Subgroups — #28328393

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291495	C₂	1	2	yes	maximal, complemented
2	#20291118	C₂	1	4	yes	maximal, complemented
3	#20291116	C₂	1	4	yes	maximal, complemented
4	D₅₃⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291031	C₂	1	2	yes	maximal, complemented
6	#20290923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16811056	C₂	1	1	yes	maximal, characteristic
8	#16811703	—	2	2	no	—
9	D₅₃⋊M₃₂	C₂²	1	1	yes	characteristic
10	#16811669	—	2	4	no	—
11	#16811667	C₂²	1	8	yes	complemented
12	D₅₃⋊#108	C₂²	1	2	yes	—
13	#16811622	—	2	2	no	—
14	#16811586	C₂²	1	1	yes	characteristic
15	#16811552	C₂²	1	2	yes	complemented
16	#16811551	C₂²	1	2	yes	complemented
17	#16811538	C₂²	1	8	yes	complemented
18	#16811510	C₂²	1	2	yes	complemented
19	C₁₆×#6157	C₂²	1	2	yes	complemented
20	C₂×#13133240	C₂²	1	2	yes	—
21	#16811056	C₄	1	2	yes	complemented
22	#16811056	C₄	1	2	yes	complemented
23	#16811056	C₂²	1	2	yes	—
24	#16811056	C₂²	1	2	yes	—
25	C₂×#13133160	C₄	1	2	yes	complemented
26	C₂×D₈₄₈	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
29	#13133305	—	2	8	no	—
30	#13133305	—	4	4	no	—
31	D₅₃⋊M₁₆	—	2	2	no	—
32	D₅₃⋊M₁₆	—	4	4	no	—
33	D₅₃⋊M₁₆	D₄	1	1	yes	characteristic
34	#13133296	D₄	1	1	yes	characteristic
35	#13133292	—	2	1	no	—
36	#13133291	—	2	1	no	—
37	D₅₃⋊C₁₆	C₂³	1	2	yes	—
38	#13133288	C₂³	1	4	yes	—
39	#13133287	—	2	2	no	—
40	#13133282	—	2	8	no	—
41	#13133271	D₄	1	2	yes	—
42	C₈×#6157	C₂³	1	2	yes	—
43	C₂×#11719949	—	2	2	no	—
44	#13133240	—	2	2	no	—
45	#13133240	—	2	2	no	—
46	#13133240	C₂×C₄	1	4	yes	—
47	#13133240	C₂³	1	4	yes	—
48	C₂×#11719924	—	2	2	no	—
49	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	#13133160	C₂×C₄	1	4	yes	—
52	#13133160	C₂×C₄	1	4	yes	—
53	D₈₄₈	C₂×C₄	1	2	yes	—
54	D₈₄₈	C₂×C₄	1	2	yes	—
55	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
61	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
62	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×Q₈×C₅₃	—	2	2	no	—
65	#11719949	—	2	4	no	—
66	#11719949	—	2	4	no	—
67	#11719949	—	4	2	no	—
68	#11719949	—	4	2	no	—
69	C₂×D₂₁₂	—	2	2	no	—
70	C₂×D₂₁₂	—	4	2	no	—
71	C₂×C₄×D₅₃	—	4	2	no	—
72	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²⋊#6157	—	2	1	no	—
74	#11719940	—	4	2	no	—
75	#11719940	D₈	1	2	yes	—
76	C₄⋊#6157	—	2	1	no	—
77	C₄×#6157	C₂×D₄	1	2	yes	—
78	#11719936	—	2	4	no	—
79	#11719936	—	4	8	no	—
80	#11719936	D₈	1	2	yes	—
81	D₅₃⋊C₈	—	4	2	no	—
82	D₅₃⋊C₈	—	4	8	no	—
83	D₅₃⋊C₈	C₂×D₄	1	2	yes	—
84	Q₁₆×C₅₃	—	2	2	no	—
85	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	#11719924	—	2	4	no	—
88	#11719924	—	2	4	no	—
89	C₂×#88322	#41	1	1	yes	characteristic
90	D₄₂₄	—	2	2	no	—
91	D₄₂₄	C₂²×C₄	1	4	yes	—
92	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
94	C₈×D₅₃	#41	1	1	yes	characteristic
95	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
96	#11719910	—	2	2	no	—
97	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	#11719908	C₂²×C₄	1	1	yes	characteristic
100	#11719908	C₂²×C₄	1	1	yes	characteristic
101	C₂²×D₅₃	—	4	2	no	—
102	C₂²×D₅₃	#135	1	1	yes	characteristic
103	C₂×#6157	—	2	2	no	—
104	Q₈×C₅₃	—	2	4	no	—
105	Q₈×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
109	D₂₁₂	—	2	4	no	—
110	D₂₁₂	—	4	4	no	—
111	D₂₁₂	—	4	4	no	—
112	D₂₁₂	—	4	2	no	—
113	C₄×D₅₃	—	4	4	no	—
114	C₄×D₅₃	—	4	4	no	—
115	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
116	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
117	C₄×D₅₃	#135	1	1	yes	characteristic
118	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
119	#88324	—	4	4	no	—
120	#88324	C₂×D₈	1	4	yes	—
121	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	#88322	C₄×D₄	1	1	yes	characteristic
124	#88322	C₂×#41	1	1	yes	characteristic
125	#33465	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
127	C₂×D₅₃	—	4	1	no	—
128	C₂×D₅₃	—	4	4	no	—
129	C₂×D₅₃	—	16	2	no	—
130	C₂×D₅₃	C₄×D₈	1	1	yes	characteristic
131	C₂×D₅₃	C₂×#135	1	1	yes	characteristic
132	#6157	—	4	2	no	—
133	C₄×C₅₃	—	4	4	no	—
134	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
135	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	Dic₅₃	C₄×D₈	1	1	yes	characteristic
137	Dic₅₃	C₂²×D₈	1	1	yes	characteristic
138	C₂×#508	—	53	1	no	—
139	#2957	—	53	2	no	—
140	#2179	—	53	4	no	—
141	#2178	—	53	4	no	—
142	C₂×#368	—	53	1	no	—
143	#2143	—	53	2	no	—
144	#2108	—	53	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
147	D₅₃	—	2	1	no	—
148	D₅₃	—	2	1	no	—
149	D₅₃	—	8	4	no	—
150	C₂×#135	—	53	2	no	—
151	#568	—	106	2	no	—
152	#508	—	53	2	no	—
153	#508	—	53	2	no	—
154	#508	—	53	2	no	—
155	#508	—	53	2	no	—
156	C₂×Q₃₂	#6157	1	1	yes	characteristic, complemented
157	C₂×SD₃₂	—	53	2	no	—
158	C₂×D₁₆	—	53	1	no	—
159	C₂×M₃₂	—	53	1	no	—
160	C₂²×C₁₆	—	53	1	no	—
161	#444	—	106	4	no	—
162	#443	—	53	8	no	—
163	C₂×#108	—	53	2	no	—
164	#421	—	106	2	no	—
165	#405	—	53	1	no	—
166	#368	—	53	2	no	—
167	#368	—	53	2	no	—
168	#359	—	53	8	no	—
169	#346	—	53	2	no	—
170	C₄×C₁₆	—	53	2	no	—
171	C₅₃	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	106	2	no	—
173	#135	—	53	4	no	—
174	#135	—	53	4	no	—
175	#135	—	106	2	no	—
176	#135	—	106	2	no	—
177	C₂×Q₁₆	C₂×#6157	1	2	yes	—
178	C₂×SD₁₆	—	106	2	no	—
179	C₂×D₈	—	53	2	no	—
180	#131	—	106	8	no	—
181	#131	—	212	4	no	—
182	C₂×M₁₆	—	53	1	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	212	4	no	—
185	C₂²×C₈	—	53	1	no	—
186	#117	—	53	1	no	—
187	Q₃₂	C₂×#6157	1	2	yes	—
188	Q₃₂	C₂×#6157	1	2	yes	—
189	SD₃₂	—	53	4	no	—
190	SD₃₂	—	53	4	no	—
191	D₁₆	—	53	2	no	—
192	D₁₆	—	53	2	no	—
193	M₃₂	—	106	1	no	—
194	M₃₂	—	106	1	no	—
195	C₂×C₁₆	—	53	1	no	—
196	C₂×C₁₆	—	53	1	no	—
197	C₂×C₁₆	—	53	1	no	—
198	C₂×C₁₆	—	53	1	no	—
199	C₂×C₁₆	—	53	1	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
202	#108	—	53	4	no	—
203	#108	—	106	2	no	—
204	#104	—	106	8	no	—
205	#97	—	53	2	no	—
206	C₄×C₈	—	53	2	no	—
207	#41	—	106	4	no	—
208	#41	—	106	4	no	—
209	#41	—	212	2	no	—
210	#41	—	212	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	106	2	no	—
213	C₂×D₄	—	212	2	no	—
214	C₂²×C₄	—	53	1	no	—
215	C₂²×C₄	—	212	2	no	—
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×#6157	1	4	yes	—
218	SD₁₆	—	106	4	no	—
219	SD₁₆	—	106	4	no	—
220	D₈	—	53	4	no	—
221	D₈	—	106	2	no	—
222	M₁₆	—	53	2	no	—
223	M₁₆	—	53	2	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	212	8	no	—
226	M₁₆	—	212	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	1	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	212	8	no	—
234	C₂×C₈	—	212	2	no	—
235	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
236	C₄⋊C₄	—	106	1	no	—
237	C₂²⋊C₄	—	106	1	no	—
238	C₄²	—	53	2	no	—
239	C₁₆	—	53	1	no	—
240	C₁₆	—	53	1	no	—
241	C₁₆	—	106	2	no	—
242	C₁₆	C₂²×#6157	1	1	yes	characteristic
243	C₁₆	C₂²×#6157	1	1	yes	characteristic
244	C₂³	—	53	1	no	—
245	C₂³	—	212	2	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	106	4	no	—
249	D₄	—	212	4	no	—
250	D₄	—	212	4	no	—
251	D₄	—	212	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	53	1	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	1	no	—
256	C₂×C₄	—	53	1	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	212	4	no	—
260	C₂×C₄	—	212	4	no	—
261	C₂×C₄	D₄×#6157	1	1	yes	characteristic
262	C₈	—	53	4	no	—
263	C₈	—	53	1	no	—
264	C₈	—	53	1	no	—
265	C₈	—	212	4	no	—
266	C₈	D₄×#6157	1	1	yes	characteristic
267	C₈	C₂³×#6157	1	1	yes	characteristic
268	C₂²	—	53	1	no	—
269	C₂²	—	53	1	no	—
270	C₂²	—	848	2	no	—
271	C₂²	—	212	1	no	—
272	C₂²	—	212	4	no	—
273	C₂²	D₈×#6157	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	53	1	no	—
276	C₄	—	53	1	no	—
277	C₄	—	212	2	no	—
278	C₄	D₈×#6157	1	1	yes	characteristic
279	C₄	C₂×D₄×#6157	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	106	1	no	—
282	C₂	—	106	1	no	—
283	C₂	—	424	4	no	—
284	C₂	C₂×D₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6157
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₈	C₂³×#6157
Fitting	C₂×Q₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	C₂×#16811056	C₂
cosocle	C₈×D₅₃	C₂⁴
hypercenter	C₂×Q₃₂	#6157
hypocenter	C₅₃	#33465
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20291495	C₂	semidirect
2. #20291118	C₂	semidirect
3. #20291116	C₂	semidirect
4. D₅₃⋊#368	C₂	semidirect
5. #20291031	C₂	semidirect
6. #20290923	C₂	semidirect
11. #16811667	C₂²	semidirect
15. #16811552	C₂²	semidirect
16. #16811551	C₂²	semidirect
17. #16811538	C₂²	semidirect
18. #16811510	C₂²	semidirect
19. C₁₆×#6157	C₂²	semidirect
21. #16811056	C₄	semidirect
22. #16811056	C₄	semidirect
25. C₂×#13133160	C₄	semidirect
26. C₂×D₈₄₈	C₄	semidirect
28. C₂×Q₃₂×C₅₃	C₄	semidirect
156. C₂×Q₃₂	#6157	semidirect
171. C₅₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#20291495	2
#20291118	2
#20291116	2
D₅₃⋊#368	2
#20291031	2
#20290923	2
C₂×#16811056	2
#33465	53