Subgroups — #27300320

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 240 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	#20272797	C₂	1	2	yes	maximal, complemented
3	#20272795	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	—	2	4	no	—
7	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
10	C₄×M₁₆×C₅₃	—	2	4	no	—
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#13133157	—	2	4	no	—
16	#13133157	—	2	4	no	—
17	C₂×#11719908	—	2	4	no	—
18	C₂³×C₄×C₅₃	—	2	1	no	—
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	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	D₄	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	C₂×C₄²×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	C₈	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#97×C₅₃	—	2	2	no	—
36	#97×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	1	no	—
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	4	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	8	no	—
53	M₁₆×C₅₃	—	2	8	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	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	#41	1	2	yes	—
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₄²×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	C₂×C₈	1	2	yes	—
70	C₄²×C₅₃	C₂×C₈	1	2	yes	—
71	#11719908	—	4	4	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	4	2	no	—
75	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
76	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	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	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	#123	1	4	yes	—
99	#12849	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	—	4	2	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₂²×C₅₃	#432	1	1	yes	characteristic
107	C₂²×C₅₃	#432	1	1	yes	characteristic
108	C₄×C₅₃	—	2	4	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	#381	1	1	yes	characteristic
112	C₄×C₅₃	#407	1	1	yes	characteristic
113	C₄×C₅₃	#431	1	1	yes	characteristic
114	C₄×C₅₃	#431	1	1	yes	characteristic
115	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
116	C₂×#349	—	53	2	no	—
117	#1376	—	53	2	no	—
118	#1375	—	53	2	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	—	4	2	no	—
122	C₂×C₅₃	#1817	1	1	yes	characteristic
123	C₂×C₅₃	#2165	1	2	yes	—
124	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
125	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
126	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
127	C₂×M₃₂	—	53	4	no	—
128	C₄×M₁₆	—	2	4	no	—
129	C₂×#97	Dic₅₃	1	1	yes	characteristic
130	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
131	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	#349	—	106	4	no	—
134	C₅₃	#12849	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	—	2	1	no	—
136	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	C₂²⋊D₅₃	1	4	yes	—
143	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	#88322	1	4	yes	—
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	—	106	4	no	—
155	#97	—	2	2	no	—
156	#97	—	2	2	no	—
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	4	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₂×#88322	1	2	yes	—
169	C₂²×C₄	#11719917	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	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	#11719917	1	2	yes	—
181	C₂×C₈	#11719917	1	2	yes	—
182	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
183	C₂×C₈	#11719945	1	2	yes	—
184	C₂×C₈	#11719945	1	2	yes	—
185	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₂×#88322	1	2	yes	—
190	C₄²	C₂×#88322	1	2	yes	—
191	C₁₆	—	212	4	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	4	2	no	—
195	C₂³	C₂²⋊#88322	1	2	yes	—
196	C₂³	C₂×#11719917	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂²⋊#88322	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	#13133176	1	1	yes	characteristic
211	C₂×C₄	#13133176	1	1	yes	characteristic
212	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
213	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	#13133200	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
225	C₂²	#16810680	1	1	yes	characteristic
226	C₂²	#16810680	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	#16810486	1	1	yes	characteristic
231	C₄	#16810587	1	1	yes	characteristic
232	C₄	#16810676	1	1	yes	characteristic
233	C₄	#16810676	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	#20274783	1	1	yes	characteristic
238	C₂	#20276527	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂²×C₈
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	#12849
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. #20272797	C₂	semidirect
3. #20272795	C₂	semidirect
134. C₅₃	#12849	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
#20272797	2
#20272795	2
C₂×C₄×M₁₆×C₅₃	2
#12849	53