Subgroups — #27537341

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	#20287817	C₂	1	2	yes	maximal, complemented
2	#20283619	C₂	1	1	yes	maximal, characteristic
3	#20276558	C₂	1	4	yes	maximal, complemented
4	#20276553	C₂	1	4	yes	maximal, complemented
5	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810406	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811423	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	2	yes	—
10	#16811041	—	2	4	no	—
11	#16811039	C₂²	1	8	yes	complemented
12	C₂×#13133157	C₂²	1	2	yes	—
13	M₃₂×D₅₃	—	2	4	no	—
14	#16810746	C₄	1	4	yes	—
15	#16810741	C₂²	1	4	yes	—
16	#16810672	C₄	1	1	yes	characteristic
17	#16810641	C₄	1	2	yes	—
18	#16810444	C₂²	1	4	yes	complemented
19	#16810406	C₂²	1	4	yes	complemented
20	#16810406	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#405	C₂²	1	1	yes	characteristic
24	C₅₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
26	C₂×#11719945	C₂×C₄	1	2	yes	—
27	#13133250	C₈	1	4	yes	—
28	#13133245	C₈	1	4	yes	—
29	#13133239	C₂×C₄	1	4	yes	—
30	#13133223	—	2	2	no	—
31	#13133223	—	2	2	no	—
32	#13133223	C₂×C₄	1	4	yes	—
33	#13133223	C₂³	1	4	yes	—
34	C₂×#11719917	C₂×C₄	1	2	yes	—
35	C₂×#11719912	—	2	2	no	—
36	C₂×C₈×D₅₃	—	2	2	no	—
37	#13133157	—	2	2	no	—
38	#13133157	C₂³	1	4	yes	—
39	#13133156	—	2	8	no	—
40	C₂×#11719908	—	2	2	no	—
41	C₁₆×D₅₃	—	2	8	no	—
42	#13133153	C₂×C₄	1	4	yes	—
43	#13133145	C₂×C₄	1	8	yes	—
44	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₅₃	D₄	1	4	yes	—
48	C₂×C₁₆×C₅₃	—	2	2	no	—
49	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
52	#97×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₅₃	—	2	2	no	—
56	#11719945	—	2	2	no	—
57	#11719945	—	2	2	no	—
58	#11719945	C₂×C₈	1	4	yes	—
59	#11719945	C₂²×C₄	1	4	yes	—
60	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
63	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	#11719917	—	2	2	no	—
73	#11719917	C₂²×C₄	1	4	yes	—
74	C₂×#88322	—	2	2	no	—
75	Dic₂₁₂	C₂×C₈	1	4	yes	—
76	D₄₂₄	C₂×C₈	1	4	yes	—
77	#11719913	C₂×C₈	1	8	yes	—
78	#11719912	—	2	4	no	—
79	#11719912	—	2	4	no	—
80	C₈×D₅₃	—	2	4	no	—
81	C₈×D₅₃	—	2	4	no	—
82	C₁₆×C₅₃	—	2	1	no	—
83	C₁₆×C₅₃	—	2	1	no	—
84	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
85	#11719908	—	2	4	no	—
86	C₂³×C₅₃	#131	1	1	yes	characteristic
87	C₂²×D₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₅₃	—	2	4	no	—
94	C₂²⋊D₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	D₂₁₂	—	2	2	no	—
97	D₂₁₂	C₂²×C₈	1	4	yes	—
98	C₄×D₅₃	—	2	4	no	—
99	C₄×D₅₃	—	2	4	no	—
100	Dic₁₀₆	—	2	2	no	—
101	Dic₁₀₆	C₂²×C₈	1	4	yes	—
102	C₈×C₅₃	C₄×D₄	1	2	yes	—
103	C₈×C₅₃	C₂²×C₈	1	4	yes	—
104	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
106	#88322	—	2	4	no	—
107	#33442	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	C₈×D₄	1	2	yes	—
110	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
111	C₂×D₅₃	—	2	4	no	—
112	C₂×D₅₃	—	8	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	C₈×D₄	1	2	yes	—
115	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
117	Dic₅₃	—	2	4	no	—
118	#3407	—	53	2	no	—
119	#2955	—	53	1	no	—
120	#2171	—	53	4	no	—
121	#2170	—	53	4	no	—
122	#2153	—	53	1	no	—
123	C₂×#350	—	53	2	no	—
124	#2108	D₅₃	1	1	yes	characteristic, complemented
125	C₂×C₅₃	—	2	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
128	D₅₃	—	4	4	no	—
129	#575	—	53	1	no	—
130	C₂×#131	—	53	2	no	—
131	#504	—	53	8	no	—
132	#504	—	106	4	no	—
133	C₂×M₃₂	—	53	2	no	—
134	C₂×M₃₂	—	106	4	no	—
135	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
136	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
137	#444	—	53	4	no	—
138	#443	—	53	4	no	—
139	#430	—	53	1	no	—
140	#421	—	53	2	no	—
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	C₂×D₅₃	1	2	yes	complemented
146	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
147	C₅₃	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	53	2	no	—
149	#137	—	53	4	no	—
150	AGL(1,ℤ/8)	—	53	4	no	—
151	#135	—	53	4	no	—
152	#131	—	53	4	no	—
153	#131	—	53	4	no	—
154	#131	—	106	2	no	—
155	#131	—	106	2	no	—
156	C₂×M₁₆	—	53	2	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	#117	C₄×D₅₃	1	1	yes	characteristic
162	M₃₂	—	53	4	no	—
163	M₃₂	—	106	2	no	—
164	M₃₂	—	106	8	no	—
165	M₃₂	D₂₁₂	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	106	2	no	—
168	C₂×C₁₆	—	106	8	no	—
169	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
172	#108	—	53	4	no	—
173	#104	—	53	8	no	—
174	#97	C₄×D₅₃	1	2	yes	—
175	C₄×C₈	C₂²×D₅₃	1	2	yes	—
176	#41	—	53	4	no	—
177	#41	—	53	4	no	—
178	#41	—	106	2	no	—
179	#41	—	106	2	no	—
180	C₂×Q₈	—	53	2	no	—
181	C₂×D₄	—	53	2	no	—
182	C₂×D₄	—	106	2	no	—
183	C₂²×C₄	—	106	2	no	—
184	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
185	Q₁₆	—	53	4	no	—
186	SD₁₆	—	53	8	no	—
187	D₈	—	53	4	no	—
188	M₁₆	—	53	4	no	—
189	M₁₆	—	106	2	no	—
190	M₁₆	—	106	4	no	—
191	M₁₆	—	106	4	no	—
192	M₁₆	C₈×D₅₃	1	4	yes	—
193	C₂×C₈	—	106	2	no	—
194	C₂×C₈	—	106	4	no	—
195	C₂×C₈	—	106	4	no	—
196	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
198	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
199	C₂×C₈	#11719945	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	C₂×C₄×D₅₃	1	2	yes	—
204	C₁₆	—	2	1	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	106	4	no	—
207	C₁₆	C₂×D₂₁₂	1	4	yes	—
208	C₂³	—	106	2	no	—
209	C₂³	#13133223	1	1	yes	characteristic
210	Q₈	—	53	4	no	—
211	Q₈	—	106	2	no	—
212	D₄	—	53	4	no	—
213	D₄	—	106	4	no	—
214	D₄	—	106	4	no	—
215	D₄	—	106	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	—	106	4	no	—
219	C₂×C₄	—	106	4	no	—
220	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
222	C₂×C₄	#13133223	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
224	C₈	—	106	4	no	—
225	C₈	C₄×D₂₁₂	1	2	yes	—
226	C₈	C₂×C₈×D₅₃	1	4	yes	—
227	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
228	C₈	C₂×#11719945	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	106	4	no	—
231	C₂²	—	424	2	no	—
232	C₂²	C₈×D₂₁₂	1	2	yes	—
233	C₂²	C₂×#13133223	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	106	4	no	—
236	C₄	C₈×D₂₁₂	1	2	yes	—
237	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	212	4	no	—
242	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#16811423	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #20287817	C₂	semidirect
3. #20276558	C₂	semidirect
4. #20276553	C₂	semidirect
5. #20276481	C₂	semidirect
6. C₂×#16810406	C₂	semidirect
7. C₅₃×#2108	C₂	semidirect
11. #16811039	C₂²	semidirect
18. #16810444	C₂²	semidirect
19. #16810406	C₂²	semidirect
20. #16810406	C₂²	semidirect
24. C₅₃×#346	C₂²	semidirect
25. C₄×C₁₆×C₅₃	C₂²	semidirect
124. #2108	D₅₃	semidirect
145. #346	C₂×D₅₃	semidirect
146. C₄×C₁₆	C₂×D₅₃	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
#20287817	2
#20283619	2
#20276558	2
#20276553	2
#20276481	2
C₂×#16810406	2
C₅₃×#2108	2
#33442	53