Subgroups — #27294185

Order 13568.

334 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	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20272679	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	1	no	—
6	#16810647	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	1	yes	characteristic
13	C₅₃×#494	C₄	1	1	yes	characteristic
14	C₅₃×#487	—	2	1	no	—
15	C₅₃×#467	—	2	1	no	—
16	C₅₃×#467	C₄	1	2	yes	—
17	C₄×Q₁₆×C₅₃	—	2	1	no	—
18	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	4	1	no	—
21	C₂×#11719908	—	2	4	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	1	no	—
25	C₂×Q₁₆×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	C₄⋊Q₈×C₅₃	—	2	1	no	—
31	#125×C₅₃	—	4	1	no	—
32	C₂²⋊Q₈×C₅₃	—	2	2	no	—
33	C₂²⋊Q₈×C₅₃	—	4	1	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	4	1	no	—
36	#117×C₅₃	—	4	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	#107×C₅₃	—	2	1	no	—
39	#107×C₅₃	C₂×C₄	1	2	yes	—
40	#103×C₅₃	—	2	2	no	—
41	#103×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₅₃	—	2	4	no	—
52	Q₁₆×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	1	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₂²⋊C₄×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	D₈	1	1	yes	characteristic
69	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	#11719917	—	8	2	no	—
72	C₂×#88322	—	4	2	no	—
73	#11719908	—	2	4	no	—
74	C₂³×C₅₃	#102	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	4	2	no	—
77	Q₈×C₅₃	—	8	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
87	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	#104	1	2	yes	—
94	C₈×C₅₃	#131	1	2	yes	—
95	#88322	—	8	2	no	—
96	#11672	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	#358	1	2	yes	—
100	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	#358	1	2	yes	—
106	C₄×C₅₃	#359	1	2	yes	—
107	C₄×C₅₃	#408	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
109	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
110	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
111	#3170	D₅₃	1	1	yes	characteristic
112	C₂×C₄×C₁₆	—	53	1	no	—
113	#1567	—	53	1	no	—
114	#1332	—	53	4	no	—
115	C₂×C₅₃	—	2	2	no	—
116	C₂×C₅₃	#1492	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
119	#533	—	2	1	no	—
120	C₂²×C₁₆	—	106	1	no	—
121	#500	Dic₅₃	1	1	yes	characteristic
122	#494	Dic₅₃	1	1	yes	characteristic
123	#487	—	2	1	no	—
124	#467	—	2	1	no	—
125	#467	Dic₅₃	1	2	yes	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
128	#423	—	106	1	no	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	C₄×C₁₆	—	53	2	no	—
131	C₄×C₁₆	—	106	1	no	—
132	#335	—	53	2	no	—
133	#335	—	106	1	no	—
134	C₅₃	#11672	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
137	C₂×M₁₆	—	212	1	no	—
138	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	—	2	1	no	—
142	#125	—	4	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	4	1	no	—
148	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₁₆	—	106	4	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₅₃	1	2	yes	—
153	C₄⋊C₈	—	106	2	no	—
154	C₄⋊C₈	—	212	1	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	#11719922	1	1	yes	characteristic
164	C₂²×C₄	#11719923	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	424	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	#11719922	1	1	yes	characteristic
185	C₄²	#11719923	1	1	yes	characteristic
186	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	C₁₆	—	106	4	no	—
189	C₂³	#13133138	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#13133138	1	1	yes	characteristic
199	C₂×C₄	#13133138	1	1	yes	characteristic
200	C₂×C₄	#13133138	1	1	yes	characteristic
201	C₂×C₄	#13133160	1	2	yes	—
202	C₂×C₄	#13133163	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	424	2	no	—
209	C₈	#13133144	1	2	yes	—
210	C₈	#13133224	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16810427	1	2	yes	—
214	C₂²	C₂×#13133138	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	#16810427	1	2	yes	—
220	C₄	#16810430	1	2	yes	—
221	C₄	#16810590	1	1	yes	characteristic
222	C₄	C₂×#13133144	1	1	yes	characteristic
223	C₄	C₂×#13133160	1	1	yes	characteristic
224	C₄	C₂×#13133163	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#20273166	1	1	yes	characteristic
227	C₂	C₂×#16810427	1	1	yes	characteristic
228	C₂	C₂×#16810430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#11672
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20272679	C₂	semidirect
134. C₅₃	#11672	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
#20273474	2
#20272679	2
C₅₃×#3170	2
#11672	53