Subgroups — #27298931

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	—	2	2	no	—
9	C₂×#13133157	C₄	1	4	yes	complemented
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅₃	—	2	2	no	—
12	C₂×C₈×Dic₅₃	—	2	2	no	—
13	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	—	2	2	no	—
17	C₂×M₃₂×C₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂³×Dic₅₃	C₈	1	2	yes	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	C₈	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	#13133157	—	2	8	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₅₃	—	2	4	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	8	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	4	no	—
59	C₂²×Dic₅₃	—	2	8	no	—
60	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₄²	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄×Dic₅₃	—	2	4	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₂×#88322	—	2	8	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	C₂×C₈	1	4	yes	—
81	C₁₆×C₅₃	—	2	4	no	—
82	#11719908	—	2	4	no	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₄×C₈	1	2	yes	—
85	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
87	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#95	1	4	yes	—
89	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	#97	1	4	yes	—
91	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	16	no	—
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	2	4	no	—
104	C₂×Dic₅₃	M₃₂	1	8	yes	complemented
105	C₈×C₅₃	#95	1	4	yes	—
106	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	#88322	—	2	4	no	—
111	#12544	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	#336	1	4	yes	—
116	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
120	C₄×C₅₃	#336	1	4	yes	—
121	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	Dic₅₃	—	2	8	no	—
126	C₂²×M₃₂	—	53	1	no	—
127	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
128	C₂²×C₄×C₈	—	53	1	no	—
129	#1374	—	53	4	no	—
130	C₂×C₅₃	—	2	4	no	—
131	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
132	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
133	C₂×C₅₃	#2112	1	2	yes	—
134	C₂×C₅₃	#2150	1	2	yes	—
135	C₂³×C₈	—	53	1	no	—
136	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂²×C₄²	—	53	1	no	—
138	C₂×M₃₂	—	2	2	no	—
139	C₂×M₃₂	—	2	2	no	—
140	C₂×M₃₂	—	53	4	no	—
141	C₂×M₃₂	—	106	2	no	—
142	C₂×M₃₂	—	106	2	no	—
143	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
144	C₂²×C₁₆	—	53	2	no	—
145	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
146	C₂×C₄×C₈	—	53	2	no	—
147	C₂×C₄×C₈	—	53	2	no	—
148	C₂×C₄×C₈	—	106	2	no	—
149	C₂×C₄×C₈	—	106	2	no	—
150	C₅₃	#12544	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	53	2	no	—
152	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	—	53	4	no	—
155	C₂²×C₈	—	106	2	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	Dic₁₀₆	1	2	yes	—
158	C₂²×C₈	C₄×D₅₃	1	2	yes	—
159	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₈	D₂₁₂	1	2	yes	—
161	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
162	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₂×C₄²	—	53	2	no	—
166	C₂×C₄²	—	53	2	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂×C₄²	—	106	2	no	—
169	M₃₂	—	2	8	no	—
170	M₃₂	—	106	8	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	53	4	no	—
174	C₂×C₁₆	—	106	2	no	—
175	C₂×C₁₆	—	106	2	no	—
176	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
177	C₄×C₈	—	106	2	no	—
178	C₄×C₈	—	106	2	no	—
179	C₄×C₈	—	106	4	no	—
180	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
181	C₂²×C₄	—	53	4	no	—
182	C₂²×C₄	—	106	8	no	—
183	C₂²×C₄	—	106	4	no	—
184	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₈×D₅₃	1	2	yes	—
186	C₂²×C₄	#11719912	1	2	yes	—
187	C₂²×C₄	#11719912	1	2	yes	—
188	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
189	C₂²×C₄	#11719919	1	2	yes	—
190	C₂²×C₄	#11719921	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	106	8	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
197	C₂×C₈	#11719919	1	2	yes	—
198	C₂×C₈	#11719919	1	4	yes	—
199	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
200	C₂×C₈	#11719921	1	2	yes	—
201	C₂×C₈	#11719921	1	4	yes	—
202	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
203	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
204	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
206	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄²	—	106	4	no	—
210	C₄²	—	106	2	no	—
211	C₄²	—	106	2	no	—
212	C₁₆	—	2	4	no	—
213	C₁₆	—	106	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₈×Dic₅₃	1	2	yes	—
216	C₂³	#13133129	1	2	yes	—
217	C₂³	#13133146	1	2	yes	—
218	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
219	C₂×C₄	—	53	8	no	—
220	C₂×C₄	—	106	16	no	—
221	C₂×C₄	—	106	4	no	—
222	C₂×C₄	—	106	4	no	—
223	C₂×C₄	#13133124	1	4	yes	—
224	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
225	C₂×C₄	#13133127	1	4	yes	—
226	C₂×C₄	#13133129	1	4	yes	—
227	C₂×C₄	#13133129	1	2	yes	—
228	C₂×C₄	#13133146	1	2	yes	—
229	C₂×C₄	#13133146	1	4	yes	—
230	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
233	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
236	C₈	—	106	4	no	—
237	C₈	#13133124	1	4	yes	—
238	C₈	C₂×#11719921	1	1	yes	characteristic
239	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
240	C₈	C₂×#11719919	1	1	yes	characteristic
241	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	#16810377	1	4	yes	—
246	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
247	C₂²	C₂×#13133129	1	1	yes	characteristic
248	C₂²	C₂×#13133146	1	1	yes	characteristic
249	C₂²	M₃₂×D₅₃	1	4	yes	—
250	C₄	—	106	8	no	—
251	C₄	#16810377	1	4	yes	—
252	C₄	C₂×#13133124	1	1	yes	characteristic
253	C₄	C₂×#13133127	1	1	yes	characteristic
254	C₄	C₂×#13133129	1	1	yes	characteristic
255	C₄	C₂×#13133146	1	1	yes	characteristic
256	C₂	—	2	4	no	—
257	C₂	C₂×#16810377	1	1	yes	characteristic
258	C₂	M₃₂×Dic₅₃	1	2	yes	—
259	C₂	#20276325	1	2	yes	—
260	C₂	#20276470	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272792	C₂	semidirect
9. C₂×#13133157	C₄	semidirect
18. C₂×M₃₂×C₅₃	C₄	semidirect
32. C₂×#11719908	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
104. C₂×Dic₅₃	M₃₂	semidirect
143. C₂×M₃₂	Dic₅₃	semidirect
150. C₅₃	#12544	semidirect
176. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂²×C₈×Dic₅₃	2
#20272792	2
C₂²×M₃₂×C₅₃	2
#12544	53