Subgroups — #27300539

Order 13568.

314 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×#16810453	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810452	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274885	C₂	1	2	yes	maximal
5	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	C₂×#13133152	—	2	2	no	—
9	#16810646	—	2	2	no	—
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810453	—	2	2	no	—
12	#16810452	—	2	4	no	—
13	C₁₆×Dic₅₃	C₄	1	2	yes	—
14	C₁₆×Dic₅₃	C₄	1	2	yes	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#368	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂²×#88322	D₄	1	1	yes	characteristic
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	#13133153	—	2	2	no	—
25	#13133153	—	2	2	no	—
26	#13133152	—	4	4	no	—
27	#13133146	—	4	4	no	—
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
31	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
36	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
37	#108×C₅₃	—	2	2	no	—
38	#108×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
41	M₁₆×C₅₃	—	4	2	no	—
42	M₁₆×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₅₃	D₈	1	2	yes	—
53	#11719917	—	4	2	no	—
54	#11719917	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#88322	D₈	1	2	yes	—
59	C₂×#88322	Q₁₆	1	2	yes	—
60	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
61	#11719908	C₄⋊C₄	1	4	yes	—
62	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	#107	1	2	yes	—
71	C₂×Dic₅₃	#107	1	2	yes	—
72	C₈×C₅₃	—	8	2	no	—
73	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	#88322	—	8	2	no	—
78	#88322	#107	1	2	yes	—
79	#88322	#107	1	2	yes	—
80	#12909	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
83	C₂²×C₅₃	#494	1	1	yes	characteristic
84	C₂²×C₅₃	#500	1	1	yes	characteristic
85	C₄×C₅₃	#384	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₄×C₅₃	#493	1	1	yes	characteristic
88	C₄×C₅₃	#500	1	1	yes	characteristic
89	Dic₅₃	—	2	2	no	—
90	Dic₅₃	#368	1	4	yes	complemented
91	C₂×#368	—	53	1	no	—
92	C₂×#368	—	53	2	no	—
93	C₂×#368	D₅₃	1	1	yes	characteristic
94	C₂×C₄×C₁₆	—	53	1	no	—
95	#1848	—	53	2	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	#1845	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#368	1	2	yes	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂×#108	—	53	2	no	—
103	C₂×#108	—	106	2	no	—
104	C₂×#108	C₂×D₅₃	1	2	yes	—
105	#423	—	106	2	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	#368	—	2	2	no	—
108	#368	—	106	4	no	—
109	#368	—	106	2	no	—
110	C₄×C₁₆	—	53	2	no	—
111	C₄×C₁₆	—	53	2	no	—
112	C₅₃	#12909	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	106	2	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
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	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	106	2	no	—
127	#108	—	106	2	no	—
128	#108	—	212	4	no	—
129	C₄⋊C₈	—	212	4	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	53	1	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	—	53	1	no	—
134	C₂²×C₄	—	53	2	no	—
135	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	212	2	no	—
139	M₁₆	—	212	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₄²	—	53	1	no	—
153	C₄²	—	53	1	no	—
154	C₄²	—	53	2	no	—
155	C₁₆	—	53	4	no	—
156	C₁₆	#11719919	1	4	yes	—
157	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
158	C₂×C₄	—	53	2	no	—
159	C₂×C₄	—	53	2	no	—
160	C₂×C₄	—	106	4	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	D₈×D₅₃	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	53	2	no	—
169	C₈	—	53	2	no	—
170	C₈	—	424	2	no	—
171	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
172	C₈	C₂×#11719919	1	1	yes	characteristic
173	C₈	#13133215	1	1	yes	characteristic
174	C₈	#13133216	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	#107×D₅₃	1	1	yes	characteristic
177	C₂²	#16811001	1	1	yes	characteristic
178	C₂²	#16811027	1	1	yes	characteristic
179	C₄	—	53	4	no	—
180	C₄	—	106	2	no	—
181	C₄	#16810492	1	1	yes	characteristic
182	C₄	#107×D₅₃	1	1	yes	characteristic
183	C₄	#16810998	1	1	yes	characteristic
184	C₄	#16811027	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#20274873	1	1	yes	characteristic
188	C₂	#368×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	#12909
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
90. Dic₅₃	#368	semidirect
112. C₅₃	#12909	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×#16810452	2
C₂×C₁₆×Dic₅₃	2
#20274885	2
C₂×C₅₃×#368	2
#12909	53