Subgroups — #27299502

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810451	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₅₃	C₂	1	2	yes	maximal
4	#20272806	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×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₅₃	—	2	2	no	—
10	#16810451	C₄	1	4	yes	complemented
11	#16810398	C₄	1	4	yes	complemented
12	#16810385	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#366	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	C₂×C₄	1	2	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	C₂×C₄	1	4	yes	—
25	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₈×Dic₅₃	Q₈	1	2	yes	—
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	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	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	2	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊Dic₅₃	—	2	4	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄×Dic₅₃	—	8	4	no	—
50	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
51	C₄×Dic₅₃	Q₁₆	1	2	yes	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	2	1	no	—
54	C₂×#88322	C₄⋊C₄	1	2	yes	—
55	C₁₆×C₅₃	C₄²	1	4	yes	complemented
56	#11719908	—	2	2	no	—
57	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
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	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	4	4	no	—
70	C₂×Dic₅₃	—	8	2	no	—
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	#107	1	4	yes	—
73	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₈×C₅₃	#117	1	1	yes	characteristic
75	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	#88322	—	2	2	no	—
78	#12674	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₅₃	#509	1	2	yes	—
83	C₂²×C₅₃	#510	1	2	yes	—
84	C₄×C₅₃	—	8	4	no	—
85	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₃	#427	1	1	yes	characteristic
87	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	—	8	4	no	—
91	C₂×#367	—	53	1	no	—
92	C₂×#366	D₅₃	1	1	yes	characteristic
93	C₂×#346	—	53	1	no	—
94	C₄×#107	—	53	2	no	—
95	#1380	—	53	2	no	—
96	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
97	C₂×C₅₃	#2159	1	2	yes	—
98	C₂×C₅₃	#2175	1	2	yes	—
99	C₂×C₅₃	#2176	1	1	yes	characteristic
100	C₂×C₅₃	#2177	1	1	yes	characteristic
101	C₂²×C₁₆	—	53	1	no	—
102	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
103	C₂×#107	—	53	2	no	—
104	C₂×#107	C₂×D₅₃	1	2	yes	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₄×C₄⋊C₄	—	106	2	no	—
107	#367	—	53	4	no	—
108	#366	Dic₅₃	1	4	yes	complemented
109	#346	—	53	4	no	—
110	#340	—	106	2	no	—
111	C₅₃	#12674	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	53	1	no	—
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	—	53	1	no	—
117	C₂×C₄²	—	212	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	106	1	no	—
120	C₂×C₁₆	—	106	1	no	—
121	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
122	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	53	4	no	—
126	#107	—	106	2	no	—
127	#107	C₂×Dic₅₃	1	4	yes	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	53	2	no	—
130	#95	—	212	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	53	2	no	—
133	C₂²×C₄	—	212	2	no	—
134	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₈	—	53	2	no	—
136	C₂×C₈	—	106	1	no	—
137	C₂×C₈	—	106	1	no	—
138	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₈	#11719947	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719949	1	1	yes	characteristic
143	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	106	4	no	—
147	C₄⋊C₄	—	212	2	no	—
148	C₄²	—	53	2	no	—
149	C₄²	—	53	2	no	—
150	C₄²	—	424	4	no	—
151	C₁₆	—	106	2	no	—
152	C₁₆	C₄×Dic₅₃	1	4	yes	complemented
153	C₂³	D₈×D₅₃	1	1	yes	characteristic
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	—	424	2	no	—
162	C₂×C₄	—	212	4	no	—
163	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	#13133179	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
167	C₂×C₄	#13133236	1	1	yes	characteristic
168	C₂×C₄	#13133240	1	1	yes	characteristic
169	C₈	—	106	2	no	—
170	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
171	C₈	#13133175	1	1	yes	characteristic
172	C₈	#13133182	1	1	yes	characteristic
173	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
174	C₂²	#107×D₅₃	1	1	yes	characteristic
175	C₂²	#16810705	1	1	yes	characteristic
176	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
177	C₂²	#16811057	1	2	yes	—
178	C₂²	#16811065	1	2	yes	—
179	C₄	—	8	4	no	—
180	C₄	—	106	4	no	—
181	C₄	—	424	4	no	—
182	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
183	C₄	#16810660	1	1	yes	characteristic
184	C₄	#16810717	1	1	yes	characteristic
185	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
186	C₂	#107×Dic₅₃	1	1	yes	characteristic
187	C₂	#20276500	1	2	yes	—
188	C₂	#20276578	1	2	yes	—
189	C₂	#20276586	1	1	yes	characteristic
190	C₂	#20276591	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
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	#12674
LCS	G ▷ 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
10. #16810451	C₄	semidirect
11. #16810398	C₄	semidirect
15. C₅₃×#366	C₄	semidirect
55. C₁₆×C₅₃	C₄²	semidirect
108. #366	Dic₅₃	semidirect
111. C₅₃	#12674	semidirect
152. 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×#16810451	2
C₂×#16810398	2
#107×Dic₅₃	2
#20272806	2
C₂×C₅₃×#366	2
#12674	53