Subgroups — C₅₃×#21736 #27229248

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2961	C₂	1	2	yes	maximal
3	C₅₃×#1994	C₂	1	2	yes	maximal
4	C₅₃×#1946	C₂	1	2	yes	maximal
5	C₅₃×#1945	C₂	1	2	yes	maximal
6	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1773	C₂	1	2	yes	maximal
8	C₅₃×#1772	C₂	1	2	yes	maximal
9	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#533	C₂²	1	2	yes	—
11	C₅₃×#501	C₄	1	8	yes	complemented
12	C₅₃×#446	—	2	8	no	—
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#427	C₂²	1	2	yes	—
16	C₅₃×#427	C₂²	1	2	yes	—
17	C₅₃×#424	C₂²	1	4	yes	—
18	C₅₃×#423	C₂²	1	2	yes	—
19	C₅₃×#405	C₂²	1	4	yes	—
20	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₄³×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#339	C₂²	1	4	yes	—
28	C₅₃×#339	C₂²	1	4	yes	—
29	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
35	#125×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²⋊Q₈×C₅₃	—	2	4	no	—
37	C₄×Q₈×C₅₃	—	2	4	no	—
38	#117×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#108×C₅₃	—	2	4	no	—
44	#108×C₅₃	—	2	4	no	—
45	#107×C₅₃	C₂×C₄	1	4	yes	—
46	#107×C₅₃	C₂×C₄	1	4	yes	—
47	#106×C₅₃	C₂×C₄	1	4	yes	—
48	#106×C₅₃	C₂×C₄	1	4	yes	—
49	C₄⋊C₈×C₅₃	—	2	8	no	—
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	#97×C₅₃	—	2	4	no	—
52	#97×C₅₃	—	2	2	no	—
53	#97×C₅₃	C₂×C₄	1	4	yes	—
54	#97×C₅₃	D₄	1	4	yes	—
55	#97×C₅₃	Q₈	1	4	yes	complemented
56	C₄×C₈×C₅₃	—	2	4	no	—
57	C₄×C₈×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
59	C₂×Q₈×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₄⋊C₄×C₅₃	—	2	8	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
82	C₂²⋊C₄×C₅₃	—	2	4	no	—
83	C₄²×C₅₃	—	2	4	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂×D₄	1	2	yes	—
89	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
92	Q₈×C₅₃	—	8	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	4	4	no	—
108	C₈×C₅₃	C₄×Q₈	1	4	yes	—
109	C₈×C₅₃	C₄×Q₈	1	4	yes	—
110	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
111	#21736	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₅₃	#533	1	1	yes	characteristic
115	C₂²×C₅₃	#533	1	1	yes	characteristic
116	C₄×C₅₃	—	2	4	no	—
117	C₄×C₅₃	—	2	4	no	—
118	C₄×C₅₃	—	4	2	no	—
119	C₄×C₅₃	—	4	8	no	—
120	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₅₃	#532	1	1	yes	characteristic
124	#3162	C₂×C₅₃	1	1	yes	characteristic
125	#2961	C₂×C₅₃	1	2	yes	—
126	#1994	C₂×C₅₃	1	2	yes	—
127	#1946	C₂×C₅₃	1	2	yes	—
128	#1945	C₂×C₅₃	1	2	yes	—
129	#1849	C₂×C₅₃	1	1	yes	characteristic
130	#1773	C₂×C₅₃	1	2	yes	—
131	#1772	C₂×C₅₃	1	2	yes	—
132	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₅₃	#2956	1	2	yes	—
137	#533	C₂²×C₅₃	1	2	yes	—
138	#501	C₄×C₅₃	1	8	yes	complemented
139	#446	—	2	8	no	—
140	C₂×#108	C₂²×C₅₃	1	2	yes	—
141	C₂×#108	C₂²×C₅₃	1	2	yes	—
142	#427	C₂²×C₅₃	1	2	yes	—
143	#427	C₂²×C₅₃	1	2	yes	—
144	#424	C₂²×C₅₃	1	4	yes	—
145	#423	C₂²×C₅₃	1	2	yes	—
146	#405	C₂²×C₅₃	1	4	yes	—
147	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
153	C₄³	C₂²×C₅₃	1	1	yes	characteristic
154	#339	C₂²×C₅₃	1	4	yes	—
155	#339	C₂²×C₅₃	1	4	yes	—
156	C₅₃	#21736	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
158	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
163	#125	C₂×C₄×C₅₃	1	4	yes	—
164	C₂²⋊Q₈	—	2	4	no	—
165	C₄×Q₈	—	2	4	no	—
166	#117	C₂³×C₅₃	1	4	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
171	#108	—	2	4	no	—
172	#108	—	2	4	no	—
173	#107	C₂×C₄×C₅₃	1	4	yes	—
174	#107	C₂×C₄×C₅₃	1	4	yes	—
175	#106	C₂×C₄×C₅₃	1	4	yes	—
176	#106	C₂×C₄×C₅₃	1	4	yes	—
177	C₄⋊C₈	—	2	8	no	—
178	C₄⋊C₈	—	2	4	no	—
179	#97	—	2	4	no	—
180	#97	—	2	2	no	—
181	#97	C₂×C₄×C₅₃	1	4	yes	—
182	#97	D₄×C₅₃	1	4	yes	—
183	#97	Q₈×C₅₃	1	4	yes	complemented
184	C₄×C₈	—	2	4	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
187	C₂×Q₈	—	2	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	#41×C₅₃	1	2	yes	—
191	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
205	C₂×C₈	#41×C₅₃	1	2	yes	—
206	C₂×C₈	#41×C₅₃	1	2	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
216	C₄²	C₂×D₄×C₅₃	1	2	yes	—
217	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
218	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
219	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
228	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	4	4	no	—
236	C₈	C₄×Q₈×C₅₃	1	4	yes	—
237	C₈	C₄×Q₈×C₅₃	1	4	yes	—
238	C₈	C₄⋊Q₈×C₅₃	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
241	C₂²	C₅₃×#533	1	1	yes	characteristic
242	C₂²	C₅₃×#533	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	8	no	—
247	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
248	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
249	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
250	C₄	C₅₃×#532	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
254	C₂	C₅₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#501	C₄	semidirect
55. #97×C₅₃	Q₈	semidirect
58. C₄×C₈×C₅₃	Q₈	semidirect
111. #21736	C₅₃	direct (N × H)
138. #501	C₄×C₅₃	semidirect
156. C₅₃	#21736	direct (N × H)
183. #97	Q₈×C₅₃	semidirect
186. C₄×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3162	2
C₅₃×#2961	2
C₅₃×#1994	2
C₅₃×#1946	2
C₅₃×#1945	2
C₅₃×#1849	2
C₅₃×#1773	2
C₅₃×#1772	2
C₄×#97×C₅₃	2
#21736	53

Subgroups — C53×#21736 #27229248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21736 #27229248