Subgroups — C₅₃×#2332 #20271269

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#391	C₂	1	2	yes	maximal, complemented
3	C₅₃×#388	C₂	1	2	yes	maximal
4	C₅₃×#388	C₂	1	2	yes	maximal
5	C₅₃×#387	C₂	1	2	yes	maximal, complemented
6	C₅₃×#387	C₂	1	2	yes	maximal, complemented
7	C₅₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈×C₅₃	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
25	#95×C₅₃	C₂²	1	2	yes	—
26	#95×C₅₃	C₂²	1	2	yes	—
27	#95×C₅₃	C₂²	1	2	yes	—
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	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	4	1	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	—	4	1	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
50	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
51	C₂²⋊C₄×C₅₃	—	2	4	no	—
52	C₂²⋊C₄×C₅₃	—	4	1	no	—
53	C₂²⋊C₄×C₅₃	—	4	1	no	—
54	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
55	C₄²×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₂³×C₅₃	—	2	2	no	—
60	C₂³×C₅₃	—	2	1	no	—
61	C₂³×C₅₃	—	2	1	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
65	Q₈×C₅₃	—	4	4	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	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	—	4	4	no	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₅₃	#41	1	4	yes	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×#41	1	2	yes	—
98	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	#2332	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₅₃	—	4	2	no	—
111	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
112	C₂×C₅₃	#518	1	1	yes	characteristic
113	C₂×C₅₃	#519	1	1	yes	characteristic
114	C₂×C₅₃	#539	1	2	yes	—
115	C₂×C₅₃	#542	1	2	yes	—
116	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
117	#391	C₂×C₅₃	1	2	yes	complemented
118	#388	C₂×C₅₃	1	2	yes	—
119	#388	C₂×C₅₃	1	2	yes	—
120	#387	C₂×C₅₃	1	2	yes	complemented
121	#387	C₂×C₅₃	1	2	yes	complemented
122	#381	C₂×C₅₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
124	C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
125	C₅₃	#2332	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
127	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
128	C₂²⋊Q₈	C₄×C₅₃	1	8	yes	complemented
129	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
137	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
138	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
141	#95	C₂²×C₅₃	1	2	yes	—
142	#95	C₂²×C₅₃	1	2	yes	—
143	#95	C₂²×C₅₃	1	2	yes	—
144	#95	C₂²×C₅₃	1	2	yes	—
145	#95	C₂²×C₅₃	1	2	yes	—
146	#95	C₂²×C₅₃	1	2	yes	—
147	#95	C₂²×C₅₃	1	2	yes	—
148	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂×Q₈	C₂×C₄×C₅₃	1	4	yes	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	C₂×C₄×C₅₃	1	4	yes	—
155	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
156	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
166	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	4	1	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
171	C₄²	—	2	4	no	—
172	C₄²	—	4	1	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	4	1	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	2	no	—
179	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
180	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
203	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
204	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
205	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
206	C₂×C₄	#41×C₅₃	1	4	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
213	C₂²	C₂×#41×C₅₃	1	2	yes	—
214	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
215	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
216	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
217	C₄	—	2	4	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₂	—	4	2	no	—
226	C₂	C₄×#41×C₅₃	1	1	yes	characteristic
227	C₂	C₅₃×#518	1	1	yes	characteristic
228	C₂	C₅₃×#519	1	1	yes	characteristic
229	C₂	C₅₃×#539	1	2	yes	—
230	C₂	C₅₃×#542	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#391	C₂	semidirect
5. C₅₃×#387	C₂	semidirect
6. C₅₃×#387	C₂	semidirect
8. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
12. C₂²⋊Q₈×C₅₃	C₄	semidirect
109. #2332	C₅₃	direct (N × H)
117. #391	C₂×C₅₃	semidirect
120. #387	C₂×C₅₃	semidirect
121. #387	C₂×C₅₃	semidirect
123. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
125. C₅₃	#2332	direct (N × H)
128. C₂²⋊Q₈	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×C₂²⋊Q₈×C₅₃	2
C₅₃×#391	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#381	2
C₄×C₄⋊C₄×C₅₃	2
C₄×C₂²⋊C₄×C₅₃	2
#2332	53

Subgroups — C53×#2332 #20271269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2332 #20271269