Subgroups — #27299403

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal
5	C₂×C₅₃×C₄⋊C₁₆	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	C₄⋊#11719908	C₄	1	4	yes	complemented
14	#16810398	—	2	4	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
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	C₈	1	4	yes	—
28	C₈×Dic₅₃	—	2	1	no	—
29	C₈×Dic₅₃	—	2	1	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	—	2	2	no	—
32	C₈×Dic₅₃	D₄	1	2	yes	—
33	C₈×Dic₅₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
46	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₄²	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₄²×C₅₃	C₄²	1	2	yes	—
57	C₄²×C₅₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₃	—	2	4	no	—
59	C₄×Dic₅₃	—	2	1	no	—
60	C₄×Dic₅₃	—	2	1	no	—
61	C₄×Dic₅₃	—	2	4	no	—
62	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₅₃	M₁₆	1	2	yes	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	C₄⋊C₄	1	2	yes	—
67	C₂×#88322	C₂×C₈	1	2	yes	—
68	C₂×#88322	C₂×C₈	1	2	yes	—
69	C₂×#88322	M₁₆	1	2	yes	—
70	C₁₆×C₅₃	—	2	4	no	—
71	#11719908	—	2	4	no	—
72	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
88	C₂×Dic₅₃	M₃₂	1	4	yes	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
94	#88322	—	2	2	no	—
95	#88322	C₄⋊C₈	1	4	yes	—
96	#12653	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₅₃	#431	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
101	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	#346	1	4	yes	complemented
104	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
106	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₅₃	#431	1	1	yes	characteristic
108	Dic₅₃	—	2	4	no	—
109	Dic₅₃	—	2	4	no	—
110	C₂×C₄⋊C₁₆	—	53	1	no	—
111	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
112	C₂×#346	—	53	2	no	—
113	C₄²×C₈	—	53	1	no	—
114	#1374	—	53	2	no	—
115	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
117	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
118	C₂×C₅₃	#2150	1	2	yes	—
119	C₂×C₅₃	#2163	1	2	yes	—
120	C₂²×C₁₆	—	53	2	no	—
121	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄³	—	53	1	no	—
129	C₄⋊C₁₆	—	53	4	no	—
130	C₄⋊C₁₆	Dic₅₃	1	4	yes	complemented
131	#346	—	106	4	no	—
132	C₅₃	#12653	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₂×C₄²	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	2	no	—
140	C₂×C₄²	—	53	2	no	—
141	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	53	4	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	—	106	2	no	—
147	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	4	no	—
151	C₄×C₈	—	106	2	no	—
152	C₄×C₈	—	106	2	no	—
153	C₄×C₈	—	106	1	no	—
154	C₄×C₈	—	106	1	no	—
155	C₄×C₈	C₄×D₅₃	1	2	yes	—
156	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
157	C₂²×C₄	—	53	2	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
160	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
171	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
173	C₂×C₈	#11719947	1	1	yes	characteristic
174	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
175	C₂×C₈	#11719949	1	1	yes	characteristic
176	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₄²	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	—	106	4	no	—
180	C₄²	—	106	4	no	—
181	C₄²	—	106	1	no	—
182	C₄²	—	106	1	no	—
183	C₄²	C₈×D₅₃	1	2	yes	—
184	C₄²	C₄×Dic₅₃	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₁₆	—	106	4	no	—
187	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	—	106	2	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
196	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
197	C₂×C₄	#13133156	1	4	yes	—
198	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄	#13133175	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
202	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
203	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	53	4	no	—
206	C₈	—	106	2	no	—
207	C₈	#13133179	1	1	yes	characteristic
208	C₈	D₄×Dic₅₃	1	1	yes	characteristic
209	C₈	#13133182	1	1	yes	characteristic
210	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
211	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
212	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
213	C₂²	#16810675	1	1	yes	characteristic
214	C₂²	M₃₂×D₅₃	1	2	yes	—
215	C₂²	C₂×#13133156	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	106	4	no	—
218	C₄	—	106	4	no	—
219	C₄	#16810398	1	4	yes	complemented
220	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
221	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
222	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
223	C₄	#16810675	1	1	yes	characteristic
224	C₂	C₄⋊C₈×Dic₅₃	1	1	yes	characteristic
225	C₂	C₂×#16810398	1	1	yes	characteristic
226	C₂	M₃₂×Dic₅₃	1	1	yes	characteristic
227	C₂	#20276469	1	2	yes	—
228	C₂	#20276517	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12653
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. C₄⋊#11719908	C₄	semidirect
17. C₅₃×C₄⋊C₁₆	C₄	semidirect
103. C₄×C₅₃	#346	semidirect
130. C₄⋊C₁₆	Dic₅₃	semidirect
132. C₅₃	#12653	semidirect
219. C₄	#16810398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810398	2
C₄×C₈×Dic₅₃	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12653	53