Subgroups — #27284206

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	#20274464	C₂	1	1	yes	maximal, characteristic
3	#20272596	C₂	1	4	yes	maximal
4	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×#13133128	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	2	yes	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#391	—	2	1	no	—
14	C₅₃×#391	—	2	1	no	—
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#391	—	2	1	no	—
17	C₅₃×#382	C₄	1	2	yes	—
18	C₅₃×#382	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	#13133128	—	2	2	no	—
24	#13133128	—	2	4	no	—
25	#13133128	Q₈	1	4	yes	complemented
26	C₂²×Q₈×C₅₃	—	2	1	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	—	4	2	no	—
29	C₄⋊Q₈×C₅₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#95×C₅₃	—	2	2	no	—
42	#95×C₅₃	—	2	2	no	—
43	#95×C₅₃	—	2	2	no	—
44	#95×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₅₃	—	8	4	no	—
80	Q₈×C₅₃	—	8	4	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	#100	1	2	yes	—
103	C₂×C₄×C₅₃	#101	1	2	yes	—
104	C₂×C₄×C₅₃	#104	1	2	yes	—
105	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
109	#88322	—	4	4	no	—
110	#88322	—	4	4	no	—
111	#9895	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	#391	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
116	C₂²×C₅₃	#421	1	1	yes	characteristic
117	C₂²×C₅₃	#421	1	1	yes	characteristic
118	C₂²×C₅₃	#421	1	1	yes	characteristic
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	—	8	2	no	—
129	#2862	D₅₃	1	1	yes	characteristic
130	#1767	—	53	1	no	—
131	#1745	—	53	1	no	—
132	#1295	—	53	4	no	—
133	C₂×C₅₃	#1527	1	1	yes	characteristic
134	C₂×C₅₃	#1531	1	1	yes	characteristic
135	C₂×C₅₃	#1533	1	1	yes	characteristic
136	C₂×C₅₃	#1537	1	1	yes	characteristic
137	C₂×C₅₃	#1986	1	1	yes	characteristic
138	C₂×C₅₃	#1995	1	1	yes	characteristic
139	C₂×C₅₃	#1997	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	106	1	no	—
143	C₂×#97	—	53	2	no	—
144	C₂×#97	—	53	2	no	—
145	#391	—	2	1	no	—
146	#391	—	2	1	no	—
147	#391	—	2	1	no	—
148	#391	—	2	1	no	—
149	#382	Dic₅₃	1	2	yes	—
150	#382	C₂×D₅₃	1	2	yes	—
151	C₄³	C₂×D₅₃	1	1	yes	characteristic
152	#336	—	106	1	no	—
153	#336	—	106	1	no	—
154	#336	—	106	1	no	—
155	C₅₃	#9895	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₄⋊Q₈	—	4	2	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₈	—	106	4	no	—
174	#97	—	53	4	no	—
175	#97	—	106	2	no	—
176	#97	—	106	4	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	212	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	Q₈×D₅₃	1	2	yes	—
213	C₄²	#11719949	1	2	yes	—
214	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
215	Q₈	—	8	4	no	—
216	Q₈	—	8	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#13133134	1	2	yes	—
239	C₂×C₄	#13133136	1	2	yes	—
240	C₂×C₄	#13133144	1	2	yes	—
241	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
242	C₂×C₄	#13133193	1	2	yes	—
243	C₂×C₄	#13133206	1	1	yes	characteristic
244	C₂×C₄	#13133215	1	1	yes	characteristic
245	C₈	—	212	4	no	—
246	C₈	—	212	4	no	—
247	C₂²	#16810530	1	1	yes	characteristic
248	C₂²	C₂×#13133134	1	1	yes	characteristic
249	C₂²	C₂×#13133136	1	1	yes	characteristic
250	C₂²	C₂×#13133144	1	1	yes	characteristic
251	C₂²	#16810640	1	1	yes	characteristic
252	C₂²	#16810640	1	1	yes	characteristic
253	C₂²	#16810640	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	2	no	—
264	C₂	#20273313	1	1	yes	characteristic
265	C₂	#20273330	1	1	yes	characteristic
266	C₂	#20273339	1	1	yes	characteristic
267	C₂	#20273353	1	1	yes	characteristic
268	C₂	#20275660	1	1	yes	characteristic
269	C₂	#20275697	1	1	yes	characteristic
270	C₂	#20275705	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2862	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2862	D₅₃
hypocenter	C₅₃	#9895
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
25. #13133128	Q₈	semidirect
155. C₅₃	#9895	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
#20274464	2
#20272596	2
C₅₃×#2862	2
#9895	53