Subgroups — #27300471

Order 13568.

342 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×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	C₄×#11719908	C₄	1	4	yes	complemented
10	C₂×#107×C₅₃	—	2	1	no	—
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#384	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄⋊Dic₅₃	—	4	1	no	—
16	C₂×C₄⋊Dic₅₃	—	4	1	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	C₂×C₄	1	4	yes	—
20	#13133151	—	2	4	no	—
21	#13133151	—	4	2	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
26	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	#107×C₅₃	—	2	4	no	—
29	#107×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	—	4	2	no	—
33	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
39	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₂×C₈×C₅₃	#41	1	2	yes	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₄⋊C₄×C₅₃	—	8	2	no	—
47	C₄⋊C₄×C₅₃	—	8	2	no	—
48	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₄²×C₅₃	Q₁₆	1	2	yes	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	8	2	no	—
53	C₄⋊Dic₅₃	—	8	2	no	—
54	#11719908	—	2	4	no	—
55	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	#107	1	2	yes	—
60	C₂×C₄×C₅₃	#107	1	2	yes	—
61	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
63	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
64	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	2	no	—
74	C₈×C₅₃	#106	1	4	yes	—
75	C₈×C₅₃	#117	1	2	yes	—
76	C₈×C₅₃	#125	1	2	yes	—
77	#12888	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₅₃	#486	1	1	yes	characteristic
80	C₂²×C₅₃	#486	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
85	C₄×C₅₃	—	16	2	no	—
86	C₄×C₅₃	#366	1	4	yes	—
87	C₄×C₅₃	#367	1	4	yes	—
88	C₄×C₅₃	#382	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₅₃	#487	1	1	yes	characteristic
91	C₄×C₅₃	#487	1	1	yes	characteristic
92	Dic₅₃	—	16	2	no	—
93	C₂×C₄×C₁₆	—	53	1	no	—
94	#1845	—	53	1	no	—
95	#1845	D₅₃	1	1	yes	characteristic
96	#1380	—	53	4	no	—
97	C₂×C₅₃	#1834	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
100	C₂×C₅₃	#2240	1	1	yes	characteristic
101	C₂×C₅₃	#2240	1	1	yes	characteristic
102	C₂×C₅₃	#2241	1	1	yes	characteristic
103	C₂×C₅₃	#2241	1	1	yes	characteristic
104	C₂²×C₁₆	—	53	2	no	—
105	C₂×#107	—	2	1	no	—
106	C₂×#107	—	53	2	no	—
107	C₂×#107	—	106	1	no	—
108	C₂×#107	C₂×D₅₃	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#384	—	106	1	no	—
112	C₄×C₁₆	—	53	4	no	—
113	C₅₃	#12888	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	212	1	no	—
121	C₂×C₄⋊C₄	—	212	1	no	—
122	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂×C₁₆	—	53	4	no	—
124	C₂×C₁₆	—	106	2	no	—
125	C₂×C₁₆	—	106	2	no	—
126	#107	—	2	4	no	—
127	#107	—	4	2	no	—
128	#107	—	106	4	no	—
129	#107	—	212	2	no	—
130	C₄×C₈	Dic₁₀₆	1	2	yes	—
131	C₄×C₈	C₄×D₅₃	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	212	2	no	—
134	C₂²×C₄	#11719922	1	1	yes	characteristic
135	C₂²×C₄	#11719922	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719919	1	2	yes	—
138	C₂×C₈	#11719923	1	2	yes	—
139	C₂×C₈	#11719924	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719947	1	2	yes	—
143	C₂×C₈	#11719949	1	2	yes	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	424	2	no	—
149	C₄⋊C₄	—	424	2	no	—
150	C₄⋊C₄	—	212	2	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄²	#11719919	1	2	yes	—
153	C₄²	#11719925	1	2	yes	—
154	C₁₆	—	106	4	no	—
155	C₂³	C₂×#11719922	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	424	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	—	424	2	no	—
162	C₂×C₄	#13133150	1	2	yes	—
163	C₂×C₄	#13133150	1	2	yes	—
164	C₂×C₄	#13133158	1	2	yes	—
165	C₂×C₄	#13133160	1	2	yes	—
166	C₂×C₄	#13133161	1	2	yes	—
167	C₂×C₄	#13133163	1	2	yes	—
168	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
169	C₂×C₄	#13133204	1	1	yes	characteristic
170	C₂×C₄	#13133206	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
174	C₈	#13133148	1	4	yes	—
175	C₈	#13133175	1	2	yes	—
176	C₈	#13133208	1	2	yes	—
177	C₂²	C₂×#13133150	1	1	yes	characteristic
178	C₂²	#16810969	1	1	yes	characteristic
179	C₂²	#16810971	1	1	yes	characteristic
180	C₂²	C₂×#13133158	1	1	yes	characteristic
181	C₂²	C₂×#13133160	1	1	yes	characteristic
182	C₂²	C₂×#13133161	1	1	yes	characteristic
183	C₂²	C₂×#13133163	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	848	2	no	—
186	C₄	#16810448	1	4	yes	—
187	C₄	#16810450	1	4	yes	—
188	C₄	#16810488	1	1	yes	characteristic
189	C₄	C₂×#13133148	1	1	yes	characteristic
190	C₄	#16810973	1	1	yes	characteristic
191	C₄	#16810975	1	1	yes	characteristic
192	C₂	#20274842	1	1	yes	characteristic
193	C₂	C₂×#16810448	1	1	yes	characteristic
194	C₂	C₂×#16810450	1	1	yes	characteristic
195	C₂	#20276938	1	1	yes	characteristic
196	C₂	#20276940	1	1	yes	characteristic
197	C₂	#20276942	1	1	yes	characteristic
198	C₂	#20276944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#12888
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
9. C₄×#11719908	C₄	semidirect
113. C₅₃	#12888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
#20274875	2
#20272806	2
C₅₃×#1845	2
#12888	53