Subgroups — #27295392

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20272773	C₂	1	2	yes	maximal, complemented
4	#20272770	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	C₂²	1	2	yes	—
10	#16810376	—	2	1	no	—
11	#16810376	C₂²	1	2	yes	—
12	C₅₃×#533	—	2	1	no	—
13	C₅₃×#500	C₂²	1	1	yes	characteristic
14	C₅₃×#500	C₂²	1	1	yes	characteristic
15	C₅₃×#499	—	2	1	no	—
16	C₅₃×#499	C₄	1	2	yes	—
17	C₅₃×#498	—	2	1	no	—
18	C₅₃×#427	—	2	1	no	—
19	C₅₃×#427	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	4	1	no	—
22	#13133157	—	4	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	4	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	1	no	—
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	#125×C₅₃	—	2	2	no	—
31	C₂²⋊Q₈×C₅₃	—	4	1	no	—
32	C₂²⋊Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	C₄×Q₈×C₅₃	—	4	1	no	—
35	#117×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	#107×C₅₃	—	2	1	no	—
38	#107×C₅₃	—	2	1	no	—
39	#107×C₅₃	C₂×C₄	1	2	yes	—
40	#107×C₅₃	C₂×C₄	1	2	yes	—
41	#106×C₅₃	—	2	2	no	—
42	#106×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×Q₈×C₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
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	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	1	no	—
64	C₄⋊C₄×C₅₃	—	4	1	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	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	8	2	no	—
73	#11719908	—	4	2	no	—
74	C₂³×C₅₃	#102	1	1	yes	characteristic
75	Q₈×C₅₃	—	8	2	no	—
76	Q₈×C₅₃	—	8	1	no	—
77	Q₈×C₅₃	—	8	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	#101	1	2	yes	—
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#103	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	#103	1	2	yes	—
94	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
96	#88322	—	8	2	no	—
97	#11942	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	#332	1	2	yes	—
101	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	8	2	no	—
106	C₄×C₅₃	—	8	2	no	—
107	C₄×C₅₃	#332	1	2	yes	—
108	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
110	C₄×C₅₃	#509	1	1	yes	characteristic
111	C₄×C₅₃	#510	1	1	yes	characteristic
112	#3160	D₅₃	1	1	yes	characteristic
113	#2150	—	53	1	no	—
114	C₂×#335	—	53	1	no	—
115	#1365	—	53	2	no	—
116	#1364	—	53	2	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
119	C₂×C₅₃	#2139	1	1	yes	characteristic
120	C₂×C₅₃	#2148	1	1	yes	characteristic
121	#533	—	2	1	no	—
122	C₂×M₃₂	—	106	1	no	—
123	#500	C₂×D₅₃	1	1	yes	characteristic
124	#500	C₂×D₅₃	1	1	yes	characteristic
125	#499	—	2	1	no	—
126	#499	Dic₅₃	1	2	yes	—
127	#498	—	2	1	no	—
128	#427	—	2	1	no	—
129	#427	Dic₅₃	1	2	yes	—
130	C₂×C₄⋊C₈	—	106	1	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	53	2	no	—
133	C₄⋊C₁₆	—	106	1	no	—
134	#335	—	53	2	no	—
135	#335	—	106	1	no	—
136	C₅₃	#11942	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	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	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	2	2	no	—
148	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
149	M₃₂	—	212	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#107	—	2	1	no	—
152	#107	—	2	1	no	—
153	#107	C₂×Dic₅₃	1	2	yes	—
154	#107	C₂×Dic₅₃	1	2	yes	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	C₄⋊C₈	—	106	2	no	—
158	C₄⋊C₈	—	212	1	no	—
159	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	#11719922	1	1	yes	characteristic
166	C₂²×C₄	#11719923	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	424	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	#11719924	1	2	yes	—
172	C₂×C₈	#11719925	1	2	yes	—
173	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	#11719922	1	1	yes	characteristic
186	C₄²	#11719923	1	1	yes	characteristic
187	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₁₆	—	212	2	no	—
190	C₂³	#13133138	1	1	yes	characteristic
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	Q₈	—	8	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#13133136	1	2	yes	—
200	C₂×C₄	#13133138	1	1	yes	characteristic
201	C₂×C₄	#13133138	1	1	yes	characteristic
202	C₂×C₄	#13133138	1	1	yes	characteristic
203	C₂×C₄	#13133141	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	424	2	no	—
210	C₈	#13133141	1	2	yes	—
211	C₈	C₂×#11719924	1	1	yes	characteristic
212	C₈	C₂×#11719925	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16810369	1	2	yes	—
216	C₂²	C₂×#13133138	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	8	2	no	—
221	C₄	—	8	2	no	—
222	C₄	#16810369	1	2	yes	—
223	C₄	C₂×#13133136	1	1	yes	characteristic
224	C₄	C₂×#13133141	1	1	yes	characteristic
225	C₄	#16811058	1	1	yes	characteristic
226	C₄	#16811063	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#16810369	1	1	yes	characteristic
229	C₂	#20276427	1	1	yes	characteristic
230	C₂	#20276464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#11942
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20272773	C₂	semidirect
4. #20272770	C₂	semidirect
136. C₅₃	#11942	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810376	2
#20272773	2
#20272770	2
C₅₃×#3160	2
#11942	53