Subgroups — #27276890

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	2	yes	maximal
2	C₂×#16810372	C₂	1	2	yes	maximal
3	#20272598	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	4	yes	—
8	#16810372	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
11	C₅₃×#391	C₄	1	2	yes	—
12	C₅₃×#389	—	2	2	no	—
13	C₅₃×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	4	4	no	—
20	C₄⋊#88322	—	4	4	no	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
25	C₄×Q₈×C₅₃	—	4	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#95×C₅₃	—	2	4	no	—
37	C₂×Q₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	4	2	no	—
39	C₂×Q₈×C₅₃	—	4	2	no	—
40	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	—	4	4	no	—
59	C₄²×C₅₃	SD₁₆	1	4	yes	—
60	C₄²×C₅₃	Q₁₆	1	4	yes	—
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	2	yes	—
64	C₂×#88322	—	4	4	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂×#88322	—	4	4	no	—
67	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₅₃	—	4	4	no	—
69	Q₈×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	#100	1	2	yes	—
82	C₂×C₄×C₅₃	#103	1	4	yes	—
83	C₂×C₄×C₅₃	#103	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₅₃	#137	1	2	yes	—
92	#88322	—	8	4	no	—
93	#8587	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#380	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
96	C₂²×C₅₃	#413	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×#103	1	2	yes	—
98	C₂²×C₅₃	#417	1	2	yes	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	—	4	4	no	—
103	C₄×C₅₃	—	4	1	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	#333	1	4	yes	complemented
106	C₄×C₅₃	#460	1	2	yes	—
107	C₄×C₅₃	#462	1	2	yes	—
108	C₄×C₅₃	#475	1	2	yes	—
109	C₄×C₅₃	#477	1	2	yes	—
110	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
111	#1842	—	53	2	no	—
112	C₂×#333	—	53	2	no	—
113	#1296	—	53	2	no	—
114	C₂×C₅₃	C₂×#333	1	1	yes	characteristic
115	C₂×C₅₃	#1550	1	2	yes	—
116	C₂×C₅₃	#1560	1	1	yes	characteristic
117	C₂×C₅₃	#1800	1	1	yes	characteristic
118	C₂×C₅₃	#1803	1	2	yes	—
119	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
121	C₂×C₄⋊C₈	—	53	4	no	—
122	C₂×C₄⋊C₈	—	106	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	#391	Dic₅₃	1	2	yes	—
125	#389	—	2	2	no	—
126	#384	C₂×D₅₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄³	C₂×D₅₃	1	1	yes	characteristic
129	#333	—	106	4	no	—
130	C₅₃	#8587	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
132	C₂²×C₈	—	106	4	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
135	C₄⋊Q₈	D₂₁₂	1	2	yes	—
136	C₄×Q₈	—	4	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
145	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	C₄⋊C₈	—	106	4	no	—
148	C₄⋊C₈	—	106	4	no	—
149	C₄⋊C₈	—	212	4	no	—
150	C₄⋊C₈	—	212	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
163	C₂×C₈	—	212	4	no	—
164	C₂×C₈	—	212	4	no	—
165	C₂×C₈	—	212	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	#11719924	1	4	yes	—
178	C₄²	#11719925	1	4	yes	—
179	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
180	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
181	C₄²	#11719945	1	2	yes	—
182	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#13133134	1	2	yes	—
197	C₂×C₄	#13133141	1	4	yes	—
198	C₂×C₄	#13133141	1	4	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
200	C₂×C₄	#13133176	1	1	yes	characteristic
201	C₂×C₄	#13133191	1	2	yes	—
202	C₂×C₄	#13133196	1	2	yes	—
203	C₂×C₄	C₂×#11719924	1	2	yes	—
204	C₂×C₄	C₂×#11719925	1	2	yes	—
205	C₂×C₄	#13133242	1	2	yes	—
206	C₂×C₄	#13133247	1	2	yes	—
207	C₈	—	424	4	no	—
208	C₂²	#16810483	1	1	yes	characteristic
209	C₂²	C₂×#13133134	1	1	yes	characteristic
210	C₂²	#16810607	1	1	yes	characteristic
211	C₂²	C₂×#13133141	1	2	yes	—
212	C₂²	#16810624	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	#16810372	1	4	yes	complemented
220	C₄	#16810823	1	2	yes	—
221	C₄	#16810833	1	2	yes	—
222	C₄	#16810908	1	2	yes	—
223	C₄	#16810918	1	2	yes	—
224	C₂	C₂×#16810372	1	1	yes	characteristic
225	C₂	#20273405	1	2	yes	—
226	C₂	#20273450	1	1	yes	characteristic
227	C₂	#20274707	1	1	yes	characteristic
228	C₂	#20274720	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#8587
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
105. C₄×C₅₃	#333	semidirect
130. C₅₃	#8587	semidirect
219. C₄	#16810372	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₂×#16810372	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
#8587	53