Subgroups — #27537708

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276541	C₂	1	4	yes	maximal, complemented
2	D₄×#11719908	C₂	1	4	yes	maximal, complemented
3	#20276522	C₂	1	2	yes	maximal, complemented
4	#20276329	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133157	C₂²	1	2	yes	complemented
8	C₂²×#11719908	C₂²	1	4	yes	—
9	C₂²×#11719908	C₂²	1	2	yes	complemented
10	C₄⋊#11719908	C₂²	1	4	yes	complemented
11	C₂²⋊#11719908	C₂²	1	4	yes	—
12	C₂²⋊#11719908	C₂²	1	2	yes	—
13	C₂²⋊#11719908	C₂²	1	2	yes	—
14	C₂²⋊#11719908	C₂²	1	4	yes	complemented
15	#16810399	C₂²	1	2	yes	complemented
16	C₄×#11719908	C₂²	1	2	yes	complemented
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#525	C₄	1	1	yes	characteristic
20	C₅₃×#435	C₄	1	4	yes	—
21	C₈×D₄×C₅₃	C₂²	1	4	yes	—
22	C₅₃×#424	C₄	1	1	yes	characteristic
23	C₅₃×#408	C₄	1	1	yes	characteristic
24	C₅₃×#407	C₄	1	1	yes	characteristic
25	C₅₃×#405	C₂²	1	1	yes	characteristic
26	#13133157	—	4	2	no	—
27	C₂×#11719908	—	4	4	no	—
28	C₂×#11719908	—	4	2	no	—
29	C₂×#11719908	C₂³	1	4	yes	—
30	C₂×#11719908	C₂³	1	4	yes	—
31	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	#131×C₅₃	—	4	2	no	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	4	2	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	#123×C₅₃	C₈	1	2	yes	—
44	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
45	#121×C₅₃	C₈	1	2	yes	—
46	#120×C₅₃	C₈	1	2	yes	—
47	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
48	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
52	#97×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
59	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₅₃	—	4	1	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
86	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	#11719908	—	2	4	no	—
89	#11719908	—	2	4	no	—
90	C₂³×C₅₃	—	2	4	no	—
91	C₂³×C₅₃	—	4	1	no	—
92	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
93	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
94	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
95	Q₈×C₅₃	—	4	1	no	—
96	D₄×C₅₃	—	2	8	no	—
97	D₄×C₅₃	—	4	1	no	—
98	D₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
117	#33491	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	4	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	4	4	no	—
125	C₂²×C₅₃	#504	1	4	yes	—
126	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	#518	1	1	yes	characteristic
132	C₄×C₅₃	#568	1	1	yes	characteristic
133	#2929	D₅₃	1	1	yes	characteristic
134	#2168	—	53	4	no	—
135	D₄×C₁₆	—	53	4	no	—
136	#2164	—	53	2	no	—
137	#2113	—	53	2	no	—
138	C₂×C₂²⋊C₁₆	—	53	2	no	—
139	C₂×C₅₃	—	2	4	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	—	4	1	no	—
142	C₂×C₅₃	#2972	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#504	1	2	yes	—
144	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
145	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#525	Dic₅₃	1	1	yes	characteristic
147	C₂×M₃₂	—	53	2	no	—
148	C₂²×C₁₆	—	53	4	no	—
149	C₂²×C₁₆	—	53	2	no	—
150	#435	Dic₅₃	1	4	yes	—
151	C₈×D₄	C₂×D₅₃	1	4	yes	—
152	#424	Dic₅₃	1	1	yes	characteristic
153	#408	Dic₅₃	1	1	yes	characteristic
154	#407	Dic₅₃	1	1	yes	characteristic
155	#405	C₂×D₅₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	53	4	no	—
157	C₂²⋊C₁₆	—	53	4	no	—
158	C₂²⋊C₁₆	—	53	2	no	—
159	C₂²⋊C₁₆	—	53	2	no	—
160	C₂²⋊C₁₆	—	53	4	no	—
161	#346	—	53	2	no	—
162	C₄×C₁₆	—	53	2	no	—
163	C₅₃	#33491	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
166	#131	—	4	2	no	—
167	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
169	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂²×C₈	—	4	2	no	—
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
175	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
176	#123	#88322	1	2	yes	—
177	C₂²⋊Q₈	#88322	1	2	yes	—
178	#121	#88322	1	2	yes	—
179	#120	#88322	1	2	yes	—
180	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
181	#117	C₂×Dic₅₃	1	1	yes	characteristic
182	M₃₂	—	212	2	no	—
183	C₂×C₁₆	—	53	4	no	—
184	C₂×C₁₆	—	53	4	no	—
185	C₂×C₁₆	—	212	2	no	—
186	C₂×C₁₆	—	212	4	no	—
187	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
188	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
189	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
190	#97	C₂×Dic₅₃	1	2	yes	—
191	C₄×C₈	C₂²×D₅₃	1	2	yes	—
192	C₂⁴	C₂×#88322	1	1	yes	characteristic
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
196	C₂×D₄	C₂×#88322	1	4	yes	—
197	C₂×D₄	C₂×#88322	1	2	yes	—
198	C₂×D₄	C₂×#88322	1	1	yes	characteristic
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×#88322	1	2	yes	—
202	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
216	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
218	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
219	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#88322	1	2	yes	—
221	C₄⋊C₄	C₂×#88322	1	4	yes	—
222	C₂²⋊C₄	C₂×#88322	1	2	yes	—
223	C₂²⋊C₄	C₂×#88322	1	4	yes	—
224	C₂²⋊C₄	C₂×#88322	1	4	yes	—
225	C₄²	C₂²×Dic₅₃	1	2	yes	—
226	C₁₆	—	106	4	no	—
227	C₁₆	—	106	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂²×#88322	1	1	yes	characteristic
231	C₂³	C₂²×#88322	1	2	yes	—
232	C₂³	C₂²×#88322	1	2	yes	—
233	Q₈	—	4	1	no	—
234	D₄	—	2	8	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	C₂²×#88322	1	4	yes	—
246	C₂×C₄	C₂²×#88322	1	2	yes	—
247	C₂×C₄	C₂²×#88322	1	2	yes	—
248	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
249	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	#13133262	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	#16811040	1	4	yes	—
263	C₂²	C₂³×#88322	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	#16811098	1	1	yes	characteristic
269	C₄	#16811390	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	#20283739	1	1	yes	characteristic
274	C₂	C₂×#16811040	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2929	C₂
prime radical	C₅₃×#525	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2929	D₅₃
hypocenter	C₅₃	#33491
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. #20276541	C₂	semidirect
2. D₄×#11719908	C₂	semidirect
3. #20276522	C₂	semidirect
4. #20276329	C₂	semidirect
5. C₂×C₂²⋊#11719908	C₂	semidirect
7. C₂×#13133157	C₂²	semidirect
9. C₂²×#11719908	C₂²	semidirect
10. C₄⋊#11719908	C₂²	semidirect
14. C₂²⋊#11719908	C₂²	semidirect
15. #16810399	C₂²	semidirect
16. C₄×#11719908	C₂²	semidirect
163. C₅₃	#33491	semidirect

Maximal subgroups

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

subgroup	index
#20276541	2
D₄×#11719908	2
#20276522	2
#20276329	2
C₂×C₂²⋊#11719908	2
C₅₃×#2929	2
#33491	53