Subgroups — #27300538

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810450	C₂	1	2	yes	maximal
2	C₂×#16810448	C₂	1	2	yes	maximal
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
4	#20274875	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133151	C₂²	1	2	yes	—
9	#16810493	—	2	1	no	—
10	#16810450	—	2	4	no	—
11	#16810448	—	2	4	no	—
12	#16810399	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#106×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#382	—	2	1	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	4	2	no	—
20	C₂×#11719908	C₂×C₄	1	4	yes	—
21	C₂×#11719908	D₄	1	4	yes	—
22	C₂×#11719908	Q₈	1	4	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133151	—	4	2	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#107×C₅₃	—	2	2	no	—
32	#107×C₅₃	—	2	2	no	—
33	#106×C₅₃	—	4	2	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	#95×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	—	4	1	no	—
39	C₂²×C₄×C₅₃	D₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	D₈	1	2	yes	—
44	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	4	1	no	—
50	C₄⋊C₄×C₅₃	—	4	1	no	—
51	C₄⋊C₄×C₅₃	—	4	1	no	—
52	C₄⋊C₄×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	Q₁₆	1	2	yes	—
55	C₄⋊Dic₅₃	—	4	2	no	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	8	4	no	—
58	#11719908	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	—	8	1	no	—
64	C₂×C₄×C₅₃	—	8	2	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	#107	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₅₃	—	8	2	no	—
74	C₂×Dic₅₃	—	8	4	no	—
75	C₈×C₅₃	#107	1	4	yes	—
76	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
78	#12908	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	#493	1	1	yes	characteristic
81	C₂²×C₅₃	#500	1	1	yes	characteristic
82	C₂²×C₅₃	#509	1	2	yes	—
83	C₂²×C₅₃	#510	1	2	yes	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	16	1	no	—
86	C₄×C₅₃	—	16	1	no	—
87	C₄×C₅₃	#384	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
89	C₄×C₅₃	#494	1	1	yes	characteristic
90	C₄×C₅₃	#500	1	1	yes	characteristic
91	Dic₅₃	—	16	2	no	—
92	C₂×#367	—	53	2	no	—
93	C₂×#366	—	53	2	no	—
94	C₂×#346	—	53	1	no	—
95	#1846	D₅₃	1	1	yes	characteristic
96	#1845	—	53	1	no	—
97	C₂×C₅₃	#1845	1	1	yes	characteristic
98	C₂×C₅₃	#2159	1	2	yes	—
99	C₂×C₅₃	#2250	1	1	yes	characteristic
100	C₂×C₅₃	#2251	1	1	yes	characteristic
101	C₂×C₅₃	#2255	1	2	yes	—
102	C₂²×C₁₆	—	53	2	no	—
103	C₂×#107	—	53	2	no	—
104	C₂×#107	—	106	1	no	—
105	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
107	C₂×#106	—	2	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#384	—	106	1	no	—
110	#382	—	2	1	no	—
111	#367	—	106	4	no	—
112	#366	—	106	4	no	—
113	#346	—	53	4	no	—
114	C₅₃	#12908	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	212	2	no	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×C₁₆	—	53	4	no	—
123	C₂×C₁₆	—	53	4	no	—
124	C₂×C₁₆	—	53	4	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	—	106	2	no	—
128	#107	—	106	2	no	—
129	#107	—	212	2	no	—
130	#106	—	4	2	no	—
131	C₄×C₈	Dic₁₀₆	1	2	yes	—
132	C₄×C₈	C₄×D₅₃	1	2	yes	—
133	#95	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	212	2	no	—
137	C₂²×C₄	#11719922	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719919	1	2	yes	—
140	C₂×C₈	#11719922	1	2	yes	—
141	C₂×C₈	#11719925	1	2	yes	—
142	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	D₄×D₅₃	1	2	yes	—
145	C₂×C₈	Q₈×D₅₃	1	2	yes	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	424	4	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄⋊C₄	—	212	2	no	—
153	C₄²	#11719919	1	2	yes	—
154	C₄²	#11719925	1	2	yes	—
155	C₁₆	—	53	8	no	—
156	C₂³	C₂×#11719922	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	424	4	no	—
164	C₂×C₄	—	424	2	no	—
165	C₂×C₄	#13133150	1	4	yes	—
166	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
167	C₂×C₄	#13133213	1	1	yes	characteristic
168	C₂×C₄	#13133215	1	1	yes	characteristic
169	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
170	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
172	C₈	#13133150	1	4	yes	—
173	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
174	C₈	#13133216	1	2	yes	—
175	C₂²	C₂×#13133150	1	1	yes	characteristic
176	C₂²	#16810999	1	1	yes	characteristic
177	C₂²	#16811026	1	1	yes	characteristic
178	C₂²	#16811058	1	2	yes	—
179	C₂²	#16811063	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	1	no	—
182	C₄	—	16	1	no	—
183	C₄	—	848	2	no	—
184	C₄	#16810492	1	1	yes	characteristic
185	C₄	C₂×#13133150	1	1	yes	characteristic
186	C₄	#16811002	1	1	yes	characteristic
187	C₄	#16811026	1	1	yes	characteristic
188	C₂	#20274874	1	1	yes	characteristic
189	C₂	#20276504	1	2	yes	—
190	C₂	#20276982	1	1	yes	characteristic
191	C₂	#20276987	1	1	yes	characteristic
192	C₂	#20277002	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#12908
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
12. #16810399	C₄	semidirect
58. #11719908	C₄⋊C₄	semidirect
114. C₅₃	#12908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810450	2
C₂×#16810448	2
C₂×#16810399	2
#20274875	2
C₅₃×#1846	2
#12908	53