Subgroups — #27300524

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 190 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×#16810448	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274873	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×#13133150	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810492	—	2	2	no	—
11	#16810449	—	2	2	no	—
12	#16810448	—	2	4	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×#11719919	—	4	4	no	—
20	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₂×#11719908	D₄	1	2	yes	—
23	C₂×#11719908	Q₈	1	2	yes	—
24	#13133151	—	2	2	no	—
25	#13133151	—	2	2	no	—
26	#13133150	—	4	4	no	—
27	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
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	#11719919	—	8	8	no	—
51	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₅₃	Q₁₆	1	2	yes	—
53	C₂×#88322	C₄⋊C₄	1	2	yes	—
54	C₂×#88322	D₈	1	2	yes	—
55	C₂×#88322	Q₁₆	1	2	yes	—
56	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
57	#11719908	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	4	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₅₃	—	8	2	no	—
68	C₂×Dic₅₃	—	8	4	no	—
69	C₂×Dic₅₃	#107	1	4	yes	—
70	C₂×Dic₅₃	D₁₆	1	4	yes	—
71	C₂×Dic₅₃	Q₃₂	1	4	yes	—
72	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	#88322	#107	1	4	yes	—
77	#12905	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₅₃	#493	1	1	yes	characteristic
80	C₂²×C₅₃	#500	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
82	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
83	C₄×C₅₃	—	16	2	no	—
84	C₄×C₅₃	#384	1	1	yes	characteristic
85	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
86	C₄×C₅₃	#494	1	1	yes	characteristic
87	C₄×C₅₃	#500	1	1	yes	characteristic
88	Dic₅₃	—	16	2	no	—
89	Dic₅₃	#366	1	8	yes	complemented
90	C₂×#366	—	53	1	no	—
91	C₂×#366	—	53	2	no	—
92	C₂×#366	D₅₃	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	53	1	no	—
94	#1845	—	53	2	no	—
95	C₂×C₅₃	#1845	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#366	1	2	yes	—
97	C₂×C₅₃	#2246	1	1	yes	characteristic
98	C₂×C₅₃	#2247	1	1	yes	characteristic
99	C₂×C₅₃	#2252	1	2	yes	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂×#107	—	53	2	no	—
103	C₂×#107	—	106	2	no	—
104	C₂×#107	C₂×D₅₃	1	2	yes	—
105	C₂×C₄×C₈	—	53	1	no	—
106	#384	—	106	2	no	—
107	#366	—	2	2	no	—
108	#366	—	106	4	no	—
109	#366	—	106	2	no	—
110	C₄×C₁₆	—	53	4	no	—
111	C₅₃	#12905	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×C₄⋊C₄	—	212	4	no	—
117	C₂×C₄²	—	53	1	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
122	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	106	2	no	—
127	#107	—	106	2	no	—
128	#107	—	212	4	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	53	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	—	53	2	no	—
137	C₂×C₈	—	53	2	no	—
138	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	424	8	no	—
147	C₄⋊C₄	—	212	2	no	—
148	C₄⋊C₄	—	212	2	no	—
149	C₄²	—	53	2	no	—
150	C₄²	—	53	2	no	—
151	C₁₆	—	53	4	no	—
152	C₁₆	#11719919	1	4	yes	complemented
153	C₂³	D₈×D₅₃	1	1	yes	characteristic
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	53	4	no	—
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	—	53	4	no	—
159	C₂×C₄	—	424	4	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₄	#13133213	1	1	yes	characteristic
163	C₂×C₄	#13133216	1	1	yes	characteristic
164	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
167	C₈	—	53	4	no	—
168	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
169	C₈	C₂×#11719919	1	1	yes	characteristic
170	C₈	#13133215	1	1	yes	characteristic
171	C₈	#13133216	1	1	yes	characteristic
172	C₂²	#107×D₅₃	1	1	yes	characteristic
173	C₂²	#16810998	1	1	yes	characteristic
174	C₂²	#16811027	1	1	yes	characteristic
175	C₂²	D₁₆×D₅₃	1	2	yes	—
176	C₂²	Q₃₂×D₅₃	1	2	yes	—
177	C₄	—	16	2	no	—
178	C₄	—	53	8	no	—
179	C₄	—	848	2	no	—
180	C₄	#16810492	1	1	yes	characteristic
181	C₄	#107×D₅₃	1	1	yes	characteristic
182	C₄	#16811001	1	1	yes	characteristic
183	C₄	#16811027	1	1	yes	characteristic
184	C₂	#20274873	1	1	yes	characteristic
185	C₂	#366×D₅₃	1	2	yes	—
186	C₂	#20276963	1	1	yes	characteristic
187	C₂	#20276966	1	1	yes	characteristic
188	C₂	#20276989	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	#12905
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
56. C₁₆×C₅₃	C₄⋊C₄	semidirect
57. #11719908	C₄⋊C₄	semidirect
89. Dic₅₃	#366	semidirect
111. C₅₃	#12905	semidirect
152. C₁₆	#11719919	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×#16810448	2
C₂×C₁₆×Dic₅₃	2
#20274873	2
C₂×C₅₃×#366	2
#12905	53