Subgroups — C₅₃×#33459 #27240971

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2951	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2175	C₂	1	4	yes	maximal, complemented
4	C₄×Q₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2139	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#520	—	2	2	no	—
11	C₅₃×#508	C₄	1	8	yes	complemented
12	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#440	—	2	4	no	—
18	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
19	C₄×D₈×C₅₃	C₂²	1	4	yes	—
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#417	—	2	2	no	—
22	C₅₃×#405	C₂²	1	1	yes	characteristic
23	C₅₃×#366	C₂²	1	2	yes	—
24	C₅₃×#366	C₂²	1	2	yes	—
25	C₅₃×#358	C₂²	1	4	yes	—
26	C₅₃×#357	C₂²	1	4	yes	—
27	C₅₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	#135×C₅₃	—	2	4	no	—
31	#135×C₅₃	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	—	2	4	no	—
38	C₄×D₄×C₅₃	—	2	4	no	—
39	C₄×D₄×C₅₃	—	4	4	no	—
40	#117×C₅₃	—	4	2	no	—
41	#117×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
44	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	M₃₂×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	—	2	2	no	—
53	#107×C₅₃	C₂³	1	4	yes	—
54	#103×C₅₃	—	2	4	no	—
55	#102×C₅₃	—	2	4	no	—
56	#97×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	#41×C₅₃	—	2	8	no	—
59	#41×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	—	4	4	no	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	2	2	no	—
68	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₅₃	—	2	8	no	—
70	D₈×C₅₃	—	2	2	no	—
71	D₈×C₅₃	C₂²×C₄	1	4	yes	—
72	M₁₆×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₃	—	4	4	no	—
84	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
85	C₄²×C₅₃	—	4	4	no	—
86	C₄²×C₅₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₅₃	—	2	2	no	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	4	no	—
93	Q₈×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	4	8	no	—
96	D₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	8	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	16	4	no	—
101	C₂×C₄×C₅₃	—	16	2	no	—
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#135	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	C₄×D₄	1	2	yes	—
110	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
112	#33459	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	—	8	2	no	—
118	C₂²×C₅₃	C₄×D₈	1	2	yes	—
119	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	—	8	4	no	—
124	C₄×C₅₃	C₄×D₈	1	2	yes	—
125	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
127	C₂×#508	C₂×C₅₃	1	1	yes	characteristic
128	#2951	C₂×C₅₃	1	2	yes	complemented
129	#2175	C₂×C₅₃	1	4	yes	complemented
130	C₄×Q₃₂	C₂×C₅₃	1	2	yes	complemented
131	C₄×D₁₆	C₂×C₅₃	1	2	yes	complemented
132	C₂×#366	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#2139	C₂×C₅₃	1	2	yes	complemented
134	#2108	C₂×C₅₃	1	1	yes	characteristic, complemented
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	—	8	4	no	—
137	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
138	C₂×C₅₃	#3415	1	1	yes	characteristic
139	C₂×C₅₃	#3417	1	1	yes	characteristic
140	C₂×#135	C₂²×C₅₃	1	2	yes	—
141	#520	—	2	2	no	—
142	#508	C₄×C₅₃	1	8	yes	complemented
143	C₂×Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
145	C₂×D₁₆	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
148	#440	—	2	4	no	—
149	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
150	C₄×D₈	C₂²×C₅₃	1	4	yes	—
151	C₂×#107	C₂²×C₅₃	1	2	yes	—
152	#417	—	2	2	no	—
153	#405	C₂²×C₅₃	1	1	yes	characteristic
154	#366	C₂²×C₅₃	1	2	yes	—
155	#366	C₂²×C₅₃	1	2	yes	—
156	#358	C₂²×C₅₃	1	4	yes	—
157	#357	C₂²×C₅₃	1	4	yes	—
158	#346	C₂²×C₅₃	1	2	yes	—
159	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
160	C₅₃	#33459	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#41	—	2	2	no	—
162	#135	—	2	4	no	—
163	#135	C₂×C₄×C₅₃	1	8	yes	complemented
164	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	C₂³×C₅₃	1	2	yes	—
167	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
169	C₄×Q₈	—	2	4	no	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	—	4	4	no	—
172	#117	—	4	2	no	—
173	#117	D₄×C₅₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	4	2	no	—
176	Q₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
177	SD₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
178	D₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
179	M₃₂	—	2	2	no	—
180	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
181	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
182	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
184	#107	—	2	2	no	—
185	#107	C₂³×C₅₃	1	4	yes	—
186	#103	—	2	4	no	—
187	#102	—	2	4	no	—
188	#97	D₄×C₅₃	1	2	yes	—
189	C₄×C₈	C₂³×C₅₃	1	2	yes	—
190	#41	—	2	8	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
201	SD₁₆	—	2	8	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×C₄×C₅₃	1	4	yes	—
204	M₁₆	—	2	2	no	—
205	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	#41×C₅₃	1	2	yes	—
209	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
214	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
217	C₄²	—	4	4	no	—
218	C₄²	C₂×D₄×C₅₃	1	2	yes	—
219	C₁₆	—	2	2	no	—
220	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
221	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	16	4	no	—
233	C₂×C₄	—	16	2	no	—
234	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
235	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
236	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
237	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
238	C₂×C₄	#135×C₅₃	1	2	yes	—
239	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	C₄×D₄×C₅₃	1	2	yes	—
242	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
243	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₄×D₈×C₅₃	1	2	yes	—
250	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	4	no	—
255	C₄	C₄×D₈×C₅₃	1	2	yes	—
256	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
257	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
261	C₂	C₅₃×#3415	1	1	yes	characteristic
262	C₂	C₅₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2951	C₂	semidirect
3. C₅₃×#2175	C₂	semidirect
4. C₄×Q₃₂×C₅₃	C₂	semidirect
5. C₄×D₁₆×C₅₃	C₂	semidirect
6. C₂×C₅₃×#366	C₂	semidirect
7. C₅₃×#2139	C₂	semidirect
8. C₅₃×#2108	C₂	semidirect
11. C₅₃×#508	C₄	semidirect
31. #135×C₅₃	C₂×C₄	semidirect
44. Q₃₂×C₅₃	C₂×C₄	semidirect
45. SD₃₂×C₅₃	C₂×C₄	semidirect
46. D₁₆×C₅₃	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
112. #33459	C₅₃	direct (N × H)
128. #2951	C₂×C₅₃	semidirect
129. #2175	C₂×C₅₃	semidirect
130. C₄×Q₃₂	C₂×C₅₃	semidirect
131. C₄×D₁₆	C₂×C₅₃	semidirect
132. C₂×#366	C₂×C₅₃	semidirect
133. #2139	C₂×C₅₃	semidirect
134. #2108	C₂×C₅₃	semidirect
142. #508	C₄×C₅₃	semidirect
160. C₅₃	#33459	direct (N × H)
163. #135	C₂×C₄×C₅₃	semidirect
176. Q₃₂	C₂×C₄×C₅₃	semidirect
177. SD₃₂	C₂×C₄×C₅₃	semidirect
178. D₁₆	C₂×C₄×C₅₃	semidirect
180. C₂×C₁₆	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#508	2
C₅₃×#2951	2
C₅₃×#2175	2
C₄×Q₃₂×C₅₃	2
C₄×D₁₆×C₅₃	2
C₂×C₅₃×#366	2
C₅₃×#2139	2
C₅₃×#2108	2
#33459	53

Subgroups — C53×#33459 #27240971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33459 #27240971