Subgroups — #27537470

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283583	C₂	1	2	yes	maximal
2	#20276586	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₅₃	C₂	1	4	yes	maximal, complemented
4	#20276499	C₂	1	1	yes	maximal, characteristic
5	#20276435	C₂	1	2	yes	maximal
6	#20276311	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811101	—	2	2	no	—
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	#16810730	—	2	4	no	—
11	Q₁₆×Dic₅₃	C₂²	1	8	yes	—
12	#16810661	C₂²	1	2	yes	—
13	#16810632	—	2	2	no	—
14	#16810578	C₂²	1	1	yes	characteristic
15	#16810451	C₂²	1	2	yes	—
16	#16810449	C₂²	1	2	yes	—
17	#16810427	C₂²	1	8	yes	—
18	#16810398	C₂²	1	2	yes	—
19	C₁₆×Dic₅₃	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₅₃	C₄	1	8	yes	complemented
22	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	D₄	1	1	yes	characteristic
26	Q₈×Dic₅₃	—	2	8	no	—
27	Q₈×Dic₅₃	—	4	4	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	4	4	no	—
30	#13133176	D₄	1	1	yes	characteristic
31	#13133157	—	2	2	no	—
32	C₂×#11719908	C₂³	1	2	yes	—
33	#13133151	C₂³	1	4	yes	—
34	#13133149	—	2	2	no	—
35	#13133141	—	2	8	no	—
36	#13133127	D₄	1	2	yes	—
37	C₈×Dic₅₃	C₂³	1	2	yes	—
38	C₂²×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₁₆×C₅₃	—	2	2	no	—
40	C₂×Q₁₆×C₅₃	—	2	4	no	—
41	C₂×Q₁₆×C₅₃	C₂×C₄	1	8	yes	—
42	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	Q₃₂×C₅₃	C₂×C₄	1	8	yes	—
45	Q₃₂×C₅₃	C₂×C₄	1	8	yes	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	—	2	8	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₅₃	—	2	8	no	—
56	Q₁₆×C₅₃	—	2	4	no	—
57	Q₁₆×C₅₃	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₅₃	—	4	2	no	—
63	C₂²⋊Dic₅₃	D₈	1	2	yes	—
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	4	8	no	—
66	C₄⋊Dic₅₃	D₈	1	2	yes	—
67	C₄×Dic₅₃	—	4	2	no	—
68	C₄×Dic₅₃	—	4	8	no	—
69	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
70	#11719917	—	2	2	no	—
71	C₂×#88322	C₂×D₄	1	2	yes	—
72	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
73	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
74	#11719908	—	2	2	no	—
75	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₅₃	—	2	8	no	—
77	Q₈×C₅₃	—	4	4	no	—
78	Q₈×C₅₃	—	4	8	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	8	no	—
81	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#135	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₅₃	—	4	4	no	—
86	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
87	C₈×C₅₃	C₄×D₄	1	2	yes	—
88	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
90	#88322	—	2	2	no	—
91	#33458	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	C₄×D₈	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₃	—	4	8	no	—
96	C₄×C₅₃	C₄×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	—	8	4	no	—
101	C₂²×Q₃₂	D₅₃	1	1	yes	characteristic
102	#2950	—	53	2	no	—
103	#2176	—	53	4	no	—
104	C₄×Q₃₂	—	53	4	no	—
105	#2158	—	53	1	no	—
106	#2141	—	53	2	no	—
107	#2108	—	53	1	no	—
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₅₃	#3417	1	2	yes	—
111	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
112	#519	—	106	2	no	—
113	C₂×Q₃₂	Dic₅₃	1	8	yes	complemented
114	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
115	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
116	C₂×M₃₂	—	53	1	no	—
117	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
118	#441	—	106	4	no	—
119	C₄×Q₁₆	—	53	8	no	—
120	#427	—	53	2	no	—
121	#419	—	106	2	no	—
122	#405	—	53	1	no	—
123	#367	—	53	2	no	—
124	#366	—	53	2	no	—
125	#358	—	53	8	no	—
126	#346	—	53	2	no	—
127	C₄×C₁₆	—	53	2	no	—
128	C₅₃	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×Dic₅₃	1	8	yes	—
133	C₂×Q₁₆	C₂²×D₅₃	1	4	yes	—
134	C₂×M₁₆	—	53	1	no	—
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄×Q₈	—	106	8	no	—
137	C₄×Q₈	—	212	4	no	—
138	#117	—	53	1	no	—
139	#117	—	106	2	no	—
140	#117	—	212	4	no	—
141	Q₃₂	C₂×Dic₅₃	1	8	yes	—
142	Q₃₂	C₂×Dic₅₃	1	8	yes	—
143	M₃₂	—	106	2	no	—
144	C₂×C₁₆	—	53	2	no	—
145	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
146	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
148	#107	—	53	4	no	—
149	#106	—	106	2	no	—
150	#103	—	106	8	no	—
151	#97	—	53	2	no	—
152	C₄×C₈	—	53	2	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×Dic₅₃	1	8	yes	—
162	M₁₆	—	106	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
165	C₂×C₈	#11719947	1	2	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
167	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	53	2	no	—
169	C₄⋊C₄	—	106	4	no	—
170	C₄⋊C₄	—	212	8	no	—
171	C₂²⋊C₄	—	53	2	no	—
172	C₂²⋊C₄	—	212	2	no	—
173	C₄²	—	53	2	no	—
174	C₄²	—	212	8	no	—
175	C₄²	—	212	2	no	—
176	C₁₆	—	106	2	no	—
177	C₁₆	C₂²×Dic₅₃	1	2	yes	—
178	C₁₆	C₂²×Dic₅₃	1	2	yes	—
179	C₂³	D₈×D₅₃	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	53	4	no	—
186	C₂×C₄	—	212	4	no	—
187	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
188	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
189	C₂×C₄	#13133236	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
191	C₈	—	106	2	no	—
192	C₈	D₄×Dic₅₃	1	2	yes	—
193	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
194	C₈	C₂×#11719947	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₅₃	1	2	yes	—
197	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	106	4	no	—
200	C₄	—	424	4	no	—
201	C₄	D₈×Dic₅₃	1	2	yes	—
202	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
203	C₄	C₂×#13133236	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
206	C₂	#20287864	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	#33458
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20276586	C₂	semidirect
3. Q₃₂×Dic₅₃	C₂	semidirect
21. C₂×Q₃₂×C₅₃	C₄	semidirect
113. C₂×Q₃₂	Dic₅₃	semidirect
128. C₅₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
#20283583	2
#20276586	2
Q₃₂×Dic₅₃	2
#20276499	2
#20276435	2
#20276311	2
C₂²×Q₃₂×C₅₃	2
#33458	53