Subgroups — #28253637

Order 13568.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811428	C₂	1	4	yes	maximal, complemented
2	#20288493	C₂	1	2	yes	maximal, complemented
3	#20285201	C₂	1	12	yes	maximal, complemented
4	#20285167	C₂	1	6	yes	maximal, complemented
5	#20285143	C₂	1	3	yes	maximal, complemented
6	C₂×#16811000	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133263	C₂²	1	4	yes	—
9	#16811428	—	2	32	no	—
10	C₂×#13133245	C₂²	1	24	yes	complemented
11	C₂×#13133239	C₂²	1	12	yes	complemented
12	C₂²×D₄₂₄	C₂²	1	12	yes	complemented
13	#16811295	—	2	4	no	—
14	D₄×D₂₁₂	—	2	12	no	—
15	#16811192	C₂²	1	12	yes	complemented
16	#16811173	C₂²	1	6	yes	complemented
17	C₂×#13133214	C₂²	1	6	yes	complemented
18	#16811012	C₂²	1	24	yes	complemented
19	#16811007	C₂²	1	12	yes	complemented
20	#16811003	C₂²	1	1	yes	characteristic, complemented
21	#16811000	C₂²	1	12	yes	complemented
22	#16811000	C₂²	1	3	yes	complemented
23	#16811000	C₂²	1	6	yes	complemented
24	#16810996	C₂²	1	6	yes	complemented
25	C₂×#131×C₅₃	C₂²	1	2	yes	—
26	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#405	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
30	#13133263	—	2	32	no	—
31	C₂×#11719949	—	2	4	no	—
32	C₂×#11719945	C₂³	1	12	yes	—
33	C₂²×D₂₁₂	—	2	12	no	—
34	C₂²×D₂₁₂	C₂³	1	12	yes	—
35	C₂×D₄×D₅₃	—	2	12	no	—
36	#13133245	—	2	96	no	—
37	#13133239	—	2	48	no	—
38	C₂×Dic₂₁₂	C₂³	1	4	yes	—
39	C₂×#11719913	C₂³	1	24	yes	—
40	C₂×D₄₂₄	—	2	48	no	—
41	C₂×D₄₂₄	C₂³	1	24	yes	—
42	C₂×D₄₂₄	C₂³	1	12	yes	—
43	#13133217	C₂³	1	2	yes	—
44	#13133214	—	2	12	no	—
45	#13133214	C₂³	1	12	yes	—
46	#13133214	C₂³	1	6	yes	—
47	#13133205	C₂³	1	12	yes	—
48	#13133192	—	2	24	no	—
49	#13133191	—	2	12	no	—
50	#13133187	—	2	24	no	—
51	C₄×D₂₁₂	—	2	12	no	—
52	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
53	#131×C₅₃	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
56	C₄×Q₈×C₅₃	D₄	1	2	yes	—
57	C₄×D₄×C₅₃	D₄	1	6	yes	—
58	#117×C₅₃	C₂³	1	3	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
60	#97×C₅₃	C₂³	1	6	yes	—
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	3	yes	—
63	C₄×C₈×C₅₃	C₂³	1	6	yes	—
64	C₂³×D₅₃	—	2	12	no	—
65	#41×C₅₃	C₂×D₄	1	8	yes	—
66	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
70	C₂×C₂²⋊D₅₃	—	2	12	no	—
71	#11719949	—	4	16	no	—
72	D₄×D₅₃	—	4	48	no	—
73	#11719945	—	2	48	no	—
74	C₂×D₂₁₂	—	2	48	no	—
75	C₂×D₂₁₂	—	2	12	no	—
76	C₂×D₂₁₂	—	2	12	no	—
77	C₂×D₂₁₂	—	2	24	no	—
78	C₂×D₂₁₂	—	8	24	no	—
79	C₂×D₂₁₂	C₂⁴	1	12	yes	—
80	C₂×C₄×D₅₃	—	2	12	no	—
81	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
82	M₁₆×C₅₃	C₂×D₄	1	24	yes	complemented
83	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
85	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
86	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
87	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
88	C₄²×C₅₃	C₂×D₄	1	6	yes	—
89	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₅₃	C₂⁴	1	3	yes	—
91	#11719921	—	2	24	no	—
92	C₄⋊Dic₅₃	—	2	4	no	—
93	Dic₂₁₂	—	2	8	no	—
94	D₄₂₄	—	2	24	no	—
95	D₄₂₄	—	2	48	no	—
96	#11719913	—	2	48	no	—
97	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
98	C₂²×D₅₃	—	2	12	no	—
99	C₂²×D₅₃	—	4	24	no	—
100	C₂²×D₅₃	—	8	24	no	—
101	Q₈×C₅₃	#127	1	4	yes	—
102	D₄×C₅₃	#127	1	12	yes	—
103	C₂×C₄×C₅₃	—	2	12	no	—
104	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₅₃	—	4	24	no	—
110	C₂×Dic₅₃	—	2	4	no	—
111	D₂₁₂	—	2	24	no	—
112	D₂₁₂	—	4	24	no	—
113	D₂₁₂	—	4	24	no	—
114	D₂₁₂	—	8	24	no	—
115	C₄×D₅₃	—	4	24	no	—
116	Dic₁₀₆	—	2	8	no	—
117	C₈×C₅₃	C₂²×D₄	1	12	yes	—
118	C₈×C₅₃	C₂²×D₄	1	4	yes	—
119	#60647	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	6	no	—
121	C₂²×C₅₃	C₂×#127	1	6	yes	—
122	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
123	C₂×D₅₃	—	4	12	no	—
124	C₂×D₅₃	—	4	24	no	—
125	C₂×D₅₃	—	16	24	no	—
126	C₄×C₅₃	—	2	6	no	—
127	C₄×C₅₃	C₂×#127	1	6	yes	—
128	C₄×C₅₃	#576	1	4	yes	complemented
129	C₄×C₅₃	C₂³×D₄	1	2	yes	—
130	Dic₅₃	—	4	4	no	—
131	C₂×#576	—	53	4	no	—
132	#3498	—	53	2	no	—
133	#3155	—	53	12	no	—
134	#3151	—	53	6	no	—
135	#3147	—	53	3	no	—
136	C₂×#493	—	53	3	no	—
137	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
138	C₂×C₅₃	—	2	6	no	—
139	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#576	1	2	yes	—
141	D₅₃	—	8	12	no	—
142	C₂×H(5,𝔽₂)	—	53	4	no	—
143	#576	—	106	32	no	—
144	C₂×AGL(1,ℤ/8)	—	53	24	no	—
145	C₂×#135	—	53	12	no	—
146	C₂²×D₈	—	53	12	no	—
147	C₂×#131	C₂×D₅₃	1	2	yes	—
148	#550	—	106	4	no	—
149	D₄²	—	106	12	no	—
150	#535	—	53	12	no	—
151	#532	—	53	6	no	—
152	C₂×#127	—	53	6	no	—
153	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
154	#496	—	53	24	no	—
155	#495	—	53	12	no	—
156	#494	—	53	1	no	—
157	#493	—	53	3	no	—
158	#493	—	53	6	no	—
159	#493	—	53	12	no	—
160	#492	—	53	6	no	—
161	#405	C₂×D₅₃	1	6	yes	complemented
162	C₄×M₁₆	C₂×D₅₃	1	3	yes	complemented
163	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	complemented
164	C₅₃	#60647	1	1	yes	characteristic, complemented, Sylow
165	H(5,𝔽₂)	—	106	32	no	—
166	C₂×#41	—	53	12	no	—
167	C₂×#41	—	106	4	no	—
168	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
169	C₂²×D₄	—	53	12	no	—
170	C₂²×D₄	—	106	12	no	—
171	C₂²×D₄	—	106	12	no	—
172	AGL(1,ℤ/8)	—	106	96	no	—
173	#135	—	106	48	no	—
174	C₂×Q₁₆	—	53	4	no	—
175	C₂×SD₁₆	—	53	24	no	—
176	C₂×D₈	—	53	12	no	—
177	C₂×D₈	—	53	24	no	—
178	C₂×D₈	—	106	48	no	—
179	#131	D₂₁₂	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₅₃	1	6	yes	—
181	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
182	C₄⋊Q₈	—	53	2	no	—
183	#127	—	53	6	no	—
184	#127	—	53	12	no	—
185	#127	—	106	12	no	—
186	#124	—	53	12	no	—
187	#121	—	106	12	no	—
188	#121	—	106	24	no	—
189	#120	—	106	24	no	—
190	C₄×Q₈	D₂₁₂	1	2	yes	—
191	C₄×D₄	—	106	12	no	—
192	C₄×D₄	D₂₁₂	1	6	yes	—
193	#117	C₂²×D₅₃	1	3	yes	—
194	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
195	#97	C₂²×D₅₃	1	6	yes	—
196	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₅₃	1	3	yes	—
198	C₄×C₈	C₂²×D₅₃	1	6	yes	—
199	C₂⁴	—	106	12	no	—
200	#41	—	106	48	no	—
201	#41	—	212	16	no	—
202	#41	C₂×D₂₁₂	1	8	yes	—
203	C₂×Q₈	—	53	4	no	—
204	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
205	C₂×D₄	—	53	12	no	—
206	C₂×D₄	—	106	12	no	—
207	C₂×D₄	—	106	48	no	—
208	C₂×D₄	—	106	24	no	—
209	C₂×D₄	—	106	12	no	—
210	C₂×D₄	—	106	12	no	—
211	C₂×D₄	—	424	24	no	—
212	C₂×D₄	—	212	48	no	—
213	C₂×D₄	C₂×D₂₁₂	1	3	yes	—
214	C₂²×C₄	—	106	12	no	—
215	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
217	Q₁₆	—	106	8	no	—
218	SD₁₆	—	106	48	no	—
219	D₈	—	106	24	no	—
220	D₈	—	106	48	no	—
221	M₁₆	C₂×D₂₁₂	1	24	yes	complemented
222	C₂×C₈	C₂×D₂₁₂	1	24	yes	complemented
223	C₂×C₈	C₂³×D₅₃	1	2	yes	—
224	C₂×C₈	C₂³×D₅₃	1	6	yes	—
225	C₄⋊C₄	—	106	4	no	—
226	C₄⋊C₄	C₂×D₂₁₂	1	6	yes	—
227	C₂²⋊C₄	—	106	24	no	—
228	C₂²⋊C₄	C₂×D₂₁₂	1	6	yes	—
229	C₄²	C₂×D₂₁₂	1	6	yes	—
230	C₄²	C₂³×D₅₃	1	1	yes	characteristic
231	C₄²	C₂³×D₅₃	1	3	yes	—
232	C₂³	—	106	12	no	—
233	C₂³	—	424	24	no	—
234	C₂³	—	212	24	no	—
235	C₂³	C₂²×D₂₁₂	1	3	yes	—
236	Q₈	—	106	8	no	—
237	Q₈	#13133214	1	4	yes	—
238	D₄	—	106	24	no	—
239	D₄	—	424	24	no	—
240	D₄	—	212	24	no	—
241	D₄	—	212	24	no	—
242	D₄	—	212	24	no	—
243	D₄	#13133214	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	106	4	no	—
246	C₂×C₄	—	212	24	no	—
247	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
248	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
249	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
250	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
251	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
252	C₈	C₂²×D₂₁₂	1	4	yes	—
253	C₈	C₂²×D₂₁₂	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	848	24	no	—
256	C₂²	—	212	24	no	—
257	C₂²	—	212	12	no	—
258	C₂²	C₂×#13133214	1	6	yes	—
259	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	212	4	no	—
262	C₄	C₂×#13133214	1	6	yes	—
263	C₄	#16811428	1	4	yes	complemented
264	C₄	C₂³×D₂₁₂	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	424	12	no	—
267	C₂	C₂²×#13133214	1	1	yes	characteristic
268	C₂	C₂×#16811428	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×D₄
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#60647
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16811428	C₂	semidirect
2. #20288493	C₂	semidirect
3. #20285201	C₂	semidirect
4. #20285167	C₂	semidirect
5. #20285143	C₂	semidirect
6. C₂×#16811000	C₂	semidirect
7. C₄×#131×C₅₃	C₂	semidirect
10. C₂×#13133245	C₂²	semidirect
11. C₂×#13133239	C₂²	semidirect
12. C₂²×D₄₂₄	C₂²	semidirect
15. #16811192	C₂²	semidirect
16. #16811173	C₂²	semidirect
17. C₂×#13133214	C₂²	semidirect
18. #16811012	C₂²	semidirect
19. #16811007	C₂²	semidirect
20. #16811003	C₂²	semidirect
21. #16811000	C₂²	semidirect
22. #16811000	C₂²	semidirect
23. #16811000	C₂²	semidirect
24. #16810996	C₂²	semidirect
27. C₅₃×#405	C₂²	semidirect
28. C₄×M₁₆×C₅₃	C₂²	semidirect
29. C₂×C₄×C₈×C₅₃	C₂²	semidirect
53. #131×C₅₃	D₄	semidirect
82. M₁₆×C₅₃	C₂×D₄	semidirect
83. C₂×C₈×C₅₃	C₂×D₄	semidirect
128. C₄×C₅₃	#576	semidirect
137. C₄×#131	D₅₃	semidirect
161. #405	C₂×D₅₃	semidirect
162. C₄×M₁₆	C₂×D₅₃	semidirect
163. C₂×C₄×C₈	C₂×D₅₃	semidirect
164. C₅₃	#60647	semidirect
179. #131	D₂₁₂	semidirect
221. M₁₆	C₂×D₂₁₂	semidirect
222. C₂×C₈	C₂×D₂₁₂	semidirect
263. C₄	#16811428	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811428	2
#20288493	2
#20285201	2
#20285167	2
#20285143	2
C₂×#16811000	2
C₄×#131×C₅₃	2
#60647	53