Subgroups — #27295366

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 224 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×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20272767	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	#16810374	—	2	1	no	—
10	#16810374	C₂²	1	2	yes	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	1	yes	characteristic
13	C₅₃×#500	C₄	1	1	yes	characteristic
14	C₅₃×#499	—	2	1	no	—
15	C₅₃×#499	C₂²	1	2	yes	—
16	C₅₃×#498	—	2	1	no	—
17	C₅₃×#427	—	2	1	no	—
18	C₅₃×#427	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	4	1	no	—
21	#13133157	—	4	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	1	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	—	2	1	no	—
29	#125×C₅₃	—	2	2	no	—
30	C₂²⋊Q₈×C₅₃	—	4	1	no	—
31	C₂²⋊Q₈×C₅₃	—	4	1	no	—
32	C₄×Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	#117×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#107×C₅₃	—	2	1	no	—
37	#107×C₅₃	—	2	1	no	—
38	#107×C₅₃	C₂×C₄	1	2	yes	—
39	#107×C₅₃	C₂×C₄	1	2	yes	—
40	#106×C₅₃	—	2	2	no	—
41	#106×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	—	4	1	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	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	2	no	—
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	2	no	—
64	C₂²⋊C₄×C₅₃	—	4	2	no	—
65	C₄²×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	1	yes	characteristic
68	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	8	2	no	—
72	#11719908	—	4	2	no	—
73	C₂³×C₅₃	#102	1	1	yes	characteristic
74	Q₈×C₅₃	—	8	2	no	—
75	Q₈×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	#101	1	2	yes	—
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#103	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	#103	1	2	yes	—
91	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	#88322	—	8	2	no	—
94	#11936	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	#332	1	2	yes	—
98	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	#332	1	2	yes	—
105	C₄×C₅₃	#360	1	2	yes	—
106	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
108	#3160	D₅₃	1	1	yes	characteristic
109	#2150	—	53	1	no	—
110	C₂×#334	—	53	1	no	—
111	#1363	—	53	4	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
114	C₂×C₅₃	#2143	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
116	#533	—	2	1	no	—
117	C₂×M₃₂	—	106	1	no	—
118	#500	Dic₅₃	1	1	yes	characteristic
119	#500	Dic₅₃	1	1	yes	characteristic
120	#499	—	2	1	no	—
121	#499	C₂×D₅₃	1	2	yes	—
122	#498	—	2	1	no	—
123	#427	—	2	1	no	—
124	#427	Dic₅₃	1	2	yes	—
125	C₂×C₄⋊C₈	—	106	1	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	53	2	no	—
128	C₄⋊C₁₆	—	106	1	no	—
129	#334	—	53	2	no	—
130	#334	—	106	1	no	—
131	C₅₃	#11936	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	212	1	no	—
133	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	4	1	no	—
139	C₂²⋊Q₈	—	4	1	no	—
140	C₄×Q₈	—	4	1	no	—
141	C₄×Q₈	—	4	1	no	—
142	#117	—	2	2	no	—
143	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
144	M₃₂	—	212	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×Dic₅₃	1	2	yes	—
149	#107	C₂×Dic₅₃	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂²×C₄	#11719922	1	1	yes	characteristic
161	C₂²×C₄	#11719923	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	424	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	#11719924	1	2	yes	—
167	C₂×C₈	#11719925	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#11719922	1	1	yes	characteristic
181	C₄²	#11719923	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₁₆	—	212	2	no	—
185	C₂³	#13133138	1	1	yes	characteristic
186	Q₈	—	8	2	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	#13133136	1	2	yes	—
193	C₂×C₄	#13133138	1	1	yes	characteristic
194	C₂×C₄	#13133138	1	1	yes	characteristic
195	C₂×C₄	#13133138	1	1	yes	characteristic
196	C₂×C₄	#13133141	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
199	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	424	2	no	—
203	C₈	#13133141	1	2	yes	—
204	C₈	C₂×#11719924	1	1	yes	characteristic
205	C₈	C₂×#11719925	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16810369	1	2	yes	—
209	C₂²	C₂×#13133138	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	2	no	—
215	C₄	#16810369	1	2	yes	—
216	C₄	#16810433	1	2	yes	—
217	C₄	C₂×#13133136	1	1	yes	characteristic
218	C₄	C₂×#13133141	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂×#16810369	1	1	yes	characteristic
221	C₂	#20276443	1	1	yes	characteristic
222	C₂	C₂×#16810433	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	#11936
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. #20272767	C₂	semidirect
131. C₅₃	#11936	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×#16810374	2
#20272767	2
C₅₃×#3160	2
#11936	53