Subgroups — #27299154

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276326	C₂	1	2	yes	maximal
2	C₂×#16810401	C₂	1	2	yes	maximal, complemented
3	#20272795	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	C₂²⋊#11719908	C₄	1	8	yes	complemented
7	#16810401	—	2	4	no	—
8	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
11	C₅₃×#405	—	2	4	no	—
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	#13133157	—	2	4	no	—
17	#13133157	—	2	4	no	—
18	C₂×#11719908	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	#117×C₅₃	—	2	2	no	—
31	#117×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#97×C₅₃	—	2	2	no	—
37	#97×C₅₃	D₄	1	4	yes	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	C₄²	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	#41	1	2	yes	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂²⋊C₄×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	M₁₆	1	2	yes	—
71	#11719908	—	2	8	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	4	2	no	—
75	C₂³×C₅₃	#97	1	2	yes	—
76	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	#97	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	C₄×D₄	1	4	yes	—
98	#12606	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	4	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	—	4	2	no	—
104	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
105	C₂²×C₅₃	#407	1	1	yes	characteristic
106	C₂²×C₅₃	#432	1	1	yes	characteristic
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
112	C₄×C₅₃	#407	1	1	yes	characteristic
113	C₄×C₅₃	#432	1	1	yes	characteristic
114	C₂×#405	D₅₃	1	1	yes	characteristic
115	#2112	—	53	2	no	—
116	C₂×#347	—	53	2	no	—
117	#1375	—	53	2	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	#1749	1	1	yes	characteristic
122	C₂×C₅₃	#2110	1	2	yes	—
123	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#117	Dic₅₃	1	1	yes	characteristic
126	C₂×M₃₂	—	53	4	no	—
127	#405	—	2	4	no	—
128	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
129	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
131	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
132	C₂²⋊C₁₆	—	53	8	no	—
133	#347	—	106	4	no	—
134	C₅₃	#12606	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	C₄×D₅₃	1	4	yes	—
142	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
152	M₃₂	—	106	4	no	—
153	M₃₂	—	106	4	no	—
154	C₂×C₁₆	—	53	8	no	—
155	#97	—	2	2	no	—
156	#97	C₂²⋊D₅₃	1	4	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₂⁴	#11719917	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	#11719917	1	1	yes	characteristic
167	C₂²×C₄	#11719917	1	2	yes	—
168	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
170	M₁₆	—	2	8	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	#11719917	1	2	yes	—
179	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
182	C₂×C₈	#11719945	1	2	yes	—
183	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	#11719917	1	2	yes	—
189	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
190	C₁₆	—	106	8	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	#13133128	1	2	yes	—
195	C₂³	C₂×#11719917	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	#13133128	1	2	yes	—
207	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂×C₄	#13133176	1	1	yes	characteristic
210	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
211	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	C₄×C₂²⋊D₅₃	1	4	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#13133128	1	1	yes	characteristic
223	C₂²	#16810587	1	1	yes	characteristic
224	C₂²	#16810680	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
229	C₄	C₄×#11719917	1	1	yes	characteristic
230	C₄	#16810587	1	1	yes	characteristic
231	C₄	#16810680	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	C₂	#20274477	1	1	yes	characteristic
236	C₂	#20276320	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12606
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₂×#16810401	C₂	semidirect
3. #20272795	C₂	semidirect
6. C₂²⋊#11719908	C₄	semidirect
134. C₅₃	#12606	semidirect

Maximal subgroups

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

subgroup	index
#20276326	2
C₂×#16810401	2
#20272795	2
C₂×C₅₃×#405	2
#12606	53