Subgroups — #27537757

Order 13568.

444 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	#20276622	C₂	1	4	yes	maximal, complemented
3	#20276621	C₂	1	4	yes	maximal, complemented
4	#20276519	C₂	1	4	yes	maximal
5	#20276518	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
7	#16811180	C₄	1	1	yes	characteristic
8	#16810753	C₄	1	2	yes	—
9	#16810753	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₂²	1	2	yes	—
11	C₈×Dic₁₀₆	C₂²	1	2	yes	—
12	#16810649	C₄	1	2	yes	—
13	#16810647	C₄	1	1	yes	characteristic
14	#16810578	C₂²	1	2	yes	—
15	C₄⋊#11719908	C₂²	1	2	yes	—
16	#16810441	C₂²	1	8	yes	—
17	C₂²⋊#11719908	C₂²	1	2	yes	—
18	#16810399	C₂²	1	2	yes	—
19	#16810398	C₂²	1	4	yes	—
20	C₁₆×Dic₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
23	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
24	C₅₃×#346	C₂²	1	2	yes	—
25	C₂×#11719917	C₂×C₄	1	2	yes	—
26	#13133217	C₈	1	1	yes	characteristic
27	#13133217	C₈	1	1	yes	characteristic
28	#13133209	C₈	1	2	yes	—
29	#13133196	C₈	1	2	yes	—
30	#13133196	C₈	1	2	yes	—
31	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
33	#13133176	C₂×C₄	1	2	yes	—
34	C₂×#11719908	C₂³	1	4	yes	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	C₄⋊#88322	C₂×C₄	1	2	yes	—
37	#13133146	C₂×C₄	1	4	yes	—
38	#13133146	C₂×C₄	1	4	yes	—
39	#13133127	C₂×C₄	1	4	yes	—
40	C₈×Dic₅₃	C₂³	1	4	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
53	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
65	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
68	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
69	#11719919	C₂×C₈	1	4	yes	—
70	#11719919	C₂×C₈	1	4	yes	—
71	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	C₂²×C₄	1	4	yes	—
74	C₁₆×C₅₃	—	2	4	no	—
75	#11719908	—	2	4	no	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
88	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
89	Dic₁₀₆	—	4	2	no	—
90	Dic₁₀₆	—	4	2	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
95	#88322	—	2	4	no	—
96	#33501	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	#519	1	1	yes	characteristic
107	C₄×C₅₃	#568	1	1	yes	characteristic
108	Dic₅₃	—	2	4	no	—
109	Dic₅₃	—	2	4	no	—
110	#2960	—	53	1	no	—
111	#2183	—	53	4	no	—
112	#2183	—	53	4	no	—
113	#2164	—	53	4	no	—
114	#2163	—	53	1	no	—
115	#2163	D₅₃	1	1	yes	characteristic
116	C₂×C₅₃	—	4	1	no	—
117	C₂×C₅₃	#2982	1	1	yes	characteristic
118	C₂×C₅₃	#3407	1	1	yes	characteristic
119	C₂×C₅₃	#3407	1	1	yes	characteristic
120	#533	—	53	1	no	—
121	#446	—	53	2	no	—
122	#446	—	53	2	no	—
123	C₈×Q₈	—	53	2	no	—
124	C₈×Q₈	—	53	2	no	—
125	#424	—	53	2	no	—
126	#423	—	53	1	no	—
127	#405	—	53	2	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄⋊C₁₆	—	53	8	no	—
130	C₄⋊C₁₆	—	53	2	no	—
131	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
132	C₂²⋊C₁₆	—	53	2	no	—
133	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
134	#346	—	53	4	no	—
135	#346	—	53	2	no	—
136	#346	C₂×D₅₃	1	2	yes	—
137	C₄×C₁₆	—	53	4	no	—
138	C₅₃	#33501	1	1	yes	characteristic, complemented, Sylow
139	C₂×M₁₆	—	53	2	no	—
140	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	53	1	no	—
143	C₄⋊Q₈	—	53	1	no	—
144	#125	—	53	2	no	—
145	C₂²⋊Q₈	—	53	2	no	—
146	C₂²⋊Q₈	—	53	2	no	—
147	C₄×Q₈	—	53	2	no	—
148	C₄×Q₈	—	53	2	no	—
149	#117	—	53	2	no	—
150	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₁₆	—	53	4	no	—
152	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
153	C₄⋊C₈	—	53	4	no	—
154	C₄⋊C₈	—	53	4	no	—
155	C₄⋊C₈	—	53	2	no	—
156	C₄⋊C₈	—	53	2	no	—
157	#97	—	53	4	no	—
158	C₄×C₈	—	53	4	no	—
159	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₄×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	—	53	2	no	—
164	C₂×Q₈	—	53	2	no	—
165	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
168	M₁₆	—	212	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	53	2	no	—
178	C₄⋊C₄	—	53	4	no	—
179	C₄⋊C₄	—	53	4	no	—
180	C₄⋊C₄	—	53	2	no	—
181	C₄⋊C₄	—	53	2	no	—
182	C₂²⋊C₄	—	53	4	no	—
183	C₄²	—	53	4	no	—
184	C₄²	C₈×D₅₃	1	1	yes	characteristic
185	C₄²	C₈×D₅₃	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₁₆	—	106	4	no	—
190	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
191	Q₈	—	212	2	no	—
192	Q₈	—	212	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	106	4	no	—
209	C₈	#13133264	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	106	4	no	—
220	C₄	—	106	4	no	—
221	C₄	#16811100	1	1	yes	characteristic
222	C₄	#16811389	1	1	yes	characteristic
223	C₂	—	4	1	no	—
224	C₂	#20283795	1	1	yes	characteristic
225	C₂	#20287817	1	1	yes	characteristic
226	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2163	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#33501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #20276622	C₂	semidirect
3. #20276621	C₂	semidirect
138. C₅₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
#20276622	2
#20276621	2
#20276519	2
#20276518	2
C₅₃×#2163	2
#33501	53