Subgroups — C₅₃×#2731 #20271668

Order 6784.

492 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	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#396	C₂	1	2	yes	maximal, complemented
4	C₅₃×#394	C₂	1	2	yes	maximal
5	C₅₃×#393	C₂	1	2	yes	maximal, complemented
6	C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#388	C₂	1	2	yes	maximal, complemented
8	C₅₃×#388	C₂	1	2	yes	maximal, complemented
9	C₅₃×#387	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₃	—	2	4	no	—
14	#124×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	#95×C₅₃	C₂²	1	2	yes	—
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	2	yes	—
32	#95×C₅₃	C₂²	1	2	yes	—
33	C₂⁴×C₅₃	C₂³	1	2	yes	—
34	C₂×Q₈×C₅₃	—	2	2	no	—
35	C₂×Q₈×C₅₃	—	4	1	no	—
36	C₂×Q₈×C₅₃	—	4	1	no	—
37	C₂×Q₈×C₅₃	—	4	1	no	—
38	C₂×Q₈×C₅₃	—	4	1	no	—
39	C₂×Q₈×C₅₃	D₄	1	4	yes	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	—	4	1	no	—
42	C₂×D₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	1	no	—
55	C₂²⋊C₄×C₅₃	—	2	4	no	—
56	C₂²⋊C₄×C₅₃	—	2	4	no	—
57	C₂²⋊C₄×C₅₃	—	2	4	no	—
58	C₂²⋊C₄×C₅₃	—	2	4	no	—
59	C₂²⋊C₄×C₅₃	—	4	2	no	—
60	C₂²⋊C₄×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	4	1	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	4	no	—
73	Q₈×C₅₃	—	4	4	no	—
74	D₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	#41	1	4	yes	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	—	4	2	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#41	1	2	yes	—
108	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	—	4	2	no	—
119	#2731	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₅₃	—	8	2	no	—
121	C₂×C₅₃	#534	1	1	yes	characteristic
122	C₂×C₅₃	#542	1	2	yes	—
123	C₂×C₅₃	#548	1	2	yes	—
124	C₂×C₅₃	#555	1	1	yes	characteristic
125	C₂×C₅₃	#563	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
127	C₂×#124	C₂×C₅₃	1	1	yes	characteristic
128	#396	C₂×C₅₃	1	2	yes	complemented
129	#394	C₂×C₅₃	1	2	yes	—
130	#393	C₂×C₅₃	1	2	yes	complemented
131	#391	C₂×C₅₃	1	1	yes	characteristic, complemented
132	#388	C₂×C₅₃	1	2	yes	complemented
133	#388	C₂×C₅₃	1	2	yes	complemented
134	#387	C₂×C₅₃	1	2	yes	complemented
135	C₅₃	#2731	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
137	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
138	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	4	no	—
140	#124	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
151	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
154	#95	C₂²×C₅₃	1	2	yes	—
155	#95	C₂²×C₅₃	1	2	yes	—
156	#95	C₂²×C₅₃	1	2	yes	—
157	#95	C₂²×C₅₃	1	2	yes	—
158	#95	C₂²×C₅₃	1	2	yes	—
159	C₂⁴	C₂³×C₅₃	1	2	yes	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	D₄×C₅₃	1	4	yes	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	4	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
198	Q₈	—	4	4	no	—
199	Q₈	—	4	4	no	—
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂×C₄	#41×C₅₃	1	4	yes	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
233	C₂²	C₂×#41×C₅₃	1	2	yes	—
234	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
235	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
236	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₂	—	8	2	no	—
246	C₂	C₅₃×#534	1	1	yes	characteristic
247	C₂	C₅₃×#542	1	2	yes	—
248	C₂	C₅₃×#548	1	2	yes	—
249	C₂	C₅₃×#555	1	1	yes	characteristic
250	C₂	C₅₃×#563	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#396	C₂	semidirect
5. C₅₃×#393	C₂	semidirect
6. C₅₃×#391	C₂	semidirect
7. C₅₃×#388	C₂	semidirect
8. C₅₃×#388	C₂	semidirect
9. C₅₃×#387	C₂	semidirect
119. #2731	C₅₃	direct (N × H)
126. C₂×C₄⋊Q₈	C₂×C₅₃	semidirect
128. #396	C₂×C₅₃	semidirect
130. #393	C₂×C₅₃	semidirect
131. #391	C₂×C₅₃	semidirect
132. #388	C₂×C₅₃	semidirect
133. #388	C₂×C₅₃	semidirect
134. #387	C₂×C₅₃	semidirect
135. C₅₃	#2731	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₅₃	2
C₂×#124×C₅₃	2
C₅₃×#396	2
C₅₃×#394	2
C₅₃×#393	2
C₅₃×#391	2
C₅₃×#388	2
C₅₃×#387	2
#2731	53

Subgroups — C53×#2731 #20271668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2731 #20271668