Subgroups — #27302702

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₂×C₅₃×#499	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	#16810399	—	2	2	no	—
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₅₃×#499	—	2	2	no	—
12	C₅₃×#499	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	C₂²	1	2	yes	—
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	4	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	Q₈	1	4	yes	—
21	#13133151	—	2	4	no	—
22	#13133151	—	4	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	#125×C₅₃	—	2	4	no	—
26	#125×C₅₃	—	4	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	#107×C₅₃	—	2	2	no	—
31	#107×C₅₃	C₂×C₄	1	4	yes	—
32	#106×C₅₃	—	2	4	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	D₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	8	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₃	SD₁₆	1	2	yes	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₄⋊Dic₅₃	—	8	4	no	—
54	#11719908	—	2	4	no	—
55	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	#102	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₅₃	—	8	2	no	—
69	C₂×Dic₅₃	—	8	4	no	—
70	C₈×C₅₃	#103	1	4	yes	—
71	C₈×C₅₃	C₄×Q₈	1	2	yes	—
72	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
75	C₂²×C₅₃	#486	1	1	yes	characteristic
76	C₂²×C₅₃	#500	1	1	yes	characteristic
77	C₂²×C₅₃	#509	1	2	yes	—
78	C₂²×C₅₃	#510	1	2	yes	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	—	8	4	no	—
81	C₄×C₅₃	#391	1	1	yes	characteristic
82	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
83	C₄×C₅₃	#487	1	1	yes	characteristic
84	C₄×C₅₃	#498	1	1	yes	characteristic
85	Dic₅₃	—	16	2	no	—
86	C₂×#499	D₅₃	1	1	yes	characteristic
87	C₂×#346	—	53	1	no	—
88	#1845	—	53	1	no	—
89	#1380	—	53	4	no	—
90	C₂×C₅₃	#1990	1	1	yes	characteristic
91	C₂×C₅₃	#2139	1	2	yes	—
92	C₂×C₅₃	#2244	1	1	yes	characteristic
93	C₂×C₅₃	#2245	1	1	yes	characteristic
94	C₂×C₅₃	#2255	1	2	yes	—
95	C₂×#125	—	2	1	no	—
96	C₂²×C₁₆	—	53	2	no	—
97	#499	—	2	2	no	—
98	#499	Dic₅₃	1	4	yes	complemented
99	C₂×#107	—	53	2	no	—
100	C₂×#107	—	106	1	no	—
101	C₂×#107	C₂×D₅₃	1	2	yes	—
102	C₂×#106	—	2	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	#384	—	106	1	no	—
105	#346	—	106	2	no	—
106	C₅₃	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	#125	—	2	4	no	—
110	#125	—	4	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	106	2	no	—
114	C₂×C₄⋊C₄	—	212	2	no	—
115	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	53	4	no	—
117	C₂×C₁₆	—	106	2	no	—
118	C₂×C₁₆	—	106	2	no	—
119	#107	—	2	2	no	—
120	#107	—	106	4	no	—
121	#107	—	212	2	no	—
122	#107	C₂×Dic₅₃	1	4	yes	—
123	#106	—	2	4	no	—
124	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	212	2	no	—
128	C₂²×C₄	#11719922	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₈	#11719924	1	2	yes	—
131	C₂×C₈	#11719925	1	2	yes	—
132	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
133	C₂×C₈	Q₈×D₅₃	1	2	yes	—
134	C₂×C₈	#11719949	1	2	yes	—
135	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	424	4	no	—
141	C₄⋊C₄	—	212	2	no	—
142	C₄⋊C₄	—	212	2	no	—
143	C₄²	#11719923	1	2	yes	—
144	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
145	C₁₆	—	106	4	no	—
146	C₂³	C₂×#11719922	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	424	4	no	—
153	C₂×C₄	—	424	2	no	—
154	C₂×C₄	#13133138	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	#13133206	1	1	yes	characteristic
157	C₂×C₄	#13133215	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
159	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
160	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
161	C₈	#13133141	1	4	yes	—
162	C₈	Q₈×Dic₅₃	1	2	yes	—
163	C₈	#13133193	1	2	yes	—
164	C₂²	C₂×#13133138	1	1	yes	characteristic
165	C₂²	#16810971	1	1	yes	characteristic
166	C₂²	#16811026	1	1	yes	characteristic
167	C₂²	#16811058	1	2	yes	—
168	C₂²	#16811063	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	8	4	no	—
171	C₄	—	848	2	no	—
172	C₄	#16810530	1	1	yes	characteristic
173	C₄	C₂×#13133141	1	1	yes	characteristic
174	C₄	#16810975	1	1	yes	characteristic
175	C₄	#16811019	1	1	yes	characteristic
176	C₂	#20275677	1	1	yes	characteristic
177	C₂	#20276427	1	2	yes	—
178	C₂	#20276958	1	1	yes	characteristic
179	C₂	#20276962	1	1	yes	characteristic
180	C₂	#20277002	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#13284
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
12. C₅₃×#499	C₄	semidirect
98. #499	Dic₅₃	semidirect
106. C₅₃	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
#20274875	2
#20272806	2
C₂×C₅₃×#499	2
#13284	53