Subgroups — #27299402

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810441	C₂	1	2	yes	maximal
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810441	C₄	1	8	yes	complemented
14	#16810399	—	2	2	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	C₂×C₄	1	4	yes	—
27	C₄×#88322	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	C₈	1	4	yes	—
31	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
44	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	2	yes	—
52	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₄²×C₅₃	M₁₆	1	2	yes	—
54	C₄×Dic₅₃	—	2	1	no	—
55	C₄×Dic₅₃	—	2	1	no	—
56	C₄×Dic₅₃	—	2	4	no	—
57	C₄×Dic₅₃	—	2	4	no	—
58	C₄×Dic₅₃	C₄²	1	2	yes	—
59	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	1	no	—
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	C₁₆×C₅₃	—	2	4	no	—
67	#11719908	—	2	4	no	—
68	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
82	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
83	C₂×Dic₅₃	M₃₂	1	4	yes	—
84	C₂×Dic₅₃	M₃₂	1	4	yes	—
85	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
86	C₈×C₅₃	C₄×D₄	1	2	yes	—
87	C₈×C₅₃	C₄×Q₈	1	2	yes	—
88	#88322	—	2	2	no	—
89	#88322	—	2	2	no	—
90	#12653	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₃	#431	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
95	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
99	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	#431	1	1	yes	characteristic
101	Dic₅₃	—	2	2	no	—
102	Dic₅₃	—	2	4	no	—
103	Dic₅₃	#346	1	4	yes	complemented
104	C₂×C₄⋊C₁₆	—	53	2	no	—
105	C₂×#346	—	53	1	no	—
106	C₂×#346	D₅₃	1	1	yes	characteristic
107	C₄²×C₈	—	53	1	no	—
108	#1374	—	53	2	no	—
109	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
111	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
112	C₂×C₅₃	#2150	1	2	yes	—
113	C₂×C₅₃	#2163	1	2	yes	—
114	C₂²×C₁₆	—	53	2	no	—
115	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	C₄³	—	53	1	no	—
123	C₄⋊C₁₆	—	53	8	no	—
124	#346	—	2	2	no	—
125	#346	—	106	2	no	—
126	C₅₃	#12653	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	—	53	2	no	—
134	C₂×C₄²	—	53	2	no	—
135	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	53	4	no	—
139	C₂×C₁₆	—	106	2	no	—
140	C₂×C₁₆	—	106	2	no	—
141	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
142	C₄×C₈	—	53	4	no	—
143	C₄×C₈	—	53	4	no	—
144	C₄×C₈	—	106	2	no	—
145	C₄×C₈	—	106	2	no	—
146	C₄×C₈	—	106	2	no	—
147	C₄×C₈	Dic₁₀₆	1	2	yes	—
148	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	2	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	C₈×D₅₃	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	—	106	1	no	—
157	C₂×C₈	—	106	1	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	2	yes	—
163	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
165	C₂×C₈	#11719945	1	2	yes	—
166	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₄²	—	53	2	no	—
168	C₄²	—	53	2	no	—
169	C₄²	—	106	4	no	—
170	C₄²	—	106	4	no	—
171	C₄²	—	106	1	no	—
172	C₄²	—	106	1	no	—
173	C₄²	#11719912	1	2	yes	—
174	C₄²	#11719919	1	2	yes	—
175	C₁₆	—	2	4	no	—
176	C₁₆	—	106	4	no	—
177	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	53	2	no	—
182	C₂×C₄	—	53	4	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	106	2	no	—
185	C₂×C₄	—	106	2	no	—
186	C₂×C₄	—	106	2	no	—
187	C₂×C₄	#13133146	1	4	yes	—
188	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
190	C₂×C₄	#13133174	1	1	yes	characteristic
191	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
194	C₈	—	106	2	no	—
195	C₈	—	106	2	no	—
196	C₈	#13133146	1	4	yes	—
197	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
198	C₈	C₄×Dic₁₀₆	1	2	yes	—
199	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
200	C₂²	C₂×#13133146	1	1	yes	characteristic
201	C₂²	#16810674	1	1	yes	characteristic
202	C₂²	M₃₂×D₅₃	1	2	yes	—
203	C₂²	M₃₂×D₅₃	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	53	4	no	—
206	C₄	—	106	4	no	—
207	C₄	—	106	2	no	—
208	C₄	C₄×#11719919	1	1	yes	characteristic
209	C₄	C₄×#11719912	1	1	yes	characteristic
210	C₄	C₂×#13133146	1	1	yes	characteristic
211	C₄	#16810674	1	1	yes	characteristic
212	C₂	C₄×#13133146	1	1	yes	characteristic
213	C₂	#346×D₅₃	1	1	yes	characteristic
214	C₂	#20276308	1	1	yes	characteristic
215	C₂	#20276470	1	2	yes	—
216	C₂	#20276516	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12653
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #16810441	C₄	semidirect
103. Dic₅₃	#346	semidirect
126. C₅₃	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810441	2
C₂×#16810399	2
C₄×C₈×Dic₅₃	2
#20272792	2
C₂×C₅₃×#346	2
#12653	53