Subgroups — #27397983

Order 13568.

810 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	#20283433	C₂	1	2	yes	maximal, complemented
2	#20275548	C₂	1	2	yes	maximal, complemented
3	#20275044	C₂	1	8	yes	maximal, complemented
4	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274466	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133224	C₂²	1	4	yes	—
8	#16810695	—	2	8	no	—
9	#16810647	C₂²	1	2	yes	—
10	C₂×#13133144	C₂²	1	8	yes	—
11	#16810590	C₂²	1	4	yes	—
12	C₂×#13133128	C₂²	1	1	yes	characteristic
13	#16810469	C₂²	1	1	yes	characteristic
14	#16810388	C₂²	1	4	yes	—
15	#16810377	C₂²	1	4	yes	—
16	C₂×#135×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#532	C₂²	1	2	yes	—
18	C₅₃×#495	C₄	1	4	yes	complemented
19	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#492	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	#13133224	—	4	8	no	—
24	C₂×#11719917	—	2	4	no	—
25	C₂×#11719917	C₂³	1	4	yes	—
26	C₂²×#88322	—	2	4	no	—
27	C₂²×#88322	C₂³	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
29	C₄⋊#88322	—	2	4	no	—
30	#13133144	—	4	8	no	—
31	C₂²⋊#88322	—	2	8	no	—
32	#13133128	D₄	1	4	yes	—
33	#13133124	—	2	2	no	—
34	C₂×#41×C₅₃	C₂³	1	4	yes	—
35	#135×C₅₃	—	4	8	no	—
36	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
41	#127×C₅₃	C₂×C₄	1	2	yes	—
42	#124×C₅₃	C₂×C₄	1	4	yes	—
43	#121×C₅₃	—	2	4	no	—
44	C₄×D₄×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	#41×C₅₃	—	2	8	no	—
50	#41×C₅₃	—	2	8	no	—
51	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	2	no	—
59	Q₁₆×C₅₃	—	4	4	no	—
60	SD₁₆×C₅₃	—	4	8	no	—
61	D₈×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂²⋊C₄×C₅₃	—	2	8	no	—
67	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₄×Dic₅₃	—	2	4	no	—
71	#11719917	—	2	8	no	—
72	#11719917	—	4	8	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	4	8	no	—
76	C₂×#88322	C₂×D₄	1	4	yes	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	8	no	—
80	D₄×C₅₃	—	2	8	no	—
81	D₄×C₅₃	—	4	8	no	—
82	D₄×C₅₃	—	8	4	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₅₃	—	4	4	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	#88322	—	2	4	no	—
97	#88322	—	4	4	no	—
98	#22282	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	4	4	no	—
101	C₂²×C₅₃	—	4	8	no	—
102	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₅₃	#540	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	4	no	—
107	C₄×C₅₃	#518	1	2	yes	—
108	C₄×C₅₃	#535	1	2	yes	—
109	Dic₅₃	—	8	2	no	—
110	#3147	D₅₃	1	1	yes	characteristic
111	#2932	—	53	2	no	—
112	#1963	—	53	2	no	—
113	#1876	—	53	8	no	—
114	#1837	—	53	1	no	—
115	#1746	—	53	1	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	—	8	4	no	—
118	C₂×C₅₃	#2328	1	1	yes	characteristic
119	C₂×C₅₃	#2957	1	2	yes	—
120	C₂×#135	C₂×D₅₃	1	4	yes	—
121	C₂×#131	—	53	4	no	—
122	#532	C₂×D₅₃	1	2	yes	—
123	#495	Dic₅₃	1	4	yes	complemented
124	#494	Dic₅₃	1	1	yes	characteristic, complemented
125	#493	Dic₅₃	1	1	yes	characteristic, complemented
126	#492	Dic₅₃	1	2	yes	complemented
127	#435	—	106	8	no	—
128	#423	—	53	2	no	—
129	C₂×#104	—	53	8	no	—
130	#408	—	53	4	no	—
131	C₂×#97	—	53	1	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	#376	—	53	1	no	—
134	#341	—	53	4	no	—
135	#336	—	53	4	no	—
136	C₅₃	#22282	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	C₂²×D₅₃	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
140	C₂×SD₁₆	C₂×Dic₅₃	1	8	yes	—
141	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
142	#131	—	212	8	no	—
143	C₂×M₁₆	—	53	4	no	—
144	C₂×M₁₆	—	106	4	no	—
145	C₂²×C₈	—	53	2	no	—
146	C₂²×C₈	—	106	4	no	—
147	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
149	#127	C₂×Dic₅₃	1	2	yes	—
150	#124	C₂×Dic₅₃	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	—	53	2	no	—
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	106	4	no	—
156	#104	—	212	8	no	—
157	C₂²⋊C₈	—	106	8	no	—
158	#97	—	53	4	no	—
159	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
162	#95	—	106	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	#11719947	1	2	yes	—
172	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	106	8	no	—
177	M₁₆	—	212	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	53	4	no	—
180	C₂×C₈	—	106	4	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	212	8	no	—
183	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
184	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	106	4	no	—
188	C₄²	D₄×D₅₃	1	2	yes	—
189	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#11719947	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	212	4	no	—
202	C₂×C₄	—	212	2	no	—
203	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
204	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
205	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	#13133262	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	106	4	no	—
211	C₈	—	212	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
216	C₂²	#16811227	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	424	2	no	—
221	C₄	#16811098	1	2	yes	—
222	C₄	#16811186	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#20277441	1	1	yes	characteristic
226	C₂	#20283632	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#22282
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20283433	C₂	semidirect
2. #20275548	C₂	semidirect
3. #20275044	C₂	semidirect
4. #20274853	C₂	semidirect
5. #20274466	C₂	semidirect
18. C₅₃×#495	C₄	semidirect
19. C₅₃×#494	C₄	semidirect
20. C₅₃×#493	C₄	semidirect
21. C₅₃×#492	C₄	semidirect
123. #495	Dic₅₃	semidirect
124. #494	Dic₅₃	semidirect
125. #493	Dic₅₃	semidirect
126. #492	Dic₅₃	semidirect
136. C₅₃	#22282	semidirect

Maximal subgroups

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

subgroup	index
#20283433	2
#20275548	2
#20275044	2
#20274853	2
#20274466	2
C₅₃×#3147	2
#22282	53