Subgroups — #27302246

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810402	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
6	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₈×D₂₁₂	—	2	2	no	—
9	C₈×D₂₁₂	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₂×#13133129	C₄	1	2	yes	—
12	#16810402	—	2	8	no	—
13	C₄×#11719908	—	2	2	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₅₃	—	2	4	no	—
21	C₂×C₈×D₅₃	—	2	2	no	—
22	C₂×C₈×D₅₃	—	2	2	no	—
23	C₂×C₈×D₅₃	—	2	4	no	—
24	C₄×D₂₁₂	—	2	2	no	—
25	C₄×D₂₁₂	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×#11719921	C₈	1	2	yes	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#11719908	D₄	1	4	yes	complemented
31	C₄⋊#88322	—	2	2	no	—
32	#13133129	—	2	4	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	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂³×D₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
45	C₂×D₂₁₂	—	2	2	no	—
46	C₂×D₂₁₂	—	2	2	no	—
47	C₂×D₂₁₂	C₁₆	1	4	yes	complemented
48	C₂×C₄×D₅₃	—	2	2	no	—
49	C₂×C₄×D₅₃	—	2	4	no	—
50	C₂×C₄×D₅₃	—	2	2	no	—
51	C₂×C₄×D₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	2	yes	—
58	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₅₃	M₁₆	1	2	yes	—
60	#11719921	—	2	4	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	4	no	—
64	C₈×D₅₃	—	4	4	no	—
65	C₈×D₅₃	—	4	2	no	—
66	C₈×D₅₃	—	4	2	no	—
67	C₁₆×C₅₃	—	4	2	no	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂²×D₅₃	—	2	2	no	—
71	C₂²×D₅₃	—	2	4	no	—
72	C₂²×D₅₃	—	2	2	no	—
73	C₂²×D₅₃	—	2	4	no	—
74	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#131	1	2	yes	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	4	no	—
86	D₂₁₂	—	2	8	no	—
87	C₄×D₅₃	—	4	2	no	—
88	C₄×D₅₃	—	4	2	no	—
89	C₄×D₅₃	—	4	4	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	#121	1	2	yes	—
92	C₈×C₅₃	#124	1	1	yes	characteristic
93	C₈×C₅₃	#127	1	1	yes	characteristic
94	#88322	—	4	2	no	—
95	#13189	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×D₄	1	2	yes	—
98	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
100	C₂²×C₅₃	#504	1	2	yes	—
101	C₂×D₅₃	—	2	4	no	—
102	C₂×D₅₃	—	2	8	no	—
103	C₂×D₅₃	—	4	2	no	—
104	C₂×D₅₃	—	4	4	no	—
105	C₂×D₅₃	—	4	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	C₂²⋊C₁₆	1	4	yes	complemented
108	C₄×C₅₃	#390	1	1	yes	characteristic
109	C₄×C₅₃	#408	1	1	yes	characteristic
110	C₄×C₅₃	#436	1	2	yes	—
111	Dic₅₃	—	4	2	no	—
112	C₂×C₈×D₄	—	53	1	no	—
113	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
114	C₂×C₂²⋊C₁₆	—	53	4	no	—
115	C₂×C₄×C₁₆	—	53	1	no	—
116	C₂×C₅₃	#1954	1	1	yes	characteristic
117	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
118	C₂×C₅₃	#2113	1	1	yes	characteristic
119	C₂×C₅₃	D₄×C₁₆	1	2	yes	—
120	C₂×C₅₃	#2169	1	2	yes	—
121	D₅₃	—	4	4	no	—
122	C₂³×C₈	—	53	2	no	—
123	C₂×C₄×D₄	—	53	1	no	—
124	C₂²×C₁₆	—	53	2	no	—
125	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
126	C₈×D₄	—	106	2	no	—
127	C₈×D₄	—	106	2	no	—
128	C₂×C₄⋊C₈	—	53	1	no	—
129	C₂×C₂²⋊C₈	—	53	2	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	2	no	—
132	C₂²⋊C₁₆	—	106	8	no	—
133	C₄×C₁₆	—	106	2	no	—
134	C₅₃	#13189	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₄	—	53	1	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	—	106	2	no	—
139	C₂²×C₈	—	106	4	no	—
140	C₂²×C₈	—	106	4	no	—
141	C₂²×C₈	—	106	2	no	—
142	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₄×D₄	—	106	2	no	—
145	C₄×D₄	—	106	2	no	—
146	C₂×C₄⋊C₄	—	53	1	no	—
147	C₂×C₂²⋊C₄	—	53	2	no	—
148	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	53	4	no	—
152	C₂×C₁₆	—	106	2	no	—
153	C₂×C₁₆	—	106	2	no	—
154	C₄⋊C₈	—	106	2	no	—
155	C₂²⋊C₈	—	106	4	no	—
156	C₄×C₈	D₂₁₂	1	2	yes	—
157	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
158	C₂⁴	—	53	2	no	—
159	C₂×D₄	—	53	4	no	—
160	C₂×D₄	—	106	2	no	—
161	C₂×D₄	—	106	2	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	106	2	no	—
164	C₂²×C₄	—	106	4	no	—
165	C₂²×C₄	—	106	2	no	—
166	C₂²×C₄	—	106	4	no	—
167	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	212	4	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	#11719921	1	2	yes	—
177	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂×C₈	D₄×D₅₃	1	2	yes	—
179	C₂×C₈	#11719949	1	2	yes	—
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	106	2	no	—
182	C₂²⋊C₄	—	106	4	no	—
183	C₄²	#11719912	1	2	yes	—
184	C₄²	#11719921	1	2	yes	—
185	C₁₆	—	4	2	no	—
186	C₁₆	—	106	4	no	—
187	C₂³	—	53	4	no	—
188	C₂³	—	106	2	no	—
189	C₂³	—	106	4	no	—
190	C₂³	—	106	2	no	—
191	C₂³	—	106	4	no	—
192	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
193	D₄	—	106	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	4	no	—
197	C₂×C₄	—	212	2	no	—
198	C₂×C₄	—	212	2	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	#13133129	1	2	yes	—
201	C₂×C₄	#13133129	1	2	yes	—
202	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
203	C₂×C₄	#13133156	1	2	yes	—
204	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
205	C₂×C₄	#13133179	1	2	yes	—
206	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
207	C₂×C₄	#13133225	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	212	2	no	—
210	C₈	#13133192	1	2	yes	—
211	C₈	#13133206	1	1	yes	characteristic
212	C₈	#13133213	1	1	yes	characteristic
213	C₂²	—	106	8	no	—
214	C₂²	—	106	4	no	—
215	C₂²	—	212	2	no	—
216	C₂²	—	212	4	no	—
217	C₂²	—	212	2	no	—
218	C₂²	C₂×#13133129	1	1	yes	characteristic
219	C₂²	#16810688	1	2	yes	—
220	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
221	C₂²	C₂×#13133156	1	1	yes	characteristic
222	C₂²	#16811041	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	212	2	no	—
225	C₄	#16810402	1	4	yes	complemented
226	C₄	#16810524	1	1	yes	characteristic
227	C₄	#16810591	1	1	yes	characteristic
228	C₄	#16810700	1	2	yes	—
229	C₂	—	212	4	no	—
230	C₂	#20275508	1	1	yes	characteristic
231	C₂	C₂×#16810402	1	1	yes	characteristic
232	C₂	#20276330	1	1	yes	characteristic
233	C₂	#20276534	1	2	yes	—
234	C₂	#20276546	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#13189
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
2. C₂×#16810402	C₂	semidirect
3. C₂×C₄×#11719908	C₂	semidirect
4. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
30. C₂×#11719908	D₄	semidirect
47. C₂×D₂₁₂	C₁₆	semidirect
107. C₄×C₅₃	C₂²⋊C₁₆	semidirect
113. C₂×C₄⋊C₁₆	D₅₃	semidirect
134. C₅₃	#13189	semidirect
225. C₄	#16810402	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₁₂	2
C₂×#16810402	2
C₂×C₄×#11719908	2
C₂×C₅₃×C₄⋊C₁₆	2
#13189	53