Subgroups — #27295346

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272761	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	complemented
9	#16810376	C₂²	1	2	yes	complemented
10	#16810374	—	2	1	no	—
11	C₂²×D₈×C₅₃	—	2	1	no	—
12	C₂²×D₈×C₅₃	C₄	1	2	yes	—
13	C₂×#127×C₅₃	—	2	1	no	—
14	C₅₃×#493	—	2	1	no	—
15	C₅₃×#493	—	2	1	no	—
16	C₅₃×#493	C₄	1	2	yes	—
17	C₅₃×#493	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	#13133157	—	4	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₂²⋊#88322	—	4	1	no	—
23	C₄×#88322	—	4	1	no	—
24	C₂²×D₄×C₅₃	—	2	2	no	—
25	C₂²×D₄×C₅₃	—	4	1	no	—
26	C₂²×D₄×C₅₃	—	4	1	no	—
27	C₂×D₈×C₅₃	—	2	2	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	—	4	2	no	—
31	C₂×D₈×C₅₃	—	4	2	no	—
32	C₂×D₈×C₅₃	—	4	4	no	—
33	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	#127×C₅₃	—	2	2	no	—
37	#127×C₅₃	—	2	2	no	—
38	#127×C₅₃	—	4	1	no	—
39	#127×C₅₃	—	4	1	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂⁴×C₅₃	—	4	2	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	4	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	8	4	no	—
55	C₂×D₄×C₅₃	—	8	4	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
59	D₈×C₅₃	—	4	4	no	—
60	D₈×C₅₃	—	4	4	no	—
61	D₈×C₅₃	—	4	4	no	—
62	D₈×C₅₃	—	4	2	no	—
63	D₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	D₈	1	2	yes	—
68	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	D₈	1	1	yes	characteristic
73	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×#88322	—	4	2	no	—
76	#11719908	—	4	2	no	—
77	C₂³×C₅₃	—	4	4	no	—
78	C₂³×C₅₃	—	8	4	no	—
79	C₂³×C₅₃	—	8	4	no	—
80	C₂³×C₅₃	—	8	4	no	—
81	C₂³×C₅₃	#102	1	1	yes	characteristic
82	D₄×C₅₃	—	4	4	no	—
83	D₄×C₅₃	—	4	4	no	—
84	D₄×C₅₃	—	8	4	no	—
85	D₄×C₅₃	—	8	4	no	—
86	D₄×C₅₃	—	8	4	no	—
87	D₄×C₅₃	—	8	4	no	—
88	D₄×C₅₃	—	8	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	#100	1	2	yes	—
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#102	1	2	yes	—
95	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	#102	1	2	yes	—
102	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	#88322	—	8	2	no	—
105	#11931	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	8	4	no	—
109	C₂²×C₅₃	—	8	4	no	—
110	C₂²×C₅₃	—	8	4	no	—
111	C₂²×C₅₃	—	16	8	no	—
112	C₂²×C₅₃	#331	1	2	yes	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	#331	1	2	yes	—
117	C₄×C₅₃	#361	1	2	yes	—
118	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
120	C₄×C₅₃	#509	1	2	yes	—
121	C₂×#493	D₅₃	1	1	yes	characteristic
122	#2150	—	53	1	no	—
123	#1564	—	53	1	no	—
124	#1361	—	53	4	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	16	4	no	—
127	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
128	C₂×C₅₃	#2140	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
130	C₂²×D₈	—	2	1	no	—
131	C₂²×D₈	Dic₅₃	1	2	yes	—
132	C₂×#127	—	2	1	no	—
133	C₂×M₃₂	—	106	1	no	—
134	#493	—	2	1	no	—
135	#493	—	2	1	no	—
136	#493	Dic₅₃	1	2	yes	—
137	#493	C₂×D₅₃	1	2	yes	—
138	#431	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₄⋊C₁₆	—	53	2	no	—
141	C₄⋊C₁₆	—	106	1	no	—
142	#335	—	53	2	no	—
143	#334	—	106	1	no	—
144	C₅₃	#11931	1	1	yes	characteristic, complemented, Sylow
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	—	4	1	no	—
147	C₂²×D₄	—	4	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	4	4	no	—
152	C₂×D₈	—	4	2	no	—
153	C₂×D₈	—	4	2	no	—
154	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
155	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	—	4	1	no	—
160	#127	—	4	1	no	—
161	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
162	M₃₂	—	212	2	no	—
163	C₂×C₁₆	—	106	2	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₂²⋊C₈	—	212	1	no	—
166	C₄×C₈	—	212	1	no	—
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂⁴	—	4	2	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	8	4	no	—
181	C₂×D₄	—	8	4	no	—
182	C₂²×C₄	#11719922	1	1	yes	characteristic
183	C₂²×C₄	#11719923	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	D₈	—	4	4	no	—
186	D₈	—	4	4	no	—
187	D₈	—	4	4	no	—
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	212	2	no	—
192	C₂×C₈	#11719922	1	2	yes	—
193	C₂×C₈	#11719923	1	2	yes	—
194	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₄²	#11719922	1	1	yes	characteristic
199	C₄²	#11719923	1	1	yes	characteristic
200	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₁₆	—	212	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	4	no	—
205	C₂³	—	8	4	no	—
206	C₂³	—	8	4	no	—
207	C₂³	#13133138	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	4	no	—
213	D₄	—	8	4	no	—
214	D₄	—	8	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	#13133134	1	2	yes	—
218	C₂×C₄	#13133138	1	1	yes	characteristic
219	C₂×C₄	#13133138	1	2	yes	—
220	C₂×C₄	#13133138	1	1	yes	characteristic
221	C₂×C₄	#13133138	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
224	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	424	2	no	—
228	C₈	#13133138	1	2	yes	—
229	C₈	C₂×#11719922	1	1	yes	characteristic
230	C₈	C₂×#11719923	1	1	yes	characteristic
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	16	8	no	—
237	C₂²	#16810367	1	2	yes	—
238	C₂²	C₂×#13133138	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	#16810367	1	2	yes	—
242	C₄	#16810436	1	2	yes	—
243	C₄	C₂×#13133134	1	1	yes	characteristic
244	C₄	C₂×#13133138	1	1	yes	characteristic
245	C₄	#16811059	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	16	4	no	—
248	C₂	C₂×#16810367	1	1	yes	characteristic
249	C₂	#20276431	1	1	yes	characteristic
250	C₂	C₂×#16810436	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	#11931
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. #20273463	C₂	semidirect
3. #20272761	C₂	semidirect
8. C₄⋊#11719908	C₂²	semidirect
9. #16810376	C₂²	semidirect
144. C₅₃	#11931	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
#20273463	2
#20272761	2
C₂×C₅₃×#493	2
#11931	53