Subgroups — #27537332

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	2	yes	maximal, complemented
2	#20283619	C₂	1	1	yes	maximal, characteristic
3	#20276556	C₂	1	4	yes	maximal, complemented
4	#20276551	C₂	1	4	yes	maximal, complemented
5	#20276344	C₂	1	2	yes	maximal, complemented
6	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811423	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	2	yes	—
10	#16811041	—	2	2	no	—
11	#16811041	—	2	2	no	—
12	#16811041	C₂²	1	4	yes	complemented
13	#16811041	C₂²	1	4	yes	complemented
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	C₂×#13133156	—	2	2	no	—
16	C₂²×#11719908	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₅₃	—	2	2	no	—
18	#16810746	C₄	1	4	yes	—
19	#16810741	C₂²	1	4	yes	—
20	#16810672	C₄	1	1	yes	characteristic
21	#16810641	C₄	1	2	yes	—
22	#16810408	C₂²	1	8	yes	complemented
23	#16810399	C₂²	1	2	yes	complemented
24	C₄×#11719908	C₂²	1	2	yes	complemented
25	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#405	C₂²	1	1	yes	characteristic
27	C₅₃×#364	C₂²	1	4	yes	complemented
28	C₂×#11719945	C₂×C₄	1	2	yes	—
29	#13133250	C₈	1	4	yes	—
30	#13133245	C₈	1	4	yes	—
31	#13133239	C₂×C₄	1	4	yes	—
32	#13133223	—	2	2	no	—
33	#13133223	—	2	2	no	—
34	#13133223	C₂×C₄	1	4	yes	—
35	#13133223	C₂³	1	4	yes	—
36	C₂×#11719917	C₂×C₄	1	2	yes	—
37	C₂×#11719912	—	2	2	no	—
38	C₂×C₈×D₅₃	—	2	2	no	—
39	#13133157	—	2	2	no	—
40	#13133156	—	2	4	no	—
41	#13133156	—	2	4	no	—
42	C₂×#11719908	D₄	1	2	yes	—
43	C₂×#11719908	D₄	1	2	yes	—
44	C₂×#11719908	C₂³	1	2	yes	—
45	C₂×#11719908	C₂³	1	1	yes	characteristic
46	C₂×#11719908	C₂³	1	1	yes	characteristic
47	C₁₆×D₅₃	—	2	4	no	—
48	C₁₆×D₅₃	—	2	4	no	—
49	#13133153	C₂×C₄	1	4	yes	—
50	#13133145	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₅₃	—	2	2	no	—
55	M₃₂×C₅₃	C₂³	1	4	yes	—
56	C₂×C₁₆×C₅₃	—	2	2	no	—
57	#97×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	—	2	2	no	—
61	#11719945	—	2	2	no	—
62	#11719945	—	2	2	no	—
63	#11719945	C₂×C₈	1	4	yes	—
64	#11719945	C₂²×C₄	1	4	yes	—
65	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₅₃	—	2	2	no	—
67	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
68	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂²⋊C₄×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
77	#11719917	—	2	2	no	—
78	#11719917	C₂²×C₄	1	4	yes	—
79	C₂×#88322	—	2	2	no	—
80	Dic₂₁₂	C₂×C₈	1	4	yes	—
81	D₄₂₄	C₂×C₈	1	4	yes	—
82	#11719913	C₂×C₈	1	8	yes	—
83	#11719912	—	2	4	no	—
84	#11719912	—	2	4	no	—
85	C₈×D₅₃	—	2	4	no	—
86	C₈×D₅₃	—	2	4	no	—
87	C₁₆×C₅₃	—	2	4	no	—
88	#11719908	—	2	2	no	—
89	#11719908	C₂×D₄	1	1	yes	characteristic
90	#11719908	C₂×D₄	1	2	yes	—
91	#11719908	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₅₃	#131	1	1	yes	characteristic
93	C₂²×D₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₅₃	—	2	4	no	—
100	C₂²⋊D₅₃	—	2	4	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	D₂₁₂	—	2	2	no	—
103	D₂₁₂	C₂²×C₈	1	4	yes	—
104	C₄×D₅₃	—	2	4	no	—
105	C₄×D₅₃	—	2	4	no	—
106	Dic₁₀₆	—	2	2	no	—
107	Dic₁₀₆	C₂²×C₈	1	4	yes	—
108	C₈×C₅₃	C₄×D₄	1	2	yes	—
109	C₈×C₅₃	C₂²×C₈	1	4	yes	—
110	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
112	#88322	—	2	4	no	—
113	#33441	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	4	1	no	—
115	C₂²×C₅₃	C₈×D₄	1	2	yes	—
116	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
117	C₂×D₅₃	—	2	4	no	—
118	C₂×D₅₃	—	4	4	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	C₈×D₄	1	2	yes	—
121	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
123	Dic₅₃	—	2	4	no	—
124	C₂×#504	—	53	2	no	—
125	#2955	—	53	1	no	—
126	#2171	—	53	4	no	—
127	#2170	—	53	4	no	—
128	#2153	D₅₃	1	1	yes	characteristic, complemented
129	#2117	—	53	2	no	—
130	#2108	—	53	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
134	D₅₃	—	4	4	no	—
135	#575	—	53	1	no	—
136	C₂×#131	—	53	2	no	—
137	#504	—	53	4	no	—
138	#504	—	53	4	no	—
139	#504	—	106	2	no	—
140	#504	—	106	2	no	—
141	C₂×M₃₂	—	53	1	no	—
142	C₂×M₃₂	—	106	2	no	—
143	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
144	C₂²×C₁₆	—	53	1	no	—
145	C₂²×C₁₆	—	106	2	no	—
146	#444	—	53	4	no	—
147	#443	—	53	4	no	—
148	#430	—	53	1	no	—
149	#421	—	53	2	no	—
150	#405	C₂×D₅₃	1	1	yes	characteristic
151	#364	C₂×D₅₃	1	4	yes	complemented
152	#350	—	53	8	no	—
153	#346	—	53	2	no	—
154	C₄×C₁₆	—	53	2	no	—
155	C₅₃	#33441	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	53	2	no	—
157	#137	—	53	4	no	—
158	AGL(1,ℤ/8)	—	53	4	no	—
159	#135	—	53	4	no	—
160	#131	—	53	4	no	—
161	#131	—	53	4	no	—
162	#131	—	106	2	no	—
163	#131	—	106	2	no	—
164	C₂×M₁₆	—	53	2	no	—
165	C₂×M₁₆	—	106	2	no	—
166	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	#117	C₄×D₅₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	106	4	no	—
172	M₃₂	—	106	4	no	—
173	M₃₂	—	106	2	no	—
174	M₃₂	C₂²×D₅₃	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	53	2	no	—
177	C₂×C₁₆	—	53	2	no	—
178	C₂×C₁₆	—	53	1	no	—
179	C₂×C₁₆	—	53	1	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	106	4	no	—
182	C₂×C₁₆	—	106	4	no	—
183	#108	—	53	4	no	—
184	#104	—	53	8	no	—
185	#97	C₄×D₅₃	1	2	yes	—
186	C₄×C₈	C₂²×D₅₃	1	2	yes	—
187	#41	—	53	4	no	—
188	#41	—	53	4	no	—
189	#41	—	106	2	no	—
190	#41	—	106	2	no	—
191	C₂×Q₈	—	53	2	no	—
192	C₂×D₄	—	53	2	no	—
193	C₂×D₄	—	106	2	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
196	Q₁₆	—	53	4	no	—
197	SD₁₆	—	53	8	no	—
198	D₈	—	53	4	no	—
199	M₁₆	—	53	4	no	—
200	M₁₆	—	106	2	no	—
201	M₁₆	—	106	4	no	—
202	M₁₆	—	106	4	no	—
203	M₁₆	C₈×D₅₃	1	4	yes	—
204	C₂×C₈	—	106	2	no	—
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
209	C₂×C₈	D₄×D₅₃	1	2	yes	—
210	C₂×C₈	#11719949	1	2	yes	—
211	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₅₃	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	53	1	no	—
217	C₁₆	—	53	2	no	—
218	C₁₆	—	53	1	no	—
219	C₁₆	—	106	2	no	—
220	C₂³	—	106	2	no	—
221	C₂³	#13133225	1	1	yes	characteristic
222	Q₈	—	53	4	no	—
223	Q₈	—	106	2	no	—
224	D₄	—	53	4	no	—
225	D₄	—	106	4	no	—
226	D₄	—	106	4	no	—
227	D₄	—	106	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	4	no	—
231	C₂×C₄	—	106	4	no	—
232	C₂×C₄	#13133179	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
234	C₂×C₄	#13133225	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
236	C₈	—	106	4	no	—
237	C₈	#13133179	1	2	yes	—
238	C₈	C₂×C₈×D₅₃	1	4	yes	—
239	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
240	C₈	C₂×#11719949	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	106	4	no	—
243	C₂²	—	212	4	no	—
244	C₂²	#16810688	1	2	yes	—
245	C₂²	C₂×#13133225	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	106	4	no	—
248	C₄	#16810688	1	2	yes	—
249	C₄	C₂×#13133179	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	212	4	no	—
254	C₂	C₂×#16810688	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#16811423	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2153	D₅₃
hypocenter	C₅₃	#33441
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. C₂×#16811041	C₂	semidirect
3. #20276556	C₂	semidirect
4. #20276551	C₂	semidirect
5. #20276344	C₂	semidirect
6. #20276312	C₂	semidirect
7. C₅₃×#2153	C₂	semidirect
12. #16811041	C₂²	semidirect
13. #16811041	C₂²	semidirect
22. #16810408	C₂²	semidirect
23. #16810399	C₂²	semidirect
24. C₄×#11719908	C₂²	semidirect
27. C₅₃×#364	C₂²	semidirect
128. #2153	D₅₃	semidirect
151. #364	C₂×D₅₃	semidirect
155. C₅₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
#20283619	2
#20276556	2
#20276551	2
#20276344	2
#20276312	2
C₅₃×#2153	2
#33441	53