Subgroups — #27300530

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 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×#16810449	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274878	C₂	1	2	yes	maximal
5	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₂×#13133148	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810488	—	2	2	no	—
11	#16810450	—	2	4	no	—
12	#16810449	—	2	2	no	—
13	C₁₆×Dic₅₃	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#366	—	2	2	no	—
17	C₂²×#88322	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₂×#11719908	D₄	1	2	yes	—
22	C₂×#11719908	Q₈	1	2	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133148	—	4	4	no	—
26	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₈×Dic₅₃	Q₈	1	2	yes	—
28	#13133124	—	4	4	no	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	2	no	—
39	C₂²×Dic₅₃	D₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
51	C₄×Dic₅₃	Q₁₆	1	2	yes	—
52	C₂×#88322	C₄⋊C₄	1	2	yes	—
53	C₂×#88322	D₈	1	2	yes	—
54	C₂×#88322	Q₁₆	1	2	yes	—
55	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
56	#11719908	C₄⋊C₄	1	4	yes	—
57	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	4	no	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	8	2	no	—
69	C₂×Dic₅₃	—	8	4	no	—
70	C₂×Dic₅₃	#107	1	4	yes	—
71	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	#88322	#107	1	4	yes	—
76	#12907	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#493	1	1	yes	characteristic
79	C₂²×C₅₃	#500	1	1	yes	characteristic
80	C₂²×C₅₃	#508	1	2	yes	—
81	C₂²×C₅₃	#508	1	2	yes	—
82	C₄×C₅₃	—	16	2	no	—
83	C₄×C₅₃	#384	1	1	yes	characteristic
84	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
85	C₄×C₅₃	#494	1	1	yes	characteristic
86	C₄×C₅₃	#500	1	1	yes	characteristic
87	Dic₅₃	—	2	4	no	—
88	Dic₅₃	—	16	2	no	—
89	C₂×#367	—	53	2	no	—
90	C₂×#366	—	53	1	no	—
91	C₂×#366	D₅₃	1	1	yes	characteristic
92	C₂×C₄×C₁₆	—	53	1	no	—
93	#1846	—	53	2	no	—
94	C₂×C₅₃	#1845	1	1	yes	characteristic
95	C₂×C₅₃	#2158	1	2	yes	—
96	C₂×C₅₃	#2249	1	1	yes	characteristic
97	C₂×C₅₃	#2249	1	1	yes	characteristic
98	C₂×C₅₃	#2253	1	2	yes	—
99	C₂²×C₁₆	—	53	1	no	—
100	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
101	C₂×#107	—	53	2	no	—
102	C₂×#107	C₂×D₅₃	1	2	yes	—
103	C₂×#106	—	106	2	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	#382	—	106	2	no	—
106	#367	—	106	4	no	—
107	#366	—	2	2	no	—
108	#366	—	106	2	no	—
109	C₄×C₁₆	—	53	4	no	—
110	C₅₃	#12907	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	106	2	no	—
115	C₂×C₄²	—	53	1	no	—
116	C₂×C₁₆	—	53	2	no	—
117	C₂×C₁₆	—	53	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
120	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
121	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	106	2	no	—
125	#107	—	106	2	no	—
126	#106	—	212	4	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	#95	—	212	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	53	2	no	—
132	C₂²×C₄	—	212	2	no	—
133	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	2	no	—
136	C₂×C₈	—	53	2	no	—
137	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
139	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	212	2	no	—
146	C₄⋊C₄	—	212	2	no	—
147	C₄²	—	53	2	no	—
148	C₄²	—	53	2	no	—
149	C₁₆	—	53	4	no	—
150	C₁₆	#11719919	1	4	yes	—
151	C₂³	D₈×D₅₃	1	1	yes	characteristic
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	424	4	no	—
158	C₂×C₄	—	424	2	no	—
159	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	#13133213	1	1	yes	characteristic
161	C₂×C₄	#13133216	1	1	yes	characteristic
162	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
163	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
164	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
165	C₈	—	53	4	no	—
166	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
167	C₈	C₂×#11719919	1	1	yes	characteristic
168	C₈	#13133215	1	1	yes	characteristic
169	C₈	#13133216	1	1	yes	characteristic
170	C₂²	#107×D₅₃	1	1	yes	characteristic
171	C₂²	#16810998	1	1	yes	characteristic
172	C₂²	#16811027	1	1	yes	characteristic
173	C₂²	#16811052	1	2	yes	—
174	C₂²	#16811056	1	2	yes	—
175	C₄	—	16	2	no	—
176	C₄	—	848	2	no	—
177	C₄	—	106	4	no	—
178	C₄	#16810492	1	1	yes	characteristic
179	C₄	#107×D₅₃	1	1	yes	characteristic
180	C₄	#16811001	1	1	yes	characteristic
181	C₄	#16811027	1	1	yes	characteristic
182	C₂	#20274873	1	1	yes	characteristic
183	C₂	#20276496	1	2	yes	—
184	C₂	#20276973	1	1	yes	characteristic
185	C₂	#20276977	1	1	yes	characteristic
186	C₂	#20276992	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	#12907
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₅₃	C₄	semidirect
110. C₅₃	#12907	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×#16810449	2
C₂×C₁₆×Dic₅₃	2
#20274878	2
C₂×C₅₃×#366	2
#12907	53