Subgroups — #27537512

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283632	C₂	1	2	yes	maximal, complemented
2	#20276601	C₂	1	4	yes	maximal, complemented
3	#20276596	C₂	1	4	yes	maximal, complemented
4	#20276510	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810430	C₂	1	2	yes	maximal, complemented
6	#20276314	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3414	C₂	1	1	yes	maximal, characteristic
8	#16811390	—	2	2	no	—
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	#16810745	—	2	4	no	—
11	#16810740	C₂²	1	8	yes	complemented
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810640	—	2	2	no	—
14	#16810578	C₂²	1	1	yes	characteristic
15	#16810453	C₂²	1	4	yes	complemented
16	#16810430	C₂²	1	4	yes	complemented
17	#16810430	C₂²	1	4	yes	complemented
18	#16810398	C₂²	1	2	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
20	C₂×#135×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#510	C₄	1	4	yes	complemented
22	C₅₃×#509	C₄	1	4	yes	complemented
23	C₅₃×#508	C₂²	1	4	yes	—
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	#13133224	—	2	8	no	—
26	#13133224	—	4	4	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	4	4	no	—
29	C₂×#11719917	D₄	1	1	yes	characteristic
30	#13133176	D₄	1	1	yes	characteristic
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂³	1	1	yes	characteristic
33	C₂×#11719908	C₂³	1	1	yes	characteristic
34	#13133153	—	2	2	no	—
35	#13133153	C₂³	1	4	yes	—
36	#13133144	—	2	8	no	—
37	#13133127	D₄	1	2	yes	—
38	C₈×Dic₅₃	C₂³	1	2	yes	—
39	C₂×#41×C₅₃	—	2	2	no	—
40	#135×C₅₃	—	2	2	no	—
41	#135×C₅₃	—	2	2	no	—
42	#135×C₅₃	C₂×C₄	1	4	yes	—
43	#135×C₅₃	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	Q₃₂×C₅₃	C₂×C₄	1	4	yes	—
49	SD₃₂×C₅₃	C₂×C₄	1	8	yes	—
50	D₁₆×C₅₃	C₂×C₄	1	4	yes	—
51	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
53	#41×C₅₃	—	2	4	no	—
54	#41×C₅₃	—	2	4	no	—
55	#41×C₅₃	—	4	2	no	—
56	#41×C₅₃	—	4	2	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₅₃	—	2	2	no	—
63	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
64	SD₁₆×C₅₃	—	2	4	no	—
65	SD₁₆×C₅₃	—	2	4	no	—
66	D₈×C₅₃	—	2	2	no	—
67	D₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	2	1	no	—
73	C₄⋊Dic₅₃	—	2	1	no	—
74	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
75	#11719917	—	2	4	no	—
76	#11719917	—	4	2	no	—
77	#11719917	—	4	8	no	—
78	#11719917	D₈	1	4	yes	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	8	no	—
81	C₂×#88322	C₂×D₄	1	2	yes	—
82	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
83	#11719908	—	2	1	no	—
84	#11719908	—	2	1	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	#135	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	4	no	—
88	Q₈×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	2	4	no	—
90	D₄×C₅₃	—	4	4	no	—
91	D₄×C₅₃	—	4	4	no	—
92	D₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	2	no	—
101	C₈×C₅₃	C₄×D₄	1	2	yes	—
102	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
104	#88322	—	4	4	no	—
105	#88322	C₂×D₈	1	4	yes	—
106	#33464	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	4	no	—
109	C₂²×C₅₃	—	16	2	no	—
110	C₂²×C₅₃	C₄×D₈	1	2	yes	—
111	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
112	C₄×C₅₃	—	4	4	no	—
113	C₄×C₅₃	C₄×D₈	1	2	yes	—
114	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
116	Dic₅₃	—	4	2	no	—
117	#3414	D₅₃	1	1	yes	characteristic
118	#2957	—	53	2	no	—
119	#2179	—	53	4	no	—
120	#2178	—	53	4	no	—
121	#2161	—	53	1	no	—
122	C₂×#359	—	53	2	no	—
123	#2108	—	53	1	no	—
124	C₂×C₅₃	—	2	1	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	8	4	no	—
127	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
128	C₂×#135	C₂×D₅₃	1	2	yes	—
129	#568	—	106	2	no	—
130	#510	Dic₅₃	1	4	yes	complemented
131	#509	Dic₅₃	1	4	yes	complemented
132	#508	C₂×D₅₃	1	4	yes	—
133	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
134	C₂²×C₁₆	—	53	1	no	—
135	#444	—	106	4	no	—
136	#443	—	53	8	no	—
137	C₂×#108	—	53	2	no	—
138	#421	—	106	2	no	—
139	#405	—	53	1	no	—
140	#368	—	53	4	no	—
141	#359	—	53	4	no	—
142	#359	—	53	4	no	—
143	#346	—	53	2	no	—
144	C₄×C₁₆	—	53	2	no	—
145	C₅₃	#33464	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	C₂×Dic₅₃	1	4	yes	—
150	#135	C₂²×D₅₃	1	4	yes	—
151	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
154	#131	—	106	8	no	—
155	#131	—	212	4	no	—
156	C₂×M₁₆	—	53	1	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	—	212	4	no	—
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	#117	—	53	1	no	—
161	Q₃₂	C₂×Dic₅₃	1	4	yes	—
162	SD₃₂	C₂×Dic₅₃	1	8	yes	—
163	D₁₆	C₂×Dic₅₃	1	4	yes	—
164	M₃₂	C₂×Dic₅₃	1	4	yes	—
165	C₂×C₁₆	—	53	1	no	—
166	C₂×C₁₆	—	53	1	no	—
167	C₂×C₁₆	—	106	2	no	—
168	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
169	#108	—	53	4	no	—
170	#108	—	106	2	no	—
171	#104	—	106	8	no	—
172	#97	—	53	2	no	—
173	C₄×C₈	—	53	2	no	—
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×Dic₅₃	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×Dic₅₃	1	4	yes	—
189	M₁₆	—	53	4	no	—
190	M₁₆	—	106	4	no	—
191	M₁₆	—	212	8	no	—
192	M₁₆	—	212	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	212	8	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
197	C₂×C₈	#11719947	1	2	yes	—
198	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
199	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	106	1	no	—
201	C₂²⋊C₄	—	106	1	no	—
202	C₄²	—	53	2	no	—
203	C₁₆	—	106	1	no	—
204	C₁₆	—	106	1	no	—
205	C₁₆	C₂²×Dic₅₃	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	#13133236	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
219	C₂×C₄	D₈×D₅₃	1	2	yes	—
220	C₂×C₄	#13133236	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
222	C₈	—	53	4	no	—
223	C₈	—	212	4	no	—
224	C₈	D₄×Dic₅₃	1	2	yes	—
225	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
226	C₈	C₂×#11719947	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	D₈×Dic₅₃	1	2	yes	—
231	C₂²	C₂×#13133236	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	—	212	2	no	—
234	C₄	D₈×Dic₅₃	1	2	yes	—
235	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
236	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3414	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#3414	D₅₃
hypocenter	C₅₃	#33464
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

Complements

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

N	H ≅ G/N	type
1. #20283632	C₂	semidirect
2. #20276601	C₂	semidirect
3. #20276596	C₂	semidirect
4. #20276510	C₂	semidirect
5. C₂×#16810430	C₂	semidirect
6. #20276314	C₂	semidirect
11. #16810740	C₂²	semidirect
15. #16810453	C₂²	semidirect
16. #16810430	C₂²	semidirect
17. #16810430	C₂²	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
21. C₅₃×#510	C₄	semidirect
22. C₅₃×#509	C₄	semidirect
130. #510	Dic₅₃	semidirect
131. #509	Dic₅₃	semidirect
145. C₅₃	#33464	semidirect

Maximal subgroups

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

subgroup	index
#20283632	2
#20276601	2
#20276596	2
#20276510	2
C₂×#16810430	2
#20276314	2
C₅₃×#3414	2
#33464	53