Subgroups — #27300531

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×#16810448	C₂	1	2	yes	maximal
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	#20274879	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×#13133151	C₂²	1	2	yes	—
9	C₂×#13133149	—	2	1	no	—
10	#16810489	—	2	1	no	—
11	#16810450	—	2	4	no	—
12	#16810448	—	2	4	no	—
13	C₄×#11719908	C₄	1	4	yes	complemented
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
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×#11719908	C₂×C₄	1	2	yes	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₂×#11719908	D₄	1	2	yes	—
22	C₂×#11719908	D₄	1	2	yes	—
23	C₂×#11719908	Q₈	1	2	yes	—
24	C₂×#11719908	Q₈	1	2	yes	—
25	#13133151	—	2	2	no	—
26	#13133151	—	2	2	no	—
27	#13133149	—	4	2	no	—
28	#13133124	—	4	2	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	—	2	2	no	—
35	#106×C₅₃	—	4	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	#95×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₃	D₈	1	2	yes	—
45	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₃	Q₁₆	1	2	yes	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	#11719908	C₄⋊C₄	1	4	yes	—
59	#11719908	C₄⋊C₄	1	4	yes	—
60	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	8	2	no	—
63	C₂×C₄×C₅₃	—	8	4	no	—
64	C₂×C₄×C₅₃	#107	1	4	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	—	8	4	no	—
73	C₈×C₅₃	#107	1	4	yes	—
74	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	#12907	—	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	#508	1	4	yes	—
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	—	16	2	no	—
85	C₄×C₅₃	#384	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₄×C₅₃	#494	1	1	yes	characteristic
88	C₄×C₅₃	#500	1	1	yes	characteristic
89	Dic₅₃	—	16	2	no	—
90	C₂×#367	—	53	2	no	—
91	C₂×#366	—	53	2	no	—
92	C₂×C₄×C₁₆	—	53	1	no	—
93	#1846	—	53	1	no	—
94	#1846	D₅₃	1	1	yes	characteristic
95	C₂×C₅₃	#1845	1	1	yes	characteristic
96	C₂×C₅₃	#2158	1	2	yes	—
97	C₂×C₅₃	#2249	1	2	yes	—
98	C₂×C₅₃	#2253	1	2	yes	—
99	C₂²×C₁₆	—	53	1	no	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂×#107	—	53	2	no	—
102	C₂×#107	C₂×D₅₃	1	2	yes	—
103	C₂×#106	—	2	1	no	—
104	C₂×#106	—	106	1	no	—
105	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	#382	—	2	1	no	—
107	#382	—	106	1	no	—
108	#367	—	106	4	no	—
109	#366	—	106	4	no	—
110	C₄×C₁₆	—	53	4	no	—
111	C₅₃	#12907	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	106	2	no	—
116	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×C₁₆	—	53	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	53	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	106	2	no	—
126	#107	—	106	2	no	—
127	#106	—	4	2	no	—
128	#106	—	212	2	no	—
129	C₄×C₈	Dic₁₀₆	1	2	yes	—
130	C₄×C₈	C₄×D₅₃	1	2	yes	—
131	#95	—	4	2	no	—
132	#95	—	212	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	212	2	no	—
135	C₂²×C₄	#11719922	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719919	1	2	yes	—
138	C₂×C₈	#11719922	1	2	yes	—
139	C₂×C₈	#11719925	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	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	212	2	no	—
149	C₄⋊C₄	—	212	2	no	—
150	C₄²	#11719919	1	2	yes	—
151	C₄²	#11719925	1	2	yes	—
152	C₁₆	—	53	4	no	—
153	C₁₆	—	53	4	no	—
154	C₂³	C₂×#11719922	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	424	4	no	—
159	C₂×C₄	—	424	2	no	—
160	C₂×C₄	#13133150	1	4	yes	—
161	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
162	C₂×C₄	#13133213	1	1	yes	characteristic
163	C₂×C₄	#13133215	1	1	yes	characteristic
164	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
165	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
167	C₈	#13133150	1	4	yes	—
168	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
170	C₈	#13133216	1	1	yes	characteristic
171	C₈	#13133216	1	1	yes	characteristic
172	C₂²	C₂×#13133150	1	1	yes	characteristic
173	C₂²	#16810999	1	1	yes	characteristic
174	C₂²	#16811026	1	1	yes	characteristic
175	C₂²	#16811053	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	848	2	no	—
179	C₄	#16810492	1	1	yes	characteristic
180	C₄	C₂×#13133150	1	1	yes	characteristic
181	C₄	#16811002	1	1	yes	characteristic
182	C₄	#16811026	1	1	yes	characteristic
183	C₂	#20274874	1	1	yes	characteristic
184	C₂	#20276497	1	2	yes	—
185	C₂	#20276974	1	2	yes	—
186	C₂	#20276991	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	#12907
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
13. C₄×#11719908	C₄	semidirect
111. 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×#16810448	2
C₂×C₄×#11719908	2
#20274879	2
C₅₃×#1846	2
#12907	53