Subgroups — #27302133

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	#20275449	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal
5	C₂×C₅₃×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810522	—	2	2	no	—
10	#16810449	—	2	2	no	—
11	#16810398	—	2	2	no	—
12	#16810385	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#367	C₄	1	4	yes	complemented
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	Q₈	1	2	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	C₈×Dic₅₃	—	2	1	no	—
26	C₈×Dic₅₃	—	2	1	no	—
27	#13133124	—	4	2	no	—
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	4	2	no	—
38	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
39	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	8	4	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₂×#88322	—	2	1	no	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	Q₁₆	1	2	yes	—
56	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
57	#11719908	—	2	2	no	—
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	4	4	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	2	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₅₃	#125	1	1	yes	characteristic
79	#88322	—	2	2	no	—
80	#13167	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₅₃	#466	1	1	yes	characteristic
83	C₂²×C₅₃	#500	1	1	yes	characteristic
84	C₂²×C₅₃	#509	1	2	yes	—
85	C₂²×C₅₃	#510	1	2	yes	—
86	C₄×C₅₃	—	8	4	no	—
87	C₄×C₅₃	#389	1	1	yes	characteristic
88	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₅₃	#467	1	1	yes	characteristic
90	C₄×C₅₃	#499	1	1	yes	characteristic
91	Dic₅₃	—	4	1	no	—
92	Dic₅₃	—	4	1	no	—
93	Dic₅₃	—	16	2	no	—
94	C₂×#367	D₅₃	1	1	yes	characteristic
95	C₂×#366	—	53	1	no	—
96	C₂×#346	—	53	1	no	—
97	#1942	—	53	2	no	—
98	#1380	—	53	2	no	—
99	C₂×C₅₃	#1942	1	1	yes	characteristic
100	C₂×C₅₃	#2175	1	2	yes	—
101	C₂×C₅₃	#2218	1	1	yes	characteristic
102	C₂×C₅₃	#2219	1	1	yes	characteristic
103	C₂×C₅₃	#2255	1	2	yes	—
104	C₂²×C₁₆	—	53	1	no	—
105	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#107	—	53	2	no	—
107	C₂×#107	C₂×D₅₃	1	2	yes	—
108	C₂×C₄×C₈	—	53	1	no	—
109	#389	—	106	2	no	—
110	#367	Dic₅₃	1	4	yes	complemented
111	#366	—	106	2	no	—
112	#346	—	106	2	no	—
113	#340	—	106	2	no	—
114	C₅₃	#13167	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	106	2	no	—
119	C₂×C₄⋊C₄	—	212	2	no	—
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	106	1	no	—
123	C₂×C₁₆	—	106	1	no	—
124	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
125	C₂×C₁₆	D₂₁₂	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	106	2	no	—
129	#107	—	106	2	no	—
130	#107	C₂×Dic₅₃	1	4	yes	—
131	C₄×C₈	—	106	1	no	—
132	C₄×C₈	—	106	1	no	—
133	#95	—	212	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	212	2	no	—
138	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	106	1	no	—
141	C₂×C₈	—	106	1	no	—
142	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
144	C₂×C₈	#11719947	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
146	C₂×C₈	#11719949	1	1	yes	characteristic
147	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	424	4	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄⋊C₄	—	212	2	no	—
153	C₄²	—	106	1	no	—
154	C₄²	—	106	1	no	—
155	C₁₆	—	106	2	no	—
156	C₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
157	C₂³	D₈×D₅₃	1	1	yes	characteristic
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	1	no	—
164	C₂×C₄	—	106	1	no	—
165	C₂×C₄	—	424	2	no	—
166	C₂×C₄	—	424	2	no	—
167	C₂×C₄	—	424	2	no	—
168	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	#13133192	1	1	yes	characteristic
170	C₂×C₄	#13133216	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
172	C₂×C₄	#13133236	1	1	yes	characteristic
173	C₂×C₄	#13133240	1	1	yes	characteristic
174	C₈	—	106	2	no	—
175	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
177	C₈	#13133197	1	1	yes	characteristic
178	C₈	#13133208	1	1	yes	characteristic
179	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
180	C₂²	#16810860	1	1	yes	characteristic
181	C₂²	#16811027	1	1	yes	characteristic
182	C₂²	#16811060	1	2	yes	—
183	C₂²	#16811062	1	2	yes	—
184	C₄	—	8	4	no	—
185	C₄	—	848	2	no	—
186	C₄	—	212	1	no	—
187	C₄	—	212	1	no	—
188	C₄	#16810522	1	1	yes	characteristic
189	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
190	C₄	#16810865	1	1	yes	characteristic
191	C₄	#16811024	1	1	yes	characteristic
192	C₂	#20275449	1	1	yes	characteristic
193	C₂	#20276580	1	2	yes	—
194	C₂	#20276819	1	1	yes	characteristic
195	C₂	#20276824	1	1	yes	characteristic
196	C₂	#20277000	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#367	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#367	D₅₃
hypocenter	C₅₃	#13167
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. C₅₃×#367	C₄	semidirect
56. C₁₆×C₅₃	C₄⋊C₄	semidirect
110. #367	Dic₅₃	semidirect
114. C₅₃	#13167	semidirect
156. C₁₆	C₄⋊Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810449	2
C₂×#16810398	2
#20275449	2
#20272806	2
C₂×C₅₃×#367	2
#13167	53