Subgroups — #28310218

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133292	C₂	1	2	yes	maximal
2	#20289754	C₂	1	2	yes	maximal, complemented
3	#20289753	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133292	—	2	4	no	—
6	C₂×#13133292	—	2	4	no	—
7	C₂×#13133292	C₄	1	8	yes	complemented
8	C₂²×#11719910	C₂²	1	4	yes	—
9	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅₃	—	2	4	no	—
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	#13133292	—	2	16	no	—
18	C₂×#11719910	—	2	4	no	—
19	C₂×#11719910	—	2	4	no	—
20	C₂×#11719910	C₂×C₄	1	8	yes	complemented
21	C₂²×C₄×D₅₃	C₈	1	2	yes	—
22	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	C₈	1	4	yes	—
28	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₅₃	D₄	1	4	yes	—
30	C₂×C₈×D₅₃	D₄	1	2	yes	—
31	C₂×C₈×D₅₃	Q₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₅₃	—	2	2	no	—
35	C₄²×D₅₃	—	2	2	no	—
36	C₄×#88322	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
47	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₅₃	—	2	4	no	—
49	C₂×C₄×D₅₃	—	2	4	no	—
50	C₂×C₄×D₅₃	—	2	4	no	—
51	C₂×C₄×D₅₃	C₄²	1	2	yes	—
52	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
56	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	C₄²	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
63	C₄²×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄×Dic₅₃	—	2	1	no	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	—	2	4	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	C₂×C₈	1	2	yes	—
71	C₂×#88322	C₂²×C₄	1	2	yes	—
72	C₂×#88322	C₂×D₄	1	1	yes	characteristic
73	C₂×#88322	C₂×D₄	1	2	yes	—
74	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
75	C₈×D₅₃	—	2	8	no	—
76	C₈×D₅₃	C₂²⋊C₄	1	8	yes	—
77	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
78	#11719910	—	2	8	no	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂²×D₅₃	—	2	2	no	—
81	C₂²×D₅₃	—	2	2	no	—
82	C₂²×D₅₃	C₄×C₈	1	2	yes	—
83	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
84	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
88	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
97	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
98	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₅₃	M₃₂	1	4	yes	complemented
100	C₄×D₅₃	—	2	16	no	—
101	C₄×D₅₃	#95	1	4	yes	—
102	C₄×D₅₃	#97	1	4	yes	—
103	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
104	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	#95	1	4	yes	—
107	#88322	—	2	2	no	—
108	#88322	C₂×C₂²⋊C₄	1	2	yes	—
109	#88322	C₂×C₄⋊C₄	1	2	yes	—
110	#12544	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
115	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	2	2	no	—
118	C₂×D₅₃	—	2	4	no	—
119	C₂×D₅₃	—	2	4	no	—
120	C₂×D₅₃	#336	1	4	yes	—
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
126	Dic₅₃	—	2	4	no	—
127	Dic₅₃	#336	1	4	yes	—
128	C₂²×M₃₂	—	53	2	no	—
129	C₂²×C₄×C₈	—	53	1	no	—
130	#1374	—	53	2	no	—
131	#1374	—	53	2	no	—
132	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
133	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
134	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
135	C₂×C₅₃	#2112	1	2	yes	—
136	C₂×C₅₃	#2150	1	2	yes	—
137	D₅₃	—	2	4	no	—
138	C₂³×C₈	—	53	1	no	—
139	C₂³×C₈	—	53	1	no	—
140	C₂²×C₄²	—	53	1	no	—
141	C₂×M₃₂	—	53	8	no	—
142	C₂×M₃₂	—	106	4	no	—
143	C₂×M₃₂	—	106	4	no	—
144	C₂²×C₁₆	—	53	4	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	—	106	4	no	—
149	C₂×C₄×C₈	#6157	1	1	yes	characteristic
150	C₅₃	#12544	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	53	1	no	—
152	C₂³×C₄	—	53	2	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	4	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	53	4	no	—
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	53	4	no	—
160	C₂²×C₈	—	106	2	no	—
161	C₂²×C₈	—	106	2	no	—
162	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
163	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	—	53	2	no	—
166	C₂×C₄²	—	106	2	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
169	M₃₂	—	106	16	no	—
170	C₂×C₁₆	—	53	8	no	—
171	C₂×C₁₆	—	106	4	no	—
172	C₂×C₁₆	—	106	4	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	106	2	no	—
175	C₄×C₈	—	106	2	no	—
176	C₄×C₈	—	106	2	no	—
177	C₂⁴	—	53	1	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	53	2	no	—
185	C₂²×C₄	—	53	2	no	—
186	C₂²×C₄	—	106	4	no	—
187	C₂²×C₄	—	106	4	no	—
188	C₂²×C₄	—	106	4	no	—
189	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
190	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	1	no	—
196	C₂×C₈	—	53	1	no	—
197	C₂×C₈	—	53	8	no	—
198	C₂×C₈	—	53	8	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	106	8	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
204	C₂×C₈	C₄×#6157	1	2	yes	—
205	C₂×C₈	C₄⋊#6157	1	2	yes	—
206	C₂×C₈	C₂²⋊#6157	1	2	yes	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	106	4	no	—
210	C₄²	—	106	1	no	—
211	C₄²	—	106	1	no	—
212	C₁₆	—	106	8	no	—
213	C₂³	—	53	2	no	—
214	C₂³	—	53	2	no	—
215	C₂³	—	53	2	no	—
216	C₂³	—	106	2	no	—
217	C₂³	—	106	2	no	—
218	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	53	2	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	53	4	no	—
225	C₂×C₄	—	53	4	no	—
226	C₂×C₄	—	53	4	no	—
227	C₂×C₄	—	53	4	no	—
228	C₂×C₄	—	53	4	no	—
229	C₂×C₄	—	106	16	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	#13133292	1	4	yes	complemented
233	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
236	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
237	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
238	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	53	2	no	—
241	C₈	—	53	2	no	—
242	C₈	—	106	2	no	—
243	C₈	(C₂×#6157)⋊C₄	1	4	yes	—
244	C₂²	—	53	4	no	—
245	C₂²	—	106	2	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	106	4	no	—
248	C₂²	—	106	4	no	—
249	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
250	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
251	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
252	C₂²	D₅₃⋊M₃₂	1	2	yes	—
253	C₂²	C₂×#13133292	1	2	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	53	4	no	—
256	C₄	—	106	4	no	—
257	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
258	C₄	#16811582	1	1	yes	characteristic
259	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
260	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
261	C₂	—	106	4	no	—
262	C₂	#20290376	1	1	yes	characteristic
263	C₂	C₄×#13133292	1	1	yes	characteristic
264	C₂	#20290922	1	1	yes	characteristic
265	C₂	#20290943	1	2	yes	—
266	C₂	#20291048	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289754	C₂	semidirect
3. #20289753	C₂	semidirect
7. C₂×#13133292	C₄	semidirect
20. C₂×#11719910	C₂×C₄	semidirect
87. C₂×C₄×C₅₃	M₃₂	semidirect
99. C₂×Dic₅₃	M₃₂	semidirect
150. C₅₃	#12544	semidirect
232. C₂×C₄	#13133292	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133292	2
#20289754	2
#20289753	2
C₂×C₄×C₈×D₅₃	2
#12544	53