Subgroups — #27537340

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287818	C₂	1	2	yes	maximal, complemented
2	#20276557	C₂	1	4	yes	maximal, complemented
3	#20276552	C₂	1	4	yes	maximal, complemented
4	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810407	C₂	1	2	yes	maximal, complemented
6	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2955	C₂	1	1	yes	maximal, characteristic
8	#16811040	—	2	4	no	—
9	#16811040	C₂²	1	8	yes	complemented
10	C₂×#13133157	—	2	4	no	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	#16810444	C₂²	1	4	yes	complemented
15	#16810407	C₂²	1	4	yes	complemented
16	#16810407	C₂²	1	4	yes	complemented
17	#16810399	C₂²	1	2	yes	complemented
18	C₄×#11719908	C₂²	1	2	yes	complemented
19	C₅₃×#575	C₄	1	1	yes	characteristic
20	C₂×#131×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#444	C₄	1	4	yes	—
22	C₅₃×#443	C₂²	1	4	yes	—
23	C₅₃×#430	C₄	1	1	yes	characteristic
24	C₅₃×#421	C₄	1	2	yes	—
25	C₅₃×#405	C₂²	1	1	yes	characteristic
26	#13133157	—	2	2	no	—
27	#13133157	—	2	8	no	—
28	#13133157	D₄	1	4	yes	—
29	#13133157	C₂³	1	4	yes	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	—	2	8	no	—
33	C₂×#11719908	C₂³	1	1	yes	characteristic
34	C₂×#11719908	C₂³	1	1	yes	characteristic
35	C₂×#11719908	C₂³	1	2	yes	—
36	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
37	#137×C₅₃	C₈	1	4	yes	—
38	AGL(1,ℤ/8)×C₅₃	C₈	1	4	yes	—
39	#135×C₅₃	C₂×C₄	1	4	yes	—
40	#131×C₅₃	—	2	2	no	—
41	#131×C₅₃	—	2	2	no	—
42	#131×C₅₃	C₂×C₄	1	4	yes	—
43	#131×C₅₃	C₂³	1	4	yes	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
50	#108×C₅₃	C₂×C₄	1	4	yes	—
51	#104×C₅₃	C₂×C₄	1	8	yes	—
52	#97×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	#41×C₅₃	—	2	2	no	—
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	C₂×C₈	1	4	yes	—
57	#41×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
64	SD₁₆×C₅₃	C₂×C₈	1	8	yes	—
65	D₈×C₅₃	C₂×C₈	1	4	yes	—
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
70	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	#11719908	—	2	4	no	—
83	#11719908	—	2	1	no	—
84	#11719908	—	2	1	no	—
85	#11719908	C₂×D₄	1	4	yes	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	#131	1	1	yes	characteristic
88	Q₈×C₅₃	—	2	2	no	—
89	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
90	D₄×C₅₃	—	2	4	no	—
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	2	2	no	—
93	D₄×C₅₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₅₃	—	2	4	no	—
103	C₈×C₅₃	C₄×D₄	1	2	yes	—
104	C₈×C₅₃	C₂²×C₈	1	4	yes	—
105	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
107	#33442	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	4	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	8	2	no	—
111	C₂²×C₅₃	C₈×D₄	1	2	yes	—
112	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	C₈×D₄	1	2	yes	—
116	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
118	#3407	—	53	2	no	—
119	#2955	D₅₃	1	1	yes	characteristic
120	#2171	—	53	4	no	—
121	#2170	—	53	4	no	—
122	#2153	—	53	1	no	—
123	C₂×#350	—	53	2	no	—
124	#2108	—	53	1	no	—
125	C₂×C₅₃	—	2	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	4	4	no	—
128	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
129	#575	Dic₅₃	1	1	yes	characteristic
130	C₂×#131	C₂×D₅₃	1	2	yes	—
131	#504	—	53	8	no	—
132	#504	—	106	4	no	—
133	C₂×M₃₂	—	53	1	no	—
134	C₂×M₃₂	—	53	2	no	—
135	C₂×M₃₂	—	106	4	no	—
136	C₂²×C₁₆	—	53	1	no	—
137	#444	Dic₅₃	1	4	yes	—
138	#443	C₂×D₅₃	1	4	yes	—
139	#430	Dic₅₃	1	1	yes	characteristic
140	#421	Dic₅₃	1	2	yes	—
141	#405	C₂×D₅₃	1	1	yes	characteristic
142	#364	—	53	4	no	—
143	#350	—	53	4	no	—
144	#350	—	53	4	no	—
145	#346	—	53	2	no	—
146	C₄×C₁₆	—	53	2	no	—
147	C₅₃	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	C₂×Dic₅₃	1	2	yes	—
149	#137	#88322	1	4	yes	—
150	AGL(1,ℤ/8)	#88322	1	4	yes	—
151	#135	C₂×Dic₅₃	1	4	yes	—
152	#131	—	2	2	no	—
153	#131	—	2	2	no	—
154	#131	C₂×Dic₅₃	1	4	yes	—
155	#131	C₂²×D₅₃	1	4	yes	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
158	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	#117	C₂×Dic₅₃	1	1	yes	characteristic
162	M₃₂	—	53	4	no	—
163	M₃₂	—	53	4	no	—
164	M₃₂	—	106	8	no	—
165	M₃₂	—	106	2	no	—
166	C₂×C₁₆	—	53	2	no	—
167	C₂×C₁₆	—	53	1	no	—
168	C₂×C₁₆	—	53	1	no	—
169	C₂×C₁₆	—	106	8	no	—
170	C₂×C₁₆	—	106	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	#108	C₂×Dic₅₃	1	4	yes	—
173	#104	C₂×Dic₅₃	1	8	yes	—
174	#97	C₂×Dic₅₃	1	2	yes	—
175	C₄×C₈	C₂²×D₅₃	1	2	yes	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×#88322	1	4	yes	—
179	#41	C₂²×Dic₅₃	1	4	yes	—
180	C₂×Q₈	C₂×#88322	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×#88322	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
185	Q₁₆	C₂×#88322	1	4	yes	—
186	SD₁₆	C₂×#88322	1	8	yes	—
187	D₈	C₂×#88322	1	4	yes	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	C₂×#88322	1	4	yes	—
192	M₁₆	C₂²×Dic₅₃	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	D₄×D₅₃	1	2	yes	—
197	C₂×C₈	#11719947	1	2	yes	—
198	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
200	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂²×Dic₅₃	1	2	yes	—
204	C₁₆	—	53	4	no	—
205	C₁₆	—	106	4	no	—
206	C₁₆	—	106	1	no	—
207	C₁₆	—	106	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#13133224	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×#88322	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×#88322	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
221	C₂×C₄	C₂²×#88322	1	2	yes	—
222	C₂×C₄	#13133224	1	1	yes	characteristic
223	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	D₄×Dic₅₃	1	2	yes	—
226	C₈	C₂²×#88322	1	4	yes	—
227	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
228	C₈	C₂×#11719947	1	1	yes	characteristic
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	D₄×#88322	1	2	yes	—
233	C₂²	C₂×#13133224	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	D₄×#88322	1	2	yes	—
237	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
238	C₄	C₂³×#88322	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	C₂	C₂×D₄×#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2955	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	C₅₃×#575	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2955	D₅₃
hypocenter	C₅₃	#33442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. #20287818	C₂	semidirect
2. #20276557	C₂	semidirect
3. #20276552	C₂	semidirect
4. #20276481	C₂	semidirect
5. C₂×#16810407	C₂	semidirect
6. #20276312	C₂	semidirect
9. #16811040	C₂²	semidirect
14. #16810444	C₂²	semidirect
15. #16810407	C₂²	semidirect
16. #16810407	C₂²	semidirect
17. #16810399	C₂²	semidirect
18. C₄×#11719908	C₂²	semidirect
147. C₅₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#20276557	2
#20276552	2
#20276481	2
C₂×#16810407	2
#20276312	2
C₅₃×#2955	2
#33442	53