Subgroups — #28300057

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290428	C₂	1	1	yes	maximal, characteristic
2	#20290428	C₂	1	1	yes	maximal, characteristic
3	#20290428	C₂	1	1	yes	maximal, characteristic
4	#20289911	C₂	1	1	yes	maximal, characteristic
5	#20289909	C₂	1	1	yes	maximal, characteristic
6	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#322×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄²×#6157	C₂²	1	1	yes	characteristic
12	(C₂×#6157)⋊C₈	C₄	1	4	yes	—
13	(C₂×#6157)⋊C₈	C₄	1	4	yes	—
14	(C₂×#6157)⋊C₈	C₄	1	4	yes	—
15	#88324⋊C₈	C₄	1	4	yes	complemented
16	#16811464	C₄	1	4	yes	complemented
17	C₈×#88324	C₄	1	4	yes	complemented
18	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	#16810345	C₄	1	1	yes	characteristic, complemented
22	#16810345	C₄	1	1	yes	characteristic, complemented
23	#16810344	C₄	1	1	yes	characteristic, complemented
24	C₅₃×#322	C₄	1	1	yes	characteristic, complemented
25	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
29	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
30	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
31	C₄×#88324	C₂×C₄	1	2	yes	—
32	C₄×#88324	C₂×C₄	1	2	yes	—
33	C₄×#88324	C₂×C₄	1	2	yes	—
34	C₄×#88324	C₂×C₄	1	2	yes	—
35	C₄×#88324	C₂×C₄	1	2	yes	—
36	C₄×#88324	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₄²×D₅₃	C₂³	1	1	yes	characteristic
41	C₄×#88322	C₂×C₄	1	1	yes	characteristic
42	C₄×#88322	C₂×C₄	1	1	yes	characteristic
43	C₄×#88322	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#6157	C₄²	1	4	yes	—
51	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×#88324	C₄²	1	4	yes	—
55	C₂×#88324	C₄²	1	4	yes	—
56	C₂×#88324	C₄²	1	4	yes	—
57	C₄×#6157	—	2	4	no	—
58	C₄×#6157	—	2	4	no	—
59	C₄×#6157	—	2	4	no	—
60	D₅₃⋊C₈	—	2	4	no	—
61	D₅₃⋊C₈	—	2	4	no	—
62	D₅₃⋊C₈	—	2	4	no	—
63	C₂×C₈×C₅₃	C₄²	1	2	yes	—
64	C₂×C₈×C₅₃	C₄²	1	2	yes	—
65	C₂×C₈×C₅₃	C₄²	1	2	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×#88322	C₄²	1	2	yes	—
71	C₂×#88322	C₄²	1	2	yes	—
72	C₂×#88322	C₄²	1	2	yes	—
73	C₈×D₅₃	—	2	4	no	—
74	C₈×D₅₃	—	2	4	no	—
75	C₈×D₅₃	—	2	4	no	—
76	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×#6157	—	2	4	no	—
78	C₂×#6157	—	2	4	no	—
79	C₂×#6157	—	2	4	no	—
80	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₄×D₅₃	#131	1	4	yes	—
87	C₄×D₅₃	#131	1	4	yes	—
88	C₄×D₅₃	#131	1	4	yes	—
89	#88324	—	2	4	no	—
90	#88324	—	2	4	no	—
91	#88324	—	2	4	no	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	#88322	—	2	2	no	—
96	#88322	—	2	2	no	—
97	#88322	—	2	2	no	—
98	#7845	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	C₄³	1	1	yes	characteristic
100	C₂×D₅₃	#405	1	2	yes	—
101	C₂×D₅₃	#405	1	2	yes	—
102	C₂×D₅₃	#405	1	2	yes	—
103	#6157	—	4	4	no	—
104	C₄×C₅₃	#405	1	2	yes	—
105	C₄×C₅₃	#405	1	2	yes	—
106	C₄×C₅₃	#405	1	2	yes	—
107	Dic₅₃	#405	1	2	yes	—
108	Dic₅₃	#405	1	2	yes	—
109	Dic₅₃	#405	1	2	yes	—
110	#1744	—	53	1	no	—
111	#1744	—	53	1	no	—
112	#1744	—	53	1	no	—
113	C₂×#322	—	53	1	no	—
114	C₂×#322	—	53	1	no	—
115	C₂×#322	—	53	1	no	—
116	C₂×C₈²	—	53	1	no	—
117	C₂×C₅₃	#1726	1	1	yes	characteristic
118	C₂×C₅₃	#1726	1	1	yes	characteristic
119	C₂×C₅₃	#1726	1	1	yes	characteristic
120	D₅₃	#1454	1	4	yes	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₄³	—	53	1	no	—
128	#336	—	53	4	no	—
129	#336	—	53	4	no	—
130	#336	—	53	4	no	—
131	#322	—	53	4	no	—
132	#322	—	53	4	no	—
133	#322	—	53	1	no	—
134	#322	—	53	1	no	—
135	#322	—	53	1	no	—
136	#322	#6157	1	1	yes	characteristic, complemented
137	C₈²	—	53	4	no	—
138	C₅₃	#7845	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₈	—	53	2	no	—
140	C₂²×C₈	—	53	2	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	—	53	2	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	53	2	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	2	no	—
147	C₂×C₄²	—	53	2	no	—
148	C₂×C₄²	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	2	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₄×C₈	—	53	2	no	—
157	C₄×C₈	—	53	2	no	—
158	C₄×C₈	—	53	1	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	C₂×#6157	1	1	yes	characteristic
162	C₄×C₈	C₂×#6157	1	1	yes	characteristic
163	C₄×C₈	C₂×#6157	1	1	yes	characteristic
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	4	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	106	4	no	—
175	C₂×C₈	—	106	4	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	106	4	no	—
178	C₂×C₈	—	106	4	no	—
179	C₂×C₈	—	106	4	no	—
180	C₂×C₈	C₄×#6157	1	2	yes	—
181	C₂×C₈	C₄×#6157	1	2	yes	—
182	C₂×C₈	C₄×#6157	1	2	yes	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	106	4	no	—
187	C₄²	—	106	4	no	—
188	C₄²	—	106	4	no	—
189	C₄²	C₂²×#6157	1	1	yes	characteristic
190	C₂³	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	4	no	—
196	C₂×C₄	—	53	4	no	—
197	C₂×C₄	—	106	4	no	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	106	4	no	—
207	C₈	—	106	4	no	—
208	C₈	—	106	4	no	—
209	C₈	—	106	2	no	—
210	C₈	—	106	2	no	—
211	C₈	—	106	2	no	—
212	C₂²	—	53	2	no	—
213	C₂²	—	53	2	no	—
214	C₂²	—	53	2	no	—
215	C₂²	C₄²×#6157	1	1	yes	characteristic
216	C₄	—	53	2	no	—
217	C₄	—	53	2	no	—
218	C₄	—	53	2	no	—
219	C₄	—	212	4	no	—
220	C₄	#16811586	1	2	yes	—
221	C₄	#16811589	1	2	yes	—
222	C₄	#16811589	1	2	yes	—
223	C₂	—	53	4	no	—
224	C₂	#20290371	1	1	yes	characteristic
225	C₂	#20290373	1	1	yes	characteristic
226	C₂	#20290373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₄³
Fitting	C₅₃×#322	C₄
prime radical	C₂²×D₅₃	C₂×C₄²
cosocle	C₄²×D₅₃	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#7845
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
15. #88324⋊C₈	C₄	semidirect
16. #16811464	C₄	semidirect
17. C₈×#88324	C₄	semidirect
21. #16810345	C₄	semidirect
22. #16810345	C₄	semidirect
23. #16810344	C₄	semidirect
24. C₅₃×#322	C₄	semidirect
136. #322	#6157	semidirect
138. C₅₃	#7845	semidirect

Maximal subgroups

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

subgroup	index
#20290428	2
#20289911	2
#20289909	2
C₈×D₅₃⋊C₈	2
#322×D₅₃	2
#7845	53