Subgroups — #27302608

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810439	C₂	1	2	yes	maximal
3	#20276435	C₂	1	2	yes	maximal
4	#20274594	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810578	—	2	1	no	—
10	#16810444	—	2	2	no	—
11	#16810439	—	2	4	no	—
12	#16810427	—	2	4	no	—
13	#16810383	—	2	1	no	—
14	C₂²×Q₁₆×C₅₃	—	2	1	no	—
15	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
17	C₅₃×#494	—	2	2	no	—
18	C₅₃×#494	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	1	no	—
21	#13133176	—	2	1	no	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	#13133153	—	2	1	no	—
26	#13133153	D₄	1	2	yes	—
27	#13133151	D₄	1	2	yes	—
28	#13133149	—	2	1	no	—
29	#13133127	—	4	1	no	—
30	C₈×Dic₅₃	—	4	1	no	—
31	C₂²×Q₈×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	—	2	4	no	—
33	C₂×Q₁₆×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	—	2	2	no	—
35	C₂×Q₁₆×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	—	2	2	no	—
38	C₂×Q₁₆×C₅₃	—	4	1	no	—
39	C₂×Q₁₆×C₅₃	—	4	1	no	—
40	C₂×Q₁₆×C₅₃	—	4	2	no	—
41	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₅₃	—	2	4	no	—
45	C₄⋊Q₈×C₅₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₅₃	—	2	4	no	—
60	Q₁₆×C₅₃	—	2	4	no	—
61	Q₁₆×C₅₃	—	4	4	no	—
62	Q₁₆×C₅₃	—	4	4	no	—
63	Q₁₆×C₅₃	—	4	2	no	—
64	Q₁₆×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	—	4	8	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₂²⋊Dic₅₃	—	4	1	no	—
79	C₄⋊Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	4	1	no	—
81	#11719917	—	2	2	no	—
82	#11719917	—	4	1	no	—
83	C₂×#88322	—	4	1	no	—
84	#11719908	—	4	2	no	—
85	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	4	no	—
87	Q₈×C₅₃	—	4	4	no	—
88	Q₈×C₅₃	—	8	2	no	—
89	Q₈×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×Dic₅₃	—	4	2	no	—
102	C₈×C₅₃	#102	1	2	yes	—
103	C₈×C₅₃	#102	1	2	yes	—
104	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₈×C₅₃	#124	1	1	yes	characteristic
107	C₈×C₅₃	#127	1	1	yes	characteristic
108	#88322	—	4	2	no	—
109	#13262	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	#418	1	1	yes	characteristic
112	C₂²×C₅₃	#459	1	1	yes	characteristic
113	C₂²×C₅₃	#460	1	1	yes	characteristic
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#390	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
118	C₄×C₅₃	#459	1	1	yes	characteristic
119	C₄×C₅₃	#460	1	1	yes	characteristic
120	Dic₅₃	—	8	2	no	—
121	C₂×#494	D₅₃	1	1	yes	characteristic
122	C₂×#364	—	53	1	no	—
123	C₂×#362	—	53	2	no	—
124	#2141	—	53	2	no	—
125	#1772	—	53	1	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	#1982	1	1	yes	characteristic
129	C₂×C₅₃	#2214	1	2	yes	—
130	C₂²×Q₁₆	—	2	1	no	—
131	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
132	C₂×C₄⋊Q₈	—	2	1	no	—
133	C₂×M₃₂	—	53	2	no	—
134	#494	—	2	2	no	—
135	#494	Dic₅₃	1	4	yes	complemented
136	C₂×#108	—	53	1	no	—
137	#427	—	53	1	no	—
138	#405	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#364	—	106	2	no	—
141	#362	—	106	4	no	—
142	#358	—	106	4	no	—
143	#339	—	106	1	no	—
144	C₅₃	#13262	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	4	1	no	—
153	C₂×Q₁₆	—	4	1	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
156	C₂×M₁₆	—	106	1	no	—
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	—	4	2	no	—
161	#117	—	106	1	no	—
162	C₂×C₄⋊C₄	—	4	2	no	—
163	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
164	M₃₂	—	106	2	no	—
165	M₃₂	—	106	2	no	—
166	C₂×C₁₆	—	106	2	no	—
167	#108	—	53	2	no	—
168	#108	—	106	1	no	—
169	#107	—	53	2	no	—
170	#106	—	106	1	no	—
171	#97	—	212	1	no	—
172	C₄×C₈	—	212	1	no	—
173	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
174	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	Q₁₆	—	2	4	no	—
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	4	no	—
187	Q₁₆	—	4	4	no	—
188	Q₁₆	—	4	2	no	—
189	Q₁₆	—	4	2	no	—
190	M₁₆	—	106	2	no	—
191	M₁₆	—	212	1	no	—
192	C₂×C₈	—	212	1	no	—
193	C₂×C₈	#11719922	1	1	yes	characteristic
194	C₂×C₈	#11719922	1	1	yes	characteristic
195	C₂×C₈	#11719923	1	2	yes	—
196	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
197	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	#11719947	1	1	yes	characteristic
200	C₂×C₈	#11719947	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	4	8	no	—
204	C₄⋊C₄	—	106	2	no	—
205	C₂²⋊C₄	—	212	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	212	1	no	—
208	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
209	C₁₆	—	212	2	no	—
210	C₂³	#13133243	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	212	2	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	#13133188	1	1	yes	characteristic
223	C₂×C₄	#13133188	1	1	yes	characteristic
224	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
225	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
226	C₂×C₄	#13133243	1	1	yes	characteristic
227	C₈	—	212	2	no	—
228	C₈	#13133138	1	2	yes	—
229	C₈	#13133138	1	2	yes	—
230	C₈	D₄×Dic₅₃	1	1	yes	characteristic
231	C₈	D₄×Dic₅₃	1	1	yes	characteristic
232	C₈	#13133204	1	1	yes	characteristic
233	C₈	#13133213	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#16810628	1	1	yes	characteristic
236	C₂²	#16810815	1	1	yes	characteristic
237	C₂²	#16810820	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	8	4	no	—
240	C₄	—	424	2	no	—
241	C₄	#16810526	1	1	yes	characteristic
242	C₄	C₂×#13133138	1	1	yes	characteristic
243	C₄	#16810815	1	1	yes	characteristic
244	C₄	#16810820	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#20275643	1	1	yes	characteristic
248	C₂	#20276793	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810815
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#13262
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
18. C₅₃×#494	C₄	semidirect
135. #494	Dic₅₃	semidirect
144. C₅₃	#13262	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
C₂×#16810439	2
#20276435	2
#20274594	2
C₂×C₅₃×#494	2
#13262	53