Subgroups — #27537280

Order 13568.

690 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×#16811041	C₂	1	2	yes	maximal, complemented
2	#20283428	C₂	1	1	yes	maximal, characteristic
3	#20276546	C₂	1	4	yes	maximal, complemented
4	#20276534	C₂	1	4	yes	maximal, complemented
5	#20276330	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	2	yes	—
9	#16811173	C₄	1	1	yes	characteristic
10	#16811041	—	2	8	no	—
11	C₂×#13133156	C₂²	1	4	yes	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
15	#16810702	C₄	1	4	yes	—
16	C₈×D₂₁₂	C₂²	1	4	yes	—
17	#16810647	C₄	1	1	yes	characteristic
18	#16810589	C₄	1	2	yes	—
19	#16810402	C₂²	1	8	yes	—
20	C₄×#11719908	C₂²	1	2	yes	—
21	C₄×#11719908	C₂²	1	2	yes	—
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₂×#11719945	C₂×C₄	1	2	yes	—
26	#13133223	—	2	4	no	—
27	#13133223	—	2	4	no	—
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂×#11719912	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₅₃	C₂³	1	4	yes	—
31	#13133217	C₈	1	1	yes	characteristic
32	#13133214	C₈	1	1	yes	characteristic
33	#13133205	C₈	1	2	yes	—
34	#13133191	C₈	1	4	yes	—
35	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
36	#13133156	—	2	8	no	—
37	C₂×#11719908	—	2	2	no	—
38	C₂×#11719908	D₄	1	4	yes	complemented
39	C₂×#11719908	C₂³	1	2	yes	—
40	C₂×#11719908	C₂³	1	1	yes	characteristic
41	C₂×#11719908	C₂³	1	1	yes	characteristic
42	C₁₆×D₅₃	—	2	8	no	—
43	C₄⋊#88322	C₂×C₄	1	4	yes	—
44	#13133129	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₅₃	—	2	4	no	—
49	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
50	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₃	C₂×C₈	1	4	yes	—
55	#11719945	—	2	4	no	—
56	#11719945	—	2	4	no	—
57	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
58	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	#11719921	C₂×C₈	1	8	yes	—
71	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
72	#11719917	—	2	4	no	—
73	C₂×#88322	C₂²×C₄	1	4	yes	—
74	#11719912	—	2	8	no	—
75	C₈×D₅₃	—	2	4	no	—
76	C₈×D₅₃	—	2	4	no	—
77	C₁₆×C₅₃	—	2	4	no	—
78	#11719908	—	2	2	no	—
79	#11719908	C₂×D₄	1	2	yes	—
80	#11719908	C₂×D₄	1	2	yes	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₅₃	—	2	8	no	—
91	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
92	D₂₁₂	—	2	4	no	—
93	D₂₁₂	—	2	8	no	—
94	C₄×D₅₃	—	2	4	no	—
95	C₄×D₅₃	—	2	4	no	—
96	Dic₁₀₆	—	2	4	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	C₄×D₄	1	2	yes	—
99	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
101	#88322	—	2	4	no	—
102	#33434	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	C₈×D₄	1	2	yes	—
105	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₅₃	—	2	8	no	—
107	C₂×D₅₃	—	2	4	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	C₈×D₄	1	2	yes	—
110	C₄×C₅₃	#504	1	4	yes	complemented
111	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
113	Dic₅₃	—	2	4	no	—
114	C₂×#504	—	53	2	no	—
115	#2931	—	53	1	no	—
116	#2169	—	53	4	no	—
117	D₄×C₁₆	—	53	4	no	—
118	#2150	D₅₃	1	1	yes	characteristic, complemented
119	#2113	—	53	2	no	—
120	C₂×C₄×C₁₆	—	53	1	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#504	1	2	yes	—
124	D₅₃	—	4	4	no	—
125	C₂×#131	—	53	2	no	—
126	#532	—	53	1	no	—
127	#504	—	106	8	no	—
128	C₂×M₃₂	—	53	4	no	—
129	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
130	C₂²×C₁₆	—	53	1	no	—
131	C₂²×C₁₆	—	53	1	no	—
132	C₂²×C₁₆	—	53	4	no	—
133	#436	—	53	4	no	—
134	C₈×D₄	—	53	4	no	—
135	#423	—	53	1	no	—
136	#408	—	53	2	no	—
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	—
139	C₂²⋊C₁₆	—	53	8	no	—
140	C₄×C₁₆	—	53	2	no	—
141	C₄×C₁₆	—	53	2	no	—
142	C₅₃	#33434	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	53	2	no	—
144	#131	—	106	4	no	—
145	#131	—	106	4	no	—
146	C₂×M₁₆	—	53	2	no	—
147	C₂×M₁₆	—	53	4	no	—
148	C₂²×C₈	—	53	4	no	—
149	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	C₄⋊Q₈	—	53	1	no	—
152	#127	—	53	1	no	—
153	#124	—	53	2	no	—
154	#121	—	53	4	no	—
155	C₄×D₄	—	53	4	no	—
156	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	106	8	no	—
159	C₂×C₁₆	—	53	4	no	—
160	C₂×C₁₆	—	53	1	no	—
161	C₂×C₁₆	—	53	1	no	—
162	C₂×C₁₆	—	53	2	no	—
163	C₂×C₁₆	—	106	8	no	—
164	C₂×C₁₆	—	106	2	no	—
165	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
166	C₄⋊C₈	—	53	4	no	—
167	C₂²⋊C₈	—	53	8	no	—
168	C₄×C₈	C₄×D₅₃	1	2	yes	—
169	C₄×C₈	C₂²×D₅₃	1	2	yes	—
170	#41	—	106	4	no	—
171	#41	—	106	4	no	—
172	C₂×Q₈	—	53	2	no	—
173	C₂×D₄	—	53	4	no	—
174	C₂×D₄	—	53	4	no	—
175	C₂×D₄	—	53	2	no	—
176	C₂²×C₄	—	53	4	no	—
177	C₂²×C₄	C₈×D₅₃	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
179	M₁₆	—	106	4	no	—
180	M₁₆	—	106	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	53	4	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	106	4	no	—
185	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₈	D₄×D₅₃	1	2	yes	—
188	C₂×C₈	#11719949	1	2	yes	—
189	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	53	4	no	—
191	C₂²⋊C₄	—	53	8	no	—
192	C₄²	C₈×D₅₃	1	1	yes	characteristic
193	C₄²	C₈×D₅₃	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₅₃	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	53	2	no	—
197	C₁₆	—	53	2	no	—
198	C₁₆	—	106	2	no	—
199	C₂³	—	53	4	no	—
200	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
201	Q₈	—	106	4	no	—
202	D₄	—	106	8	no	—
203	D₄	—	106	8	no	—
204	D₄	—	106	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	53	4	no	—
207	C₂×C₄	—	106	4	no	—
208	C₂×C₄	—	106	4	no	—
209	C₂×C₄	#13133179	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133225	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	106	4	no	—
217	C₈	#13133179	1	2	yes	—
218	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
219	C₈	C₂×#11719949	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	106	8	no	—
222	C₂²	—	106	4	no	—
223	C₂²	#16810688	1	2	yes	—
224	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	106	4	no	—
227	C₄	#16810688	1	2	yes	—
228	C₄	#16811041	1	4	yes	complemented
229	C₄	C₂×#13133179	1	1	yes	characteristic
230	C₄	C₂×#13133225	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	212	4	no	—
233	C₂	C₂×#16810688	1	1	yes	characteristic
234	C₂	C₂×#16811041	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	#16811173	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#33434
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#16811041	C₂	semidirect
3. #20276546	C₂	semidirect
4. #20276534	C₂	semidirect
5. #20276330	C₂	semidirect
6. C₂×C₄×#11719908	C₂	semidirect
7. C₅₃×#2150	C₂	semidirect
38. C₂×#11719908	D₄	semidirect
110. C₄×C₅₃	#504	semidirect
118. #2150	D₅₃	semidirect
142. C₅₃	#33434	semidirect
228. C₄	#16811041	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
#20283428	2
#20276546	2
#20276534	2
#20276330	2
C₂×C₄×#11719908	2
C₅₃×#2150	2
#33434	53