Subgroups — #28300045

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290430	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133272	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	2	yes	maximal
4	C₂×#16811464	C₂	1	2	yes	maximal
5	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133272	C₄	1	4	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#16811489	C₄	1	4	yes	—
10	#88324⋊C₈	C₄	1	8	yes	complemented
11	#16811464	C₄	1	8	yes	complemented
12	C₂×C₄×#88322	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	C₂×C₄	1	4	yes	—
20	C₂²×#88324	C₂×C₄	1	4	yes	—
21	#13133272	—	2	8	no	—
22	C₄×#88324	C₂×C₄	1	4	yes	—
23	C₄×#88324	C₂×C₄	1	4	yes	—
24	C₄×#88324	C₂×C₄	1	4	yes	—
25	C₄×#88324	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₄×#88322	—	2	2	no	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄²	1	2	yes	—
44	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×#88324	—	2	8	no	—
47	C₂×#88324	—	2	8	no	—
48	C₂×#88324	C₄²	1	8	yes	—
49	C₂×#88324	C₄²	1	8	yes	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	C₄²	1	2	yes	—
53	C₂×C₈×C₅₃	C₄²	1	2	yes	—
54	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	—	2	1	no	—
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	1	no	—
60	C₄×Dic₅₃	—	2	4	no	—
61	C₄×Dic₅₃	M₁₆	1	8	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
63	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	C₄²	1	2	yes	—
67	C₂×#88322	C₄²	1	2	yes	—
68	C₂×#88322	M₁₆	1	2	yes	—
69	C₂×#88322	M₁₆	1	2	yes	—
70	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	#97	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	#97	1	4	yes	—
83	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
84	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
85	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
86	C₂×Dic₅₃	#131	1	4	yes	—
87	#88324	—	2	8	no	—
88	#88324	—	2	8	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	#97	1	4	yes	complemented
91	#88322	—	2	2	no	—
92	#88322	#97	1	4	yes	complemented
93	#7842	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	C₄³	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×#97	1	2	yes	—
96	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
97	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
98	C₂²×C₅₃	#405	1	2	yes	—
99	C₄×C₅₃	—	2	4	no	—
100	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
101	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
102	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
103	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
104	Dic₅₃	—	2	4	no	—
105	Dic₅₃	C₄×M₁₆	1	4	yes	—
106	Dic₅₃	#405	1	4	yes	—
107	#1744	—	53	1	no	—
108	C₄×#97	—	53	1	no	—
109	C₄²×C₈	—	53	1	no	—
110	C₂×#322	—	53	2	no	—
111	C₂×#322	—	53	2	no	—
112	C₂×C₅₃	#1451	1	2	yes	—
113	C₂×C₅₃	#1451	1	2	yes	—
114	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
115	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
116	C₂×C₅₃	#1726	1	1	yes	characteristic
117	C₂×#97	—	53	4	no	—
118	C₂×C₄×C₈	—	53	2	no	—
119	C₂×C₄×C₈	—	53	2	no	—
120	C₂×C₄×C₈	—	53	1	no	—
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	#6157	1	1	yes	characteristic
126	C₄³	—	53	1	no	—
127	#336	—	53	4	no	—
128	#322	—	53	8	no	—
129	#322	—	53	8	no	—
130	C₅₃	#7842	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	4	no	—
133	C₂²×C₈	—	53	4	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
136	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
137	C₂×C₄²	—	53	2	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	2	no	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
142	#97	—	106	8	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	53	4	no	—
145	C₄×C₈	—	53	4	no	—
146	C₄×C₈	—	53	4	no	—
147	C₄×C₈	—	53	4	no	—
148	C₄×C₈	—	53	4	no	—
149	C₄×C₈	—	53	4	no	—
150	C₄×C₈	—	106	2	no	—
151	C₄×C₈	—	106	2	no	—
152	C₄×C₈	—	106	2	no	—
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	C₄×#6157	1	2	yes	—
157	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	53	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	8	no	—
164	C₂×C₈	—	53	8	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	106	8	no	—
167	C₂×C₈	—	106	8	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	#11719936	1	2	yes	—
171	C₂×C₈	C₄×#6157	1	2	yes	—
172	C₂×C₈	C₄×#6157	1	2	yes	—
173	C₂×C₈	#11719940	1	2	yes	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	53	8	no	—
177	C₄²	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	—	106	4	no	—
180	C₄²	—	106	1	no	—
181	C₄²	—	106	1	no	—
182	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	4	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	—	106	2	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	#13133271	1	4	yes	—
193	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
196	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
197	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	53	4	no	—
201	C₈	—	106	8	no	—
202	C₈	—	106	8	no	—
203	C₈	—	106	2	no	—
204	C₈	#13133272	1	4	yes	complemented
205	C₂²	C₄²×#6157	1	1	yes	characteristic
206	C₂²	C₂×#13133271	1	2	yes	—
207	C₂²	C₄×#11719940	1	1	yes	characteristic
208	C₂²	C₄×#11719936	1	1	yes	characteristic
209	C₂²	#16811586	1	2	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	53	4	no	—
212	C₄	—	53	4	no	—
213	C₄	—	106	4	no	—
214	C₄	C₂×#13133272	1	1	yes	characteristic
215	C₄	#16811582	1	1	yes	characteristic
216	C₄	C₄×#11719940	1	1	yes	characteristic
217	C₄	C₄×#11719936	1	1	yes	characteristic
218	C₂	#20289924	1	2	yes	—
219	C₂	#20289925	1	2	yes	—
220	C₂	C₄×#13133271	1	1	yes	characteristic
221	C₂	C₄×#13133272	1	1	yes	characteristic
222	C₂	#20290371	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7842
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. #88324⋊C₈	C₄	semidirect
11. #16811464	C₄	semidirect
90. C₈×C₅₃	#97	semidirect
92. #88322	#97	semidirect
130. C₅₃	#7842	semidirect
204. C₈	#13133272	semidirect

Maximal subgroups

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

subgroup	index
#20290430	2
C₄×#13133272	2
C₂×#88324⋊C₈	2
C₂×#16811464	2
C₄×C₈×Dic₅₃	2
#7842	53