Subgroups — #27299158

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810405	C₂	1	2	yes	maximal
2	C₂×#16810401	C₂	1	2	yes	maximal, complemented
3	#20272797	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	#16810405	C₄	1	8	yes	complemented
7	#16810401	—	2	4	no	—
8	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₅₃	—	2	4	no	—
12	C₂×#97×C₅₃	C₄	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	C₂²	1	1	yes	characteristic
16	#13133157	—	2	4	no	—
17	#13133157	C₂×C₄	1	8	yes	complemented
18	C₂×#11719908	—	2	4	no	—
19	C₂³×C₄×C₅₃	—	2	1	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×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	4	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₅₃	—	2	8	no	—
54	M₁₆×C₅₃	—	2	8	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	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	C₄²	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	M₁₆	1	2	yes	—
72	#11719908	—	2	8	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	1	no	—
75	C₂³×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	#97	1	2	yes	—
77	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	#97	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	C₄×D₄	1	4	yes	—
100	#12607	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	4	2	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
107	C₂²×C₅₃	#407	1	1	yes	characteristic
108	C₂²×C₅₃	#432	1	1	yes	characteristic
109	C₄×C₅₃	—	2	4	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₅₃	#407	1	1	yes	characteristic
115	C₄×C₅₃	#432	1	1	yes	characteristic
116	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
117	C₂×#349	—	53	2	no	—
118	C₂×#347	—	53	2	no	—
119	#1376	—	53	2	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	—	2	1	no	—
122	C₂×C₅₃	—	4	2	no	—
123	C₂×C₅₃	#1749	1	1	yes	characteristic
124	C₂×C₅₃	#2165	1	2	yes	—
125	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
126	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
127	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
128	C₂×M₃₂	—	53	4	no	—
129	C₄×M₁₆	—	2	4	no	—
130	C₂×#97	Dic₅₃	1	1	yes	characteristic
131	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
132	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	#349	—	53	8	no	—
135	#347	—	106	4	no	—
136	C₅₃	#12607	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	2	1	no	—
138	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
145	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
154	M₃₂	—	53	8	no	—
155	M₃₂	—	106	4	no	—
156	C₂×C₁₆	—	106	4	no	—
157	#97	—	2	2	no	—
158	#97	C₂²⋊D₅₃	1	4	yes	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₂⁴	#11719917	1	1	yes	characteristic
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	—	2	1	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	#11719917	1	1	yes	characteristic
171	C₂²×C₄	#11719917	1	2	yes	—
172	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	#11719917	1	2	yes	—
183	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
186	C₂×C₈	#11719945	1	2	yes	—
187	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
188	C₄²	—	2	4	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	#11719917	1	2	yes	—
192	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
193	C₁₆	—	106	8	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#13133128	1	2	yes	—
198	C₂³	C₂×#11719917	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	#13133128	1	2	yes	—
211	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133176	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	C₄×C₂²⋊D₅₃	1	4	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂×#13133128	1	1	yes	characteristic
227	C₂²	#16810587	1	1	yes	characteristic
228	C₂²	#16810680	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₄	C₄×#11719917	1	1	yes	characteristic
234	C₄	#16810587	1	1	yes	characteristic
235	C₄	#16810680	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	#20274477	1	1	yes	characteristic
240	C₂	#20276527	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#12607
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810401	C₂	semidirect
3. #20272797	C₂	semidirect
6. #16810405	C₄	semidirect
17. #13133157	C₂×C₄	semidirect
136. C₅₃	#12607	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810405	2
C₂×#16810401	2
#20272797	2
C₂×C₄×M₁₆×C₅₃	2
#12607	53