Subgroups — #27273514

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20272546	C₂	1	4	yes	maximal, complemented
3	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	2	no	—
5	#16810680	—	2	2	no	—
6	#16810676	C₄	1	4	yes	—
7	#16810676	C₂²	1	4	yes	—
8	#16810647	—	2	2	no	—
9	#16810587	—	2	2	no	—
10	C₄×#11719917	C₄	1	4	yes	—
11	C₂³×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₃	—	2	2	no	—
14	C₂×#97×C₅₃	—	2	2	no	—
15	C₂×#97×C₅₃	C₄	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₂²⋊#88322	—	2	4	no	—
22	#13133128	—	2	4	no	—
23	C₄×#88322	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	8	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#97×C₅₃	—	2	4	no	—
40	#97×C₅₃	—	2	4	no	—
41	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	—	2	8	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
64	C₄²×C₅₃	C₄²	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	#11719917	—	4	8	no	—
73	C₂×#88322	—	2	8	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	#95	1	1	yes	characteristic
78	C₂³×C₅₃	#95	1	1	yes	characteristic
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	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	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#95	1	2	yes	—
96	C₂×C₄×C₅₃	#95	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
102	C₈×C₅₃	—	2	8	no	—
103	#88322	—	4	8	no	—
104	#7861	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	8	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	H(3,ℤ/4)	1	2	yes	—
113	C₂²×C₅₃	#338	1	2	yes	—
114	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
115	C₂²×C₅₃	#423	1	2	yes	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
123	C₄×C₅₃	#407	1	2	yes	—
124	#2920	—	53	2	no	—
125	C₂²×#97	D₅₃	1	1	yes	characteristic
126	#1275	—	53	4	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	#1728	1	1	yes	characteristic
131	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
133	C₂³×C₈	Dic₅₃	1	2	yes	—
134	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
135	#432	—	106	2	no	—
136	#432	—	106	2	no	—
137	#431	—	53	4	no	—
138	#431	—	53	4	no	—
139	#423	—	106	2	no	—
140	#407	—	106	2	no	—
141	C₄×M₁₆	—	53	4	no	—
142	C₂×#97	—	2	2	no	—
143	C₂×#97	—	2	2	no	—
144	C₂×#97	Dic₅₃	1	2	yes	—
145	C₂×#97	C₂×D₅₃	1	2	yes	—
146	C₅₃	#7861	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
148	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂×M₁₆	—	106	4	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	8	no	—
154	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
155	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
156	C₂×C₄²	C₄×D₅₃	1	2	yes	—
157	C₂×C₄²	C₄×D₅₃	1	2	yes	—
158	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	C₄⋊C₈	—	106	4	no	—
164	C₄⋊C₈	—	106	4	no	—
165	C₂²⋊C₈	—	106	4	no	—
166	C₂²⋊C₈	—	106	4	no	—
167	#97	—	2	4	no	—
168	#97	—	2	4	no	—
169	#97	—	106	4	no	—
170	C₄×C₈	—	53	8	no	—
171	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	#11719919	1	2	yes	—
178	C₂²×C₄	#11719919	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	#11719921	1	2	yes	—
183	C₂²×C₄	#11719921	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
189	M₁₆	—	212	8	no	—
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	106	8	no	—
195	C₂×C₈	#11719917	1	8	yes	complemented
196	C₄²	C₄×Dic₅₃	1	2	yes	—
197	C₄²	#11719919	1	2	yes	—
198	C₄²	#11719919	1	2	yes	—
199	C₄²	#11719921	1	2	yes	—
200	C₄²	#11719921	1	2	yes	—
201	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
202	C₄²	C₂×C₄×D₅₃	1	2	yes	—
203	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	#13133124	1	1	yes	characteristic
208	C₂³	#13133124	1	1	yes	characteristic
209	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#13133124	1	2	yes	—
224	C₂×C₄	#13133124	1	2	yes	—
225	C₂×C₄	#13133124	1	1	yes	characteristic
226	C₂×C₄	#13133124	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719921	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719919	1	2	yes	—
231	C₂×C₄	C₂×#11719917	1	4	yes	—
232	C₈	—	2	8	no	—
233	C₈	—	212	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	#16810380	1	2	yes	—
242	C₂²	#16810381	1	2	yes	—
243	C₂²	C₂×#13133124	1	1	yes	characteristic
244	C₂²	#16810647	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	C₄×#11719917	1	2	yes	—
252	C₄	#16810587	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#20274409	1	1	yes	characteristic
257	C₂	C₂×#16810380	1	1	yes	characteristic
258	C₂	C₂×#16810381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	#7861
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20272546	C₂	semidirect
63. C₂×C₈×C₅₃	M₁₆	semidirect
146. C₅₃	#7861	semidirect
195. C₂×C₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20272546	2
C₂²×#97×C₅₃	2
#7861	53