Subgroups — #27299415

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810443	C₂	1	2	yes	maximal
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20274401	C₂	1	1	yes	maximal, characteristic
4	#20272796	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810469	C₄	1	1	yes	characteristic
13	#16810443	C₄	1	8	yes	complemented
14	#16810399	—	2	1	no	—
15	C₄×#11719908	—	2	1	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#346	—	2	1	no	—
19	C₄×C₁₆×C₅₃	—	2	1	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	C₂×C₄	1	4	yes	—
26	C₂×#11719908	—	2	2	no	—
27	#13133128	C₈	1	4	yes	—
28	#13133127	—	2	2	no	—
29	#13133127	C₈	1	4	yes	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
32	#13133124	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	2	no	—
37	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	2	1	no	—
46	C₂²×Dic₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	1	no	—
60	C₄×Dic₅₃	C₄²	1	2	yes	—
61	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	C₄²	1	2	yes	—
65	C₂×#88322	C₂×C₈	1	2	yes	—
66	C₂×#88322	C₂×C₈	1	2	yes	—
67	C₂×#88322	C₂×C₈	1	2	yes	—
68	C₁₆×C₅₃	—	2	4	no	—
69	#11719908	—	2	4	no	—
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
86	C₈×C₅₃	C₄×D₄	1	2	yes	—
87	C₈×C₅₃	C₄×Q₈	1	2	yes	—
88	#88322	—	2	2	no	—
89	#88322	C₄×C₈	1	4	yes	—
90	#12656	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₅₃	#431	1	1	yes	characteristic
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	#431	1	1	yes	characteristic
101	Dic₅₃	—	4	2	no	—
102	Dic₅₃	—	4	2	no	—
103	C₂×#364	—	53	2	no	—
104	C₄×M₃₂	—	53	1	no	—
105	C₄×M₃₂	D₅₃	1	1	yes	characteristic
106	#1726	—	53	1	no	—
107	#1376	—	53	2	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₅₃	#2153	1	2	yes	—
112	C₂×M₃₂	—	53	2	no	—
113	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
114	C₂×#97	—	53	1	no	—
115	C₂×#97	—	53	1	no	—
116	C₂×#97	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#376	—	53	1	no	—
121	#364	—	53	8	no	—
122	#346	—	2	1	no	—
123	#346	—	106	1	no	—
124	C₄×C₁₆	—	2	1	no	—
125	C₄×C₁₆	—	106	1	no	—
126	C₅₃	#12656	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
134	M₃₂	—	2	2	no	—
135	M₃₂	—	53	4	no	—
136	M₃₂	—	106	2	no	—
137	M₃₂	C₄×D₅₃	1	4	yes	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	106	2	no	—
140	#97	—	53	4	no	—
141	#97	—	53	4	no	—
142	#97	—	106	2	no	—
143	C₄×C₈	—	53	4	no	—
144	C₄×C₈	—	106	2	no	—
145	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
146	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
149	#95	—	106	2	no	—
150	C₂²×C₄	—	106	1	no	—
151	C₂²×C₄	—	106	1	no	—
152	C₂²×C₄	#11719912	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	106	1	no	—
159	C₂×C₈	—	106	1	no	—
160	C₂×C₈	C₈×D₅₃	1	2	yes	—
161	C₂×C₈	#11719912	1	2	yes	—
162	C₂×C₈	#11719919	1	1	yes	characteristic
163	C₂×C₈	#11719919	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
166	C₂×C₈	#11719945	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	C₄²	—	53	2	no	—
169	C₄²	—	53	2	no	—
170	C₄²	—	106	1	no	—
171	C₄²	—	106	1	no	—
172	C₄²	C₈×D₅₃	1	2	yes	—
173	C₄²	#11719919	1	1	yes	characteristic
174	C₄²	#11719919	1	1	yes	characteristic
175	C₁₆	—	2	4	no	—
176	C₁₆	—	106	4	no	—
177	C₂³	C₂×#11719912	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	106	2	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	212	2	no	—
182	C₂×C₄	—	212	2	no	—
183	C₂×C₄	#13133146	1	2	yes	—
184	C₂×C₄	#13133146	1	2	yes	—
185	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
187	C₂×C₄	#13133174	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
191	C₈	—	53	4	no	—
192	C₈	—	106	2	no	—
193	C₈	#13133146	1	2	yes	—
194	C₈	#13133146	1	2	yes	—
195	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
196	C₈	C₄×Dic₁₀₆	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄×#11719912	1	1	yes	characteristic
199	C₂²	C₂×#13133146	1	1	yes	characteristic
200	C₂²	#16810674	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	212	2	no	—
204	C₄	—	212	2	no	—
205	C₄	C₄×#11719919	1	1	yes	characteristic
206	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
207	C₄	C₂×#13133146	1	1	yes	characteristic
208	C₄	#16810674	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×#13133146	1	1	yes	characteristic
212	C₂	#20276480	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12656
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #16810443	C₄	semidirect
126. C₅₃	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810443	2
C₄×#13133157	2
#20274401	2
#20272796	2
C₄×M₃₂×C₅₃	2
#12656	53