Subgroups — #28327915

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	2	yes	maximal
2	#20290925	C₂	1	4	yes	maximal, complemented
3	#20290924	C₂	1	2	yes	maximal, complemented
4	#20290921	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#13133292	C₂	1	2	yes	maximal, complemented
6	C₂×#16811512	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290916	C₂	1	2	yes	maximal, complemented
8	#20283137	C₂	1	1	yes	maximal, characteristic
9	#16811682	C₄	1	16	yes	complemented
10	C₂×#13133292	C₂²	1	4	yes	—
11	C₂×#13133291	C₂²	1	2	yes	—
12	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	—
13	C₂²×#11719910	C₂²	1	2	yes	—
14	#16811512	C₂²	1	1	yes	characteristic
15	#16811512	C₂²	1	2	yes	—
16	#16811512	C₂²	1	1	yes	characteristic
17	#16811512	C₂²	1	2	yes	—
18	#16811511	C₂²	1	4	yes	—
19	#16811509	C₂²	1	2	yes	—
20	C₄×#11719910	C₂²	1	4	yes	—
21	C₂×#13133225	C₂²	1	1	yes	characteristic
22	C₂×#13133223	C₄	1	1	yes	characteristic
23	#16811091	C₄	1	1	yes	characteristic
24	#16810580	C₂²	1	2	yes	—
25	#16810579	C₂²	1	1	yes	characteristic
26	#16810577	C₄	1	2	yes	—
27	#16810575	C₄	1	2	yes	—
28	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅₃	C₂²	1	2	yes	—
31	C₅₃×#405	C₄	1	1	yes	characteristic
32	#13133292	C₂×C₄	1	16	yes	complemented
33	#13133291	C₂×C₄	1	8	yes	complemented
34	D₅₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
35	C₂×#11719910	C₂×C₄	1	8	yes	complemented
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	4	yes	—
39	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
40	#13133225	C₂×C₄	1	8	yes	—
41	#13133223	C₈	1	8	yes	—
42	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719912	C₂×C₄	1	2	yes	—
44	C₂×#11719912	C₂×C₄	1	2	yes	—
45	C₂²×#88322	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂³	1	2	yes	—
48	#13133182	C₈	1	2	yes	—
49	#13133179	C₈	1	2	yes	—
50	#13133177	C₈	1	4	yes	—
51	#13133174	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×D₅₃	C₂×C₄	1	2	yes	—
54	#13133128	C₂×C₄	1	2	yes	—
55	#13133127	C₂×C₄	1	2	yes	—
56	C₄×#88322	C₂³	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#97×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
69	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
70	#11719945	C₄²	1	4	yes	—
71	#11719945	C₂×C₈	1	4	yes	—
72	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
76	M₁₆×C₅₃	C₄²	1	4	yes	—
77	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
84	#11719921	C₂×C₈	1	4	yes	—
85	#11719919	C₂×C₈	1	4	yes	—
86	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
87	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	#11719917	C₂×C₈	1	4	yes	—
91	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
92	C₂×#88322	C₂²×C₄	1	4	yes	—
93	C₂×#88322	C₂²×C₄	1	2	yes	—
94	C₂×#88322	C₂⁴	1	1	yes	characteristic
95	#11719912	C₄²	1	8	yes	—
96	#11719912	C₂×C₈	1	8	yes	—
97	C₈×D₅₃	C₂×C₈	1	8	yes	—
98	C₈×D₅₃	C₂²×C₄	1	8	yes	—
99	#11719910	C₂²×C₄	1	4	yes	—
100	#11719910	C₂²×C₄	1	4	yes	—
101	#11719910	C₂²×C₄	1	8	yes	—
102	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂²⋊D₅₃	C₄×C₈	1	8	yes	—
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	D₂₁₂	C₄×C₈	1	4	yes	—
115	C₄×D₅₃	—	4	4	no	—
116	C₄×D₅₃	C₂×C₄²	1	4	yes	—
117	C₄×D₅₃	C₂²×C₈	1	4	yes	—
118	Dic₁₀₆	C₄×C₈	1	4	yes	—
119	C₈×C₅₃	C₂×C₄²	1	4	yes	—
120	C₈×C₅₃	C₂²×C₈	1	2	yes	—
121	C₈×C₅₃	C₂²×C₈	1	2	yes	—
122	#88322	C₂×C₄²	1	2	yes	—
123	#88322	C₂²×C₈	1	4	yes	—
124	#88322	C₂³×C₄	1	2	yes	—
125	#33326	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
128	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₅₃	—	2	4	no	—
130	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
131	C₄×C₅₃	—	2	4	no	—
132	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	—	2	2	no	—
137	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
138	C₂×#504	—	53	2	no	—
139	C₄×#131	—	53	1	no	—
140	#2108	—	53	2	no	—
141	#2108	—	53	4	no	—
142	C₄×M₃₂	—	53	2	no	—
143	C₄×M₃₂	—	53	1	no	—
144	C₂×#346	—	53	2	no	—
145	C₂×#346	—	53	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
148	C₂×C₅₃	#3407	1	2	yes	—
149	D₅₃	—	2	4	no	—
150	C₂×#131	—	53	1	no	—
151	C₂×#131	—	53	1	no	—
152	C₄×#41	—	53	1	no	—
153	#504	—	53	16	no	—
154	C₂×M₃₂	—	53	2	no	—
155	C₂×M₃₂	—	53	4	no	—
156	C₂²×C₁₆	—	53	2	no	—
157	C₂²×C₁₆	—	53	4	no	—
158	#405	—	53	1	no	—
159	#405	—	53	2	no	—
160	#405	—	53	2	no	—
161	#405	#6157	1	1	yes	characteristic
162	C₄×M₁₆	—	53	1	no	—
163	C₄×M₁₆	—	53	2	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	53	2	no	—
166	#346	—	53	1	no	—
167	#346	—	53	2	no	—
168	#346	—	53	1	no	—
169	#346	—	53	4	no	—
170	#346	—	53	2	no	—
171	C₄×C₁₆	—	53	4	no	—
172	C₄×C₁₆	—	53	2	no	—
173	C₅₃	#33326	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	53	1	no	—
175	#131	—	53	8	no	—
176	#131	—	53	8	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	53	2	no	—
179	C₂×M₁₆	—	53	2	no	—
180	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
185	C₄×Q₈	—	53	2	no	—
186	C₄×D₄	—	53	2	no	—
187	C₄×D₄	—	53	4	no	—
188	#117	—	53	2	no	—
189	#117	C₂×#6157	1	1	yes	characteristic
190	C₂×C₄²	—	53	1	no	—
191	C₂×C₄²	—	53	2	no	—
192	M₃₂	—	53	8	no	—
193	M₃₂	—	53	16	no	—
194	C₂×C₁₆	—	53	2	no	—
195	C₂×C₁₆	—	53	4	no	—
196	C₂×C₁₆	—	53	2	no	—
197	C₂×C₁₆	—	53	8	no	—
198	C₂×C₁₆	—	53	16	no	—
199	#97	—	53	2	no	—
200	#97	—	53	2	no	—
201	#97	C₂×#6157	1	2	yes	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	2	no	—
208	C₄×C₈	C₂×#6157	1	2	yes	—
209	#41	—	53	4	no	—
210	#41	—	53	4	no	—
211	C₂×Q₈	—	53	1	no	—
212	C₂×D₄	—	53	2	no	—
213	C₂×D₄	—	53	1	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	53	4	no	—
217	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
218	M₁₆	—	53	4	no	—
219	M₁₆	—	53	8	no	—
220	M₁₆	—	53	8	no	—
221	M₁₆	C₄×#6157	1	4	yes	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	8	no	—
227	C₂×C₈	—	53	8	no	—
228	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
229	C₂×C₈	C₂²×#6157	1	2	yes	—
230	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
231	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
232	C₄⋊C₄	—	53	4	no	—
233	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
234	C₂²⋊C₄	—	53	4	no	—
235	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
236	C₄²	—	53	4	no	—
237	C₄²	—	53	2	no	—
238	C₄²	—	53	1	no	—
239	C₄²	—	53	1	no	—
240	C₄²	C₂²×#6157	1	2	yes	—
241	C₁₆	—	53	4	no	—
242	C₁₆	—	53	4	no	—
243	C₁₆	—	53	8	no	—
244	C₂³	—	53	2	no	—
245	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
246	Q₈	—	53	4	no	—
247	D₄	—	53	8	no	—
248	D₄	—	53	4	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	4	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	—	212	2	no	—
256	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
257	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
258	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
260	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
261	C₈	—	53	2	no	—
262	C₈	—	53	2	no	—
263	C₈	—	53	4	no	—
264	C₈	C₂×C₄×#6157	1	4	yes	—
265	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
266	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	53	4	no	—
269	C₂²	—	106	4	no	—
270	C₂²	C₄×D₅₃⋊C₈	1	2	yes	—
271	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	53	4	no	—
274	C₄	—	106	2	no	—
275	C₄	—	106	2	no	—
276	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
277	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
278	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	106	4	no	—
281	C₂	C₂×C₄×D₅₃⋊C₈	1	1	yes	characteristic
282	C₂	#20291683	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#33326
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290925	C₂	semidirect
3. #20290924	C₂	semidirect
4. #20290921	C₂	semidirect
5. C₄×#13133292	C₂	semidirect
6. C₂×#16811512	C₂	semidirect
7. #20290916	C₂	semidirect
9. #16811682	C₄	semidirect
32. #13133292	C₂×C₄	semidirect
33. #13133291	C₂×C₄	semidirect
34. D₅₃⋊C₁₆	C₂×C₄	semidirect
35. C₂×#11719910	C₂×C₄	semidirect
173. C₅₃	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811682	2
#20290925	2
#20290924	2
#20290921	2
C₄×#13133292	2
C₂×#16811512	2
#20290916	2
#20283137	2
#33326	53