Subgroups — #27290005

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	2	yes	maximal
2	C₂×#16810388	C₂	1	2	yes	maximal
3	#20274409	C₂	1	2	yes	maximal
4	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	4	yes	—
6	C₂×#13133153	—	2	4	no	—
7	C₂×#13133153	—	2	4	no	—
8	C₂×#13133153	C₄	1	8	yes	complemented
9	#16810388	—	2	4	no	—
10	#16810378	—	2	4	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#97×C₅₃	—	2	4	no	—
15	C₂×#97×C₅₃	C₄	1	4	yes	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	4	no	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	C₂×C₄	1	8	yes	—
22	C₂²×#88322	—	2	4	no	—
23	#13133153	—	2	16	no	—
24	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	4	yes	—
34	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
35	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	#97×C₅₃	—	2	8	no	—
41	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	2	yes	—
54	C₂²×C₄×C₅₃	#41	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	C₄²	1	4	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	#11719917	—	2	8	no	—
69	#11719917	—	2	8	no	—
70	#11719917	—	4	4	no	—
71	#11719917	—	4	4	no	—
72	C₂×#88322	—	2	8	no	—
73	C₂×#88322	—	4	4	no	—
74	C₂×#88322	—	4	4	no	—
75	C₂³×C₅₃	—	2	4	no	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
78	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	4	8	no	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₄⋊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	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	#121	1	2	yes	—
96	C₂×C₄×C₅₃	#121	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₅₃	#124	1	2	yes	—
99	C₂×C₄×C₅₃	#125	1	2	yes	—
100	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₈×C₅₃	—	2	4	no	—
103	C₈×C₅₃	#95	1	8	yes	—
104	#88322	—	4	8	no	—
105	#10865	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₅₃	#380	1	1	yes	characteristic
112	C₂²×C₅₃	#384	1	1	yes	characteristic
113	C₂²×C₅₃	#389	1	2	yes	—
114	C₂²×C₅₃	#391	1	2	yes	—
115	C₂²×C₅₃	#430	1	4	yes	—
116	C₄×C₅₃	—	4	4	no	—
117	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
118	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₅₃	#380	1	1	yes	characteristic
120	C₄×C₅₃	#382	1	1	yes	characteristic
121	C₄×C₅₃	#389	1	2	yes	—
122	C₄×C₅₃	#390	1	2	yes	—
123	C₂²×#108	—	53	2	no	—
124	C₂²×#97	D₅₃	1	1	yes	characteristic
125	C₂×#341	—	53	2	no	—
126	#1728	—	53	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
130	C₂×C₅₃	#1778	1	2	yes	—
131	C₂×C₅₃	#1811	1	2	yes	—
132	C₂×C₅₃	#1849	1	2	yes	—
133	C₂²×M₁₆	—	53	4	no	—
134	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
136	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	C₂×#108	—	53	8	no	—
138	C₂×#108	—	106	4	no	—
139	C₂×#108	—	106	4	no	—
140	C₂×#97	—	2	4	no	—
141	C₂×#97	Dic₅₃	1	4	yes	—
142	#341	—	106	4	no	—
143	#336	—	106	4	no	—
144	C₅₃	#10865	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	Dic₁₀₆	1	2	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×M₁₆	—	53	8	no	—
148	C₂×M₁₆	—	106	4	no	—
149	C₂×M₁₆	—	106	4	no	—
150	C₂×M₁₆	—	106	4	no	—
151	C₂×M₁₆	—	106	4	no	—
152	C₂×M₁₆	—	106	4	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	106	4	no	—
156	C₂²×C₈	Dic₁₀₆	1	2	yes	—
157	C₂²×C₈	Dic₁₀₆	1	2	yes	—
158	C₂²×C₈	C₄×D₅₃	1	4	yes	—
159	C₂²×C₈	D₂₁₂	1	2	yes	—
160	C₂²×C₈	D₂₁₂	1	2	yes	—
161	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	D₂₁₂	1	2	yes	—
167	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
168	#108	—	106	16	no	—
169	#97	—	2	8	no	—
170	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
174	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
177	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	#11719945	1	2	yes	—
181	C₂²×C₄	#11719945	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
184	M₁₆	—	106	8	no	—
185	M₁₆	—	106	8	no	—
186	M₁₆	—	212	4	no	—
187	M₁₆	—	212	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	106	8	no	—
193	C₂×C₈	—	212	4	no	—
194	C₂×C₈	—	212	4	no	—
195	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
196	C₂×C₈	#11719919	1	8	yes	—
197	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
198	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
199	C₂×C₈	#11719921	1	8	yes	—
200	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂³	C₄×Dic₁₀₆	1	2	yes	—
207	C₂³	#13133196	1	2	yes	—
208	C₂³	#13133217	1	1	yes	characteristic
209	C₂³	#13133217	1	1	yes	characteristic
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719921	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719919	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	#13133176	1	1	yes	characteristic
220	C₂×C₄	#13133176	1	1	yes	characteristic
221	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
222	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
223	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
224	C₂×C₄	#13133191	1	2	yes	—
225	C₂×C₄	#13133191	1	2	yes	—
226	C₂×C₄	#13133196	1	2	yes	—
227	C₂×C₄	#13133205	1	2	yes	—
228	C₂×C₄	#13133209	1	2	yes	—
229	C₂×C₄	#13133214	1	1	yes	characteristic
230	C₂×C₄	#13133217	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	212	8	no	—
233	C₈	#13133124	1	8	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂²	#16810483	1	1	yes	characteristic
240	C₂²	#16810493	1	1	yes	characteristic
241	C₂²	#16810521	1	2	yes	—
242	C₂²	#16810529	1	2	yes	—
243	C₂²	#16810672	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	C₂×#13133124	1	1	yes	characteristic
246	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
247	C₄	#16810483	1	1	yes	characteristic
248	C₄	#16810489	1	1	yes	characteristic
249	C₄	#16810521	1	2	yes	—
250	C₄	#16810525	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	2	2	no	—
253	C₂	#20272945	1	1	yes	characteristic
254	C₂	#20274621	1	2	yes	—
255	C₂	#20274757	1	2	yes	—
256	C₂	#20274890	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	#10865
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
8. C₂×#13133153	C₄	semidirect
144. C₅₃	#10865	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810388	2
#20274409	2
C₂²×#97×C₅₃	2
#10865	53