Subgroups — #27284248

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	4	yes	maximal
4	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
14	C₅₃×#384	—	2	1	no	—
15	C₅₃×#384	—	2	1	no	—
16	C₅₃×#384	C₄	1	2	yes	—
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	Q₈	1	4	yes	complemented
24	#13133128	—	4	2	no	—
25	C₂²×Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	4	4	no	—
27	C₄⋊Q₈×C₅₃	—	4	4	no	—
28	C₄⋊Q₈×C₅₃	—	4	2	no	—
29	C₄⋊Q₈×C₅₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	—	4	4	no	—
44	C₂×Q₈×C₅₃	—	4	4	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	4	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	D₈	1	2	yes	—
67	C₄²×C₅₃	SD₁₆	1	2	yes	—
68	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
69	C₄²×C₅₃	#41	1	2	yes	—
70	C₂×#88322	—	2	4	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₅₃	—	8	8	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	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	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	#100	1	2	yes	—
92	C₂×C₄×C₅₃	#101	1	2	yes	—
93	C₂×C₄×C₅₃	#102	1	2	yes	—
94	C₂×C₄×C₅₃	#102	1	2	yes	—
95	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₅₃	#137	1	2	yes	—
103	C₂×C₄×C₅₃	#137	1	2	yes	—
104	#88322	—	4	4	no	—
105	#88322	—	8	2	no	—
106	#9905	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	#391	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
111	C₂²×C₅₃	#418	1	1	yes	characteristic
112	C₂²×C₅₃	#419	1	1	yes	characteristic
113	C₂²×C₅₃	#419	1	1	yes	characteristic
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	8	4	no	—
120	C₄×C₅₃	#474	1	2	yes	—
121	C₄×C₅₃	#475	1	2	yes	—
122	C₄×C₅₃	#489	1	2	yes	—
123	C₄×C₅₃	#501	1	2	yes	—
124	#2868	D₅₃	1	1	yes	characteristic
125	#1831	—	53	1	no	—
126	#1767	—	53	1	no	—
127	#1296	—	53	4	no	—
128	C₂×C₅₃	#1545	1	1	yes	characteristic
129	C₂×C₅₃	#1546	1	1	yes	characteristic
130	C₂×C₅₃	#1556	1	1	yes	characteristic
131	C₂×C₅₃	#1558	1	1	yes	characteristic
132	C₂×C₅₃	#1988	1	1	yes	characteristic
133	C₂×C₅₃	#1993	1	1	yes	characteristic
134	C₂×C₅₃	#1997	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	—	2	1	no	—
136	C₂×C₄⋊Q₈	—	2	1	no	—
137	C₂×C₄⋊Q₈	Dic₅₃	1	2	yes	—
138	C₂×C₄⋊C₈	—	53	2	no	—
139	C₂×C₄⋊C₈	—	53	2	no	—
140	C₂×#97	—	106	1	no	—
141	#384	—	2	1	no	—
142	#384	—	2	1	no	—
143	#384	Dic₅₃	1	2	yes	—
144	#384	C₂×D₅₃	1	2	yes	—
145	C₄³	C₂×D₅₃	1	1	yes	characteristic
146	#336	—	106	1	no	—
147	#336	—	106	1	no	—
148	#336	—	106	1	no	—
149	C₅₃	#9905	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	—	106	2	no	—
153	C₄⋊Q₈	—	4	4	no	—
154	C₄⋊Q₈	—	4	4	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
163	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	C₄⋊C₈	—	53	4	no	—
171	C₄⋊C₈	—	106	2	no	—
172	C₄⋊C₈	—	106	4	no	—
173	#97	—	212	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂×C₈	—	106	4	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	—	212	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#11719922	1	2	yes	—
204	C₄²	#11719923	1	2	yes	—
205	C₄²	Q₈×D₅₃	1	2	yes	—
206	C₄²	#11719949	1	2	yes	—
207	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#13133134	1	2	yes	—
223	C₂×C₄	#13133136	1	2	yes	—
224	C₂×C₄	#13133138	1	2	yes	—
225	C₂×C₄	#13133138	1	2	yes	—
226	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
227	C₂×C₄	#13133193	1	2	yes	—
228	C₂×C₄	#13133206	1	1	yes	characteristic
229	C₂×C₄	#13133215	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
232	C₂×C₄	#13133243	1	2	yes	—
233	C₂×C₄	#13133248	1	2	yes	—
234	C₂×C₄	#13133248	1	2	yes	—
235	C₈	—	424	2	no	—
236	C₈	—	212	4	no	—
237	C₂²	#16810530	1	1	yes	characteristic
238	C₂²	C₂×#13133134	1	1	yes	characteristic
239	C₂²	C₂×#13133136	1	1	yes	characteristic
240	C₂²	C₂×#13133138	1	1	yes	characteristic
241	C₂²	#16810628	1	1	yes	characteristic
242	C₂²	#16810632	1	1	yes	characteristic
243	C₂²	#16810632	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#16810900	1	2	yes	—
251	C₄	#16810905	1	2	yes	—
252	C₄	#16810985	1	2	yes	—
253	C₄	#16811029	1	2	yes	—
254	C₂	#20273383	1	1	yes	characteristic
255	C₂	#20273387	1	1	yes	characteristic
256	C₂	#20273432	1	1	yes	characteristic
257	C₂	#20273441	1	1	yes	characteristic
258	C₂	#20275668	1	1	yes	characteristic
259	C₂	#20275689	1	1	yes	characteristic
260	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×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#9905
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
23. C₄⋊#88322	Q₈	semidirect
149. C₅₃	#9905	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274568	2
#20272598	2
C₅₃×#2868	2
#9905	53