Subgroups — #27302264

Order 13568.

446 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	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810407	C₂	1	4	yes	maximal, complemented
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	#16810407	—	2	8	no	—
10	#16810399	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂×#131×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#532	C₄	1	1	yes	characteristic
14	C₅₃×#436	—	2	1	no	—
15	C₅₃×#436	—	2	1	no	—
16	C₈×D₄×C₅₃	—	2	1	no	—
17	C₈×D₄×C₅₃	—	2	1	no	—
18	C₅₃×#423	C₄	1	1	yes	characteristic
19	C₅₃×#408	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#13133157	—	2	2	no	—
22	#13133157	D₄	1	4	yes	complemented
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	4	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
27	#131×C₅₃	—	2	4	no	—
28	#131×C₅₃	—	2	4	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
35	#127×C₅₃	C₈	1	1	yes	characteristic
36	#124×C₅₃	C₈	1	2	yes	—
37	#121×C₅₃	—	2	1	no	—
38	#121×C₅₃	—	2	1	no	—
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	#41×C₅₃	—	2	4	no	—
51	#41×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	2	1	no	—
55	C₂×D₄×C₅₃	—	2	1	no	—
56	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	4	no	—
62	M₁₆×C₅₃	—	4	2	no	—
63	M₁₆×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	4	4	no	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
81	#11719908	—	2	4	no	—
82	#11719908	—	4	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	4	no	—
86	D₄×C₅₃	—	2	4	no	—
87	D₄×C₅₃	—	4	2	no	—
88	D₄×C₅₃	—	4	4	no	—
89	D₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#131	1	2	yes	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	4	2	no	—
106	C₈×C₅₃	#121	1	2	yes	—
107	C₈×C₅₃	#124	1	1	yes	characteristic
108	C₈×C₅₃	#127	1	1	yes	characteristic
109	#13193	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₅₃	#436	1	2	yes	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	#350	1	2	yes	complemented
122	C₄×C₅₃	#390	1	1	yes	characteristic
123	C₄×C₅₃	#408	1	1	yes	characteristic
124	C₄×C₅₃	C₈×D₄	1	2	yes	—
125	#2931	D₅₃	1	1	yes	characteristic
126	C₂×C₄⋊C₁₆	—	53	1	no	—
127	C₂×#350	—	53	4	no	—
128	C₄×M₃₂	—	53	1	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	—	8	2	no	—
131	C₂×C₅₃	#1954	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
133	C₂×C₅₃	#2117	1	1	yes	characteristic
134	C₂×#131	C₂×D₅₃	1	2	yes	—
135	#532	Dic₅₃	1	1	yes	characteristic
136	C₂×M₃₂	—	53	2	no	—
137	C₂²×C₁₆	—	53	2	no	—
138	#436	—	2	1	no	—
139	#436	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	—	2	1	no	—
142	#423	Dic₅₃	1	1	yes	characteristic
143	#408	Dic₅₃	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	106	1	no	—
146	C₄⋊C₁₆	—	106	1	no	—
147	#350	—	106	8	no	—
148	#346	—	106	1	no	—
149	C₄×C₁₆	—	106	1	no	—
150	C₅₃	#13193	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	C₂×Dic₅₃	1	2	yes	—
152	#131	—	2	4	no	—
153	#131	—	2	4	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₄⋊Q₈	#88322	1	1	yes	characteristic
160	#127	#88322	1	1	yes	characteristic
161	#124	#88322	1	2	yes	—
162	#121	—	2	1	no	—
163	#121	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	M₃₂	—	53	4	no	—
168	M₃₂	—	106	2	no	—
169	C₂×C₁₆	—	106	2	no	—
170	C₂×C₁₆	—	106	4	no	—
171	C₂×C₁₆	—	106	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	#41	—	2	4	no	—
181	#41	—	2	4	no	—
182	C₂×Q₈	C₂×#88322	1	2	yes	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂×#88322	1	2	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	#11719917	1	1	yes	characteristic
189	C₂²×C₄	#11719917	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₈	D₄×D₅₃	1	2	yes	—
200	C₂×C₈	#11719947	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	C₂×#88322	1	1	yes	characteristic
208	C₄²	C₂×#88322	1	1	yes	characteristic
209	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₁₆	—	106	4	no	—
212	C₁₆	—	212	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×#11719917	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
231	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
232	C₂×C₄	#13133224	1	2	yes	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	#13133188	1	2	yes	—
237	C₈	#13133204	1	1	yes	characteristic
238	C₈	#13133213	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
245	C₂²	#16810701	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	#16810407	1	2	yes	complemented
251	C₄	#16810526	1	1	yes	characteristic
252	C₄	#16810590	1	1	yes	characteristic
253	C₄	D₄×#88322	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	2	no	—
256	C₂	#20275510	1	1	yes	characteristic
257	C₂	C₂×#16810407	1	1	yes	characteristic
258	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₅₃×#532	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#13193
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11719908	C₂	semidirect
2. C₂×#16810407	C₂	semidirect
3. C₄×#13133157	C₂	semidirect
22. #13133157	D₄	semidirect
121. C₄×C₅₃	#350	semidirect
150. C₅₃	#13193	semidirect
250. C₄	#16810407	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810407	2
C₄×#13133157	2
C₅₃×#2931	2
#13193	53