Subgroups — C₅₃×#33860 #27241372

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3417	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3293	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2222	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2214	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2195	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#578	—	2	2	no	—
10	C₅₃×#578	C₂²	1	4	yes	—
11	C₅₃×#510	—	2	4	no	—
12	C₅₃×#510	C₂²	1	4	yes	—
13	C₅₃×#508	—	2	2	no	—
14	C₅₃×#508	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₅₃	—	2	2	no	—
16	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#504	C₂²	1	2	yes	—
19	C₅₃×#494	C₂²	1	2	yes	—
20	C₅₃×#473	—	2	2	no	—
21	C₅₃×#473	C₂²	1	4	yes	—
22	C₅₃×#456	—	2	4	no	—
23	C₅₃×#443	C₂²	1	2	yes	—
24	C₅₃×#443	C₂²	1	1	yes	characteristic
25	C₅₃×#368	C₂²	1	1	yes	characteristic
26	C₅₃×#364	C₂²	1	1	yes	characteristic
27	C₅₃×#362	C₂²	1	4	yes	—
28	C₅₃×#359	C₂²	1	2	yes	—
29	C₅₃×#350	C₂²	1	2	yes	—
30	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	2₋¹⁺⁴×C₅₃	—	2	4	no	—
32	#137×C₅₃	—	2	4	no	—
33	#137×C₅₃	—	2	4	no	—
34	#137×C₅₃	—	4	4	no	—
35	#137×C₅₃	—	4	4	no	—
36	#135×C₅₃	—	2	4	no	—
37	#135×C₅₃	—	4	2	no	—
38	#135×C₅₃	D₄	1	1	yes	characteristic
39	#135×C₅₃	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₅₃	—	2	2	no	—
41	C₂×Q₁₆×C₅₃	—	2	4	no	—
42	C₂×Q₁₆×C₅₃	—	4	2	no	—
43	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
44	#131×C₅₃	—	2	2	no	—
45	#131×C₅₃	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₅₃	—	2	2	no	—
47	Q₃₂×C₅₃	—	2	4	no	—
48	Q₃₂×C₅₃	—	2	2	no	—
49	Q₃₂×C₅₃	—	2	2	no	—
50	Q₃₂×C₅₃	—	4	2	no	—
51	Q₃₂×C₅₃	D₄	1	1	yes	characteristic
52	SD₃₂×C₅₃	—	2	4	no	—
53	SD₃₂×C₅₃	D₄	1	2	yes	—
54	D₁₆×C₅₃	D₄	1	1	yes	characteristic
55	M₃₂×C₅₃	—	2	2	no	—
56	M₃₂×C₅₃	C₂³	1	2	yes	—
57	C₂×C₁₆×C₅₃	—	2	2	no	—
58	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
60	#108×C₅₃	D₄	1	1	yes	characteristic
61	#108×C₅₃	C₂³	1	2	yes	—
62	#104×C₅₃	—	2	2	no	—
63	#104×C₅₃	D₄	1	2	yes	—
64	#101×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	#41×C₅₃	—	2	2	no	—
67	#41×C₅₃	—	4	4	no	—
68	#41×C₅₃	—	4	4	no	—
69	#41×C₅₃	—	8	4	no	—
70	#41×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×Q₈×C₅₃	—	2	4	no	—
72	C₂×Q₈×C₅₃	—	4	4	no	—
73	C₂×Q₈×C₅₃	—	4	4	no	—
74	Q₁₆×C₅₃	—	2	4	no	—
75	Q₁₆×C₅₃	—	2	4	no	—
76	Q₁₆×C₅₃	—	4	4	no	—
77	Q₁₆×C₅₃	—	4	4	no	—
78	Q₁₆×C₅₃	—	4	2	no	—
79	Q₁₆×C₅₃	—	4	2	no	—
80	Q₁₆×C₅₃	D₈	1	1	yes	characteristic
81	Q₁₆×C₅₃	C₂×D₄	1	2	yes	—
82	SD₁₆×C₅₃	—	2	2	no	—
83	SD₁₆×C₅₃	—	4	4	no	—
84	SD₁₆×C₅₃	—	4	4	no	—
85	SD₁₆×C₅₃	D₈	1	2	yes	—
86	D₈×C₅₃	D₈	1	1	yes	characteristic
87	D₈×C₅₃	C₂×D₄	1	2	yes	—
88	M₁₆×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	—	2	2	no	—
90	M₁₆×C₅₃	—	4	2	no	—
91	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	8	2	no	—
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₅₃	—	2	2	no	—
99	C₁₆×C₅₃	—	2	1	no	—
100	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	2	no	—
103	Q₈×C₅₃	—	4	4	no	—
104	Q₈×C₅₃	—	4	4	no	—
105	Q₈×C₅₃	—	8	4	no	—
106	Q₈×C₅₃	C₂×D₈	1	2	yes	—
107	D₄×C₅₃	—	2	2	no	—
108	D₄×C₅₃	—	2	2	no	—
109	D₄×C₅₃	—	4	2	no	—
110	D₄×C₅₃	—	8	4	no	—
111	D₄×C₅₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	8	4	no	—
117	C₂×C₄×C₅₃	—	8	4	no	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
125	#33860	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	#576	1	1	yes	characteristic
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	D₄²	1	1	yes	characteristic
134	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
135	#3417	C₂×C₅₃	1	2	yes	complemented
136	#3293	C₂×C₅₃	1	2	yes	complemented
137	#2247	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#2222	C₂×C₅₃	1	2	yes	complemented
139	#2214	C₂×C₅₃	1	2	yes	complemented
140	#2195	C₂×C₅₃	1	4	yes	complemented
141	#2178	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#2170	C₂×C₅₃	1	1	yes	characteristic, complemented
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	4	2	no	—
145	C₂×C₅₃	—	8	2	no	—
146	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
147	#578	—	2	2	no	—
148	#578	C₂²×C₅₃	1	4	yes	—
149	#510	—	2	4	no	—
150	#510	C₂²×C₅₃	1	4	yes	—
151	#508	—	2	2	no	—
152	#508	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×Q₃₂	—	2	2	no	—
154	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
155	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
156	#504	C₂²×C₅₃	1	2	yes	—
157	#494	C₂²×C₅₃	1	2	yes	—
158	#473	—	2	2	no	—
159	#473	C₂²×C₅₃	1	4	yes	—
160	#456	—	2	4	no	—
161	#443	C₂²×C₅₃	1	2	yes	—
162	#443	C₂²×C₅₃	1	1	yes	characteristic
163	#368	C₂²×C₅₃	1	1	yes	characteristic
164	#364	C₂²×C₅₃	1	1	yes	characteristic
165	#362	C₂²×C₅₃	1	4	yes	—
166	#359	C₂²×C₅₃	1	2	yes	—
167	#350	C₂²×C₅₃	1	2	yes	—
168	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
169	C₅₃	#33860	1	1	yes	characteristic, direct product factor, Sylow
170	2₋¹⁺⁴	—	2	4	no	—
171	#137	—	2	4	no	—
172	#137	—	2	4	no	—
173	#137	—	4	4	no	—
174	#137	—	4	4	no	—
175	#135	—	2	4	no	—
176	#135	—	4	2	no	—
177	#135	D₄×C₅₃	1	1	yes	characteristic
178	#135	C₂³×C₅₃	1	2	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	4	2	no	—
182	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
183	#131	—	2	2	no	—
184	#131	C₂³×C₅₃	1	2	yes	—
185	C₄⋊Q₈	—	2	2	no	—
186	Q₃₂	—	2	4	no	—
187	Q₃₂	—	2	2	no	—
188	Q₃₂	—	2	2	no	—
189	Q₃₂	—	4	2	no	—
190	Q₃₂	D₄×C₅₃	1	1	yes	characteristic
191	SD₃₂	—	2	4	no	—
192	SD₃₂	D₄×C₅₃	1	2	yes	—
193	D₁₆	D₄×C₅₃	1	1	yes	characteristic
194	M₃₂	—	2	2	no	—
195	M₃₂	C₂³×C₅₃	1	2	yes	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
199	#108	D₄×C₅₃	1	1	yes	characteristic
200	#108	C₂³×C₅₃	1	2	yes	—
201	#104	—	2	2	no	—
202	#104	D₄×C₅₃	1	2	yes	—
203	#101	—	2	4	no	—
204	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
205	#41	—	2	2	no	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	#41	—	8	4	no	—
209	#41	C₂×D₄×C₅₃	1	2	yes	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	4	no	—
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	2	no	—
218	Q₁₆	—	4	2	no	—
219	Q₁₆	D₈×C₅₃	1	1	yes	characteristic
220	Q₁₆	C₂×D₄×C₅₃	1	2	yes	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	—	4	4	no	—
224	SD₁₆	D₈×C₅₃	1	2	yes	—
225	D₈	D₈×C₅₃	1	1	yes	characteristic
226	D₈	C₂×D₄×C₅₃	1	2	yes	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
235	C₄⋊C₄	—	8	2	no	—
236	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	Q₈	C₂×D₈×C₅₃	1	2	yes	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	4	no	—
250	D₄	C₂×D₈×C₅₃	1	2	yes	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
261	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
262	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
263	C₈	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₅₃×#576	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	D₄²×C₅₃	1	1	yes	characteristic
272	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	—	8	2	no	—
276	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3417	C₂	semidirect
2. C₅₃×#3293	C₂	semidirect
3. C₅₃×#2247	C₂	semidirect
4. C₅₃×#2222	C₂	semidirect
5. C₅₃×#2214	C₂	semidirect
6. C₅₃×#2195	C₂	semidirect
7. C₅₃×#2178	C₂	semidirect
8. C₅₃×#2170	C₂	semidirect
125. #33860	C₅₃	direct (N × H)
135. #3417	C₂×C₅₃	semidirect
136. #3293	C₂×C₅₃	semidirect
137. #2247	C₂×C₅₃	semidirect
138. #2222	C₂×C₅₃	semidirect
139. #2214	C₂×C₅₃	semidirect
140. #2195	C₂×C₅₃	semidirect
141. #2178	C₂×C₅₃	semidirect
142. #2170	C₂×C₅₃	semidirect
169. C₅₃	#33860	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3417	2
C₅₃×#3293	2
C₅₃×#2247	2
C₅₃×#2222	2
C₅₃×#2214	2
C₅₃×#2195	2
C₅₃×#2178	2
C₅₃×#2170	2
#33860	53

Subgroups — C53×#33860 #27241372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33860 #27241372