Subgroups — #27299306

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810424	C₂	1	2	yes	maximal
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₅₃	—	2	1	no	—
9	#16810424	C₄	1	8	yes	complemented
10	#16810399	—	2	2	no	—
11	#16810386	—	2	1	no	—
12	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	—	2	1	no	—
14	C₄×D₈×C₅₃	—	2	2	no	—
15	C₄×D₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#102×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	—	4	1	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	4	yes	complemented
24	#13133151	—	2	2	no	—
25	#13133151	C₂×C₄	1	4	yes	complemented
26	#13133124	—	4	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂³×C₄×C₅₃	—	4	1	no	—
29	C₂×D₈×C₅₃	—	2	1	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₅₃	—	2	4	no	—
36	C₄×D₄×C₅₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	1	no	—
41	#107×C₅₃	C₂×C₄	1	2	yes	—
42	#102×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂⁴×C₅₃	—	4	1	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	4	no	—
48	C₂×D₄×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	—	8	4	no	—
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	2	no	—
57	D₈×C₅₃	—	2	2	no	—
58	D₈×C₅₃	—	2	4	no	—
59	D₈×C₅₃	C₄²	1	4	yes	complemented
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₂²⋊C₄×C₅₃	—	4	4	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	SD₁₆	1	2	yes	—
71	C₄⋊Dic₅₃	—	2	4	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄×Dic₅₃	—	8	2	no	—
74	#11719908	—	2	4	no	—
75	C₂³×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	—	4	4	no	—
77	C₂³×C₅₃	—	8	2	no	—
78	C₂³×C₅₃	—	8	1	no	—
79	C₂³×C₅₃	—	8	1	no	—
80	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
81	D₄×C₅₃	—	2	4	no	—
82	D₄×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	8	8	no	—
89	C₂×C₄×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×Dic₅₃	—	4	4	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₈×C₅₃	#117	1	1	yes	characteristic
102	C₈×C₅₃	C₄×D₄	1	2	yes	—
103	#12637	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	4	4	no	—
105	C₂²×C₅₃	—	8	2	no	—
106	C₂²×C₅₃	—	8	2	no	—
107	C₂²×C₅₃	—	8	4	no	—
108	C₂²×C₅₃	—	8	4	no	—
109	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₂²×C₅₃	C₄×D₈	1	2	yes	—
111	C₂²×C₅₃	#509	1	1	yes	characteristic
112	C₂²×C₅₃	#509	1	2	yes	—
113	C₂²×C₅₃	#510	1	1	yes	characteristic
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	8	2	no	—
117	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	#416	1	1	yes	characteristic
119	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
120	Dic₅₃	—	8	4	no	—
121	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
122	C₂×#357	—	53	2	no	—
123	C₂×#346	—	53	1	no	—
124	C₄×#107	—	53	1	no	—
125	#1380	—	53	2	no	—
126	C₂×C₅₃	—	8	4	no	—
127	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
128	C₂×C₅₃	#2139	1	1	yes	characteristic
129	C₂×C₅₃	#2140	1	1	yes	characteristic
130	C₂×C₅₃	#2175	1	2	yes	—
131	C₂×C₅₃	#2177	1	2	yes	—
132	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂×C₄×D₄	—	2	1	no	—
134	C₂²×C₁₆	—	53	2	no	—
135	C₄×D₈	—	2	2	no	—
136	C₄×D₈	Dic₅₃	1	4	yes	complemented
137	C₂×#107	—	53	2	no	—
138	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
139	C₂×#102	—	2	1	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	106	1	no	—
142	#357	—	53	8	no	—
143	#346	—	106	2	no	—
144	#340	—	106	1	no	—
145	C₅₃	#12637	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	—	2	1	no	—
147	C₂³×C₄	—	4	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	1	no	—
150	C₂×D₈	C₄×D₅₃	1	4	yes	—
151	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
152	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
154	C₄×D₄	—	2	4	no	—
155	C₄×D₄	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	106	2	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₄²	—	212	1	no	—
160	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂×C₁₆	—	53	4	no	—
162	C₂×C₁₆	—	106	2	no	—
163	C₂×C₁₆	—	106	2	no	—
164	#107	—	2	1	no	—
165	#107	—	53	4	no	—
166	#107	—	106	2	no	—
167	#107	C₂×Dic₅₃	1	2	yes	—
168	#102	—	2	4	no	—
169	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	#95	—	212	1	no	—
172	C₂⁴	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	8	4	no	—
180	C₂²×C₄	—	212	2	no	—
181	C₂²×C₄	#11719922	1	1	yes	characteristic
182	C₂²×C₄	#11719922	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	D₈	C₄×Dic₅₃	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
190	C₂×C₈	#11719945	1	2	yes	—
191	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	106	4	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	424	2	no	—
199	C₄²	#11719923	1	2	yes	—
200	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
201	C₁₆	—	106	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	1	no	—
207	C₂³	C₂×#11719922	1	1	yes	characteristic
208	D₄	—	2	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	8	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	—	424	2	no	—
219	C₂×C₄	—	212	4	no	—
220	C₂×C₄	#13133138	1	2	yes	—
221	C₂×C₄	#13133138	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
224	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
225	C₂×C₄	#13133237	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
228	C₈	#13133176	1	1	yes	characteristic
229	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
230	C₂²	—	4	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₂×#13133138	1	1	yes	characteristic
236	C₂²	C₄×#11719922	1	2	yes	—
237	C₂²	#16811058	1	1	yes	characteristic
238	C₂²	#16811059	1	2	yes	—
239	C₂²	#16811063	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	424	4	no	—
244	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₄	#16810619	1	1	yes	characteristic
246	C₄	C₄×#11719923	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×#13133138	1	1	yes	characteristic
249	C₂	#20276427	1	1	yes	characteristic
250	C₂	#20276431	1	1	yes	characteristic
251	C₂	#20276579	1	2	yes	—
252	C₂	#20276589	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	#12637
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810399	C₂	semidirect
3. C₄×#13133151	C₂	semidirect
4. #20272806	C₂	semidirect
9. #16810424	C₄	semidirect
15. C₄×D₈×C₅₃	C₄	semidirect
23. C₂×#11719908	C₂×C₄	semidirect
25. #13133151	C₂×C₄	semidirect
59. D₈×C₅₃	C₄²	semidirect
136. C₄×D₈	Dic₅₃	semidirect
145. C₅₃	#12637	semidirect
186. D₈	C₄×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810424	2
C₂×#16810399	2
C₄×#13133151	2
#20272806	2
C₂×C₄×D₈×C₅₃	2
#12637	53