106	—	53	4	no	—
111	C₂×#106	C₂×D₅₃	1	4	yes	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	—	53	2	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#389	—	106	4	no	—
117	#389	—	106	4	no	—
118	#384	—	53	2	no	—
119	#384	C₂×D₅₃	1	2	yes	—
120	#382	—	53	4	no	—
121	C₄³	—	53	1	no	—
122	C₅₃	#33311	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	53	2	no	—
124	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
125	#125	—	106	16	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	53	4	no	—
128	C₂×C₄⋊C₄	—	106	4	no	—
129	C₂×C₄⋊C₄	—	106	4	no	—
130	C₂×C₄⋊C₄	—	106	4	no	—
131	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	1	no	—
134	C₂×C₄²	—	53	4	no	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	#107	—	106	16	no	—
137	#106	—	2	8	no	—
138	#106	—	106	8	no	—
139	C₄×C₈	—	53	8	no	—
140	C₄×C₈	—	53	8	no	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	Dic₁₀₆	1	4	yes	—
143	#95	—	106	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	—	106	4	no	—
148	C₂²×C₄	D₄×D₅₃	1	2	yes	—
149	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
150	C₂×C₈	—	53	4	no	—
151	C₂×C₈	—	53	4	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
154	C₂×C₈	D₄×D₅₃	1	4	yes	—
155	C₂×C₈	Q₈×D₅₃	1	4	yes	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	106	8	no	—
159	C₄⋊C₄	—	212	8	no	—
160	C₄⋊C₄	—	212	8	no	—
161	C₄⋊C₄	—	212	8	no	—
162	C₄²	—	53	4	no	—
163	C₄²	—	53	4	no	—
164	C₄²	—	106	8	no	—
165	C₄²	C₂×Dic₁₀₆	1	2	yes	—
166	C₄²	Q₈×D₅₃	1	2	yes	—
167	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	53	4	no	—
171	C₂×C₄	—	53	4	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	212	8	no	—
175	C₂×C₄	—	212	4	no	—
176	C₂×C₄	#13133216	1	4	yes	—
177	C₂×C₄	#13133238	1	8	yes	—
178	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
179	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
181	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
182	C₈	—	53	8	no	—
183	C₈	#13133216	1	8	yes	—
184	C₂²	#127×D₅₃	1	1	yes	characteristic
185	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
186	C₂²	C₂×#13133216	1	1	yes	characteristic
187	C₂²	C₂×#13133238	1	4	yes	—
188	C₄	—	8	4	no	—
189	C₄	—	106	4	no	—
190	C₄	—	106	4	no	—
191	C₄	—	424	4	no	—
192	C₄	#16811023	1	8	yes	complemented
193	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
194	C₄	C₂×#13133216	1	2	yes	—
195	C₂	#20283112	1	1	yes	characteristic
196	C₂	#20285138	1	2	yes	—
197	C₂	C₂×#16811023	1	2	yes	—
198	C₂	#20285224	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	#33311
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
90. C₄×C₅₃	#499	semidirect
122. C₅₃	#33311	semidirect
192. C₄	#16811023	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811023	2
#20283076	2
#20274876	2
#20274874	2
#20274872	2
C₄×C₈×Dic₅₃	2
C₅₃×#1844	2
#33311	53