Subgroups — #28303530

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290647	C₂	1	2	yes	maximal
2	#20289545	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88324	—	2	2	no	—
5	C₂×C₄⋊#88324	—	2	2	no	—
6	C₂×C₄⋊#88324	C₂²	1	4	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	—	2	4	no	—
13	C₄⋊#88324	—	2	8	no	—
14	C₄⋊#88324	—	4	4	no	—
15	C₄⋊#88324	—	4	4	no	—
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
23	C₄×#88322	C₈	1	4	yes	complemented
24	C₈×Dic₅₃	—	2	4	no	—
25	C₈×Dic₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×#88324	—	4	4	no	—
37	C₂×#88324	—	4	4	no	—
38	C₂×#88324	—	4	4	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
42	C₄²×C₅₃	C₂×C₈	1	2	yes	—
43	C₄²×C₅₃	M₁₆	1	2	yes	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	D₈	1	2	yes	—
48	C₄×Dic₅₃	SD₁₆	1	4	yes	—
49	C₄×Dic₅₃	Q₁₆	1	2	yes	—
50	C₄×Dic₅₃	#41	1	4	yes	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	C₂×C₈	1	4	yes	—
54	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	#106	1	2	yes	—
58	C₂×C₄×C₅₃	#107	1	2	yes	—
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₂×C₄×C₅₃	#108	1	2	yes	—
61	C₂×C₄×C₅₃	#117	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
65	C₂×Dic₅₃	—	2	1	no	—
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	—	2	1	no	—
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	2	1	no	—
71	C₂×Dic₅₃	#106	1	2	yes	—
72	C₂×Dic₅₃	#107	1	2	yes	—
73	C₂×Dic₅₃	#124	1	2	yes	—
74	C₂×Dic₅₃	#125	1	2	yes	—
75	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
77	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	#88324	—	8	4	no	—
79	C₈×C₅₃	—	2	4	no	—
80	#88322	—	2	4	no	—
81	#8784	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	#382	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₅₃	#423	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₅₃	#334	1	4	yes	—
90	C₄×C₅₃	#335	1	4	yes	—
91	C₄×C₅₃	#431	1	2	yes	—
92	C₄×C₅₃	#432	1	2	yes	—
93	Dic₅₃	—	2	2	no	—
94	Dic₅₃	—	2	2	no	—
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	#486	1	2	yes	—
97	Dic₅₃	#487	1	2	yes	—
98	#1842	—	53	2	no	—
99	C₄²×C₈	—	53	1	no	—
100	#1277	—	53	4	no	—
101	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
103	C₂×C₅₃	#1566	1	1	yes	characteristic
104	C₂×C₅₃	#1567	1	1	yes	characteristic
105	C₂×C₅₃	#1831	1	1	yes	characteristic
106	C₂×C₅₃	#1834	1	1	yes	characteristic
107	C₂×C₅₃	#1837	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	53	4	no	—
109	C₂×C₄⋊C₈	—	106	2	no	—
110	C₂×C₄⋊C₈	—	106	2	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	—	53	2	no	—
114	C₂×C₄×C₈	#6157	1	1	yes	characteristic
115	C₄³	—	53	1	no	—
116	C₅₃	#8784	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	—	106	4	no	—
119	C₂²×C₈	C₂×#6157	1	2	yes	—
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	53	1	no	—
123	C₂×C₄²	—	53	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	53	1	no	—
126	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
127	C₄⋊C₈	—	106	8	no	—
128	C₄⋊C₈	—	212	4	no	—
129	C₄⋊C₈	—	212	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	53	4	no	—
132	C₄×C₈	—	106	4	no	—
133	C₄×C₈	—	106	4	no	—
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	53	1	no	—
138	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	53	4	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	106	2	no	—
146	C₂×C₈	—	212	4	no	—
147	C₂×C₈	—	212	4	no	—
148	C₂×C₈	—	212	4	no	—
149	C₂×C₈	#11719936	1	4	yes	—
150	C₄²	—	53	4	no	—
151	C₄²	—	53	2	no	—
152	C₄²	—	53	2	no	—
153	C₄²	—	53	4	no	—
154	C₄²	—	106	2	no	—
155	C₄²	—	106	2	no	—
156	C₄²	—	106	2	no	—
157	C₄²	D₅₃⋊C₈	1	2	yes	—
158	C₄²	#11719940	1	2	yes	—
159	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	53	2	no	—
162	C₂×C₄	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	53	2	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	106	1	no	—
168	C₂×C₄	—	106	1	no	—
169	C₂×C₄	—	106	1	no	—
170	C₂×C₄	—	106	1	no	—
171	C₂×C₄	—	106	1	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	#13133284	1	2	yes	—
174	C₂×C₄	#13133284	1	2	yes	—
175	C₂×C₄	#13133285	1	2	yes	—
176	C₂×C₄	#13133286	1	2	yes	—
177	C₂×C₄	#13133287	1	2	yes	—
178	C₂×C₄	#13133288	1	2	yes	—
179	C₂×C₄	#13133296	1	2	yes	—
180	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
181	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
182	C₂×C₄	C₂×#11719936	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	106	4	no	—
185	C₈	—	424	4	no	—
186	C₂²	#16811567	1	1	yes	characteristic
187	C₂²	C₂×#13133284	1	1	yes	characteristic
188	C₂²	#16811627	1	1	yes	characteristic
189	C₂²	C₂×#13133285	1	1	yes	characteristic
190	C₂²	C₂×#13133286	1	1	yes	characteristic
191	C₂²	C₂×#13133287	1	1	yes	characteristic
192	C₂²	C₂×#13133288	1	1	yes	characteristic
193	C₄	—	53	2	no	—
194	C₄	—	53	2	no	—
195	C₄	—	106	2	no	—
196	C₄	—	106	2	no	—
197	C₄	—	106	2	no	—
198	C₄	#16811485	1	4	yes	—
199	C₄	#16811487	1	4	yes	—
200	C₄	#16811642	1	2	yes	—
201	C₄	#16811644	1	2	yes	—
202	C₂	C₂×#16811485	1	1	yes	characteristic
203	C₂	C₂×#16811487	1	1	yes	characteristic
204	C₂	#20290270	1	1	yes	characteristic
205	C₂	#20290273	1	1	yes	characteristic
206	C₂	#20290632	1	1	yes	characteristic
207	C₂	#20290636	1	1	yes	characteristic
208	C₂	#20290639	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
23. C₄×#88322	C₈	semidirect
116. C₅₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#20290647	2
#20289545	2
C₄×C₈×Dic₅₃	2
#8784	53