Subgroups — C₅₃×#23837 #27231349

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₅₃×#2084	C₂	1	4	yes	maximal
3	C₅₃×#2058	C₂	1	4	yes	maximal
4	C₄×#106×C₅₃	C₂	1	4	yes	maximal
5	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
8	C₂×#106×C₅₃	C₂²	1	8	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#391	—	2	2	no	—
11	C₅₃×#389	—	2	2	no	—
12	C₅₃×#389	C₂²	1	4	yes	—
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#340	C₂²	1	4	yes	—
18	C₅₃×#340	C₂²	1	4	yes	—
19	C₅₃×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₅₃	—	2	8	no	—
23	C₄×Q₈×C₅₃	—	4	8	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₅₃	—	2	4	no	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#106×C₅₃	—	2	16	no	—
37	C₄×C₈×C₅₃	Q₈	1	8	yes	complemented
38	#95×C₅₃	—	2	4	no	—
39	#95×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	2	8	no	—
41	C₂×Q₈×C₅₃	—	4	4	no	—
42	C₂×Q₈×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	8	no	—
51	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
52	C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	8	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₄⋊C₄×C₅₃	#41	1	8	yes	—
59	C₄²×C₅₃	—	2	4	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₄²×C₅₃	—	4	8	no	—
63	C₄²×C₅₃	SD₁₆	1	8	yes	—
64	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
65	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	16	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	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	8	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	4	4	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
86	C₈×C₅₃	—	2	8	no	—
87	#23837	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	#525	1	2	yes	—
89	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
91	C₂²×C₅₃	#575	1	1	yes	characteristic
92	C₂²×C₅₃	#575	1	1	yes	characteristic
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	8	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#498	1	4	yes	—
99	C₄×C₅₃	#533	1	4	yes	—
100	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
101	#2084	C₂×C₅₃	1	4	yes	—
102	#2058	C₂×C₅₃	1	4	yes	—
103	C₄×#106	C₂×C₅₃	1	4	yes	—
104	#1744	C₂×C₅₃	1	1	yes	characteristic
105	C₂×C₅₃	#2399	1	1	yes	characteristic
106	C₂×C₅₃	#3101	1	2	yes	—
107	C₂×C₅₃	#3103	1	2	yes	—
108	C₂×C₅₃	C₂×#498	1	1	yes	characteristic
109	C₂×C₅₃	#3162	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
111	C₂×C₄×Q₈	—	2	2	no	—
112	C₂×#106	C₂²×C₅₃	1	8	yes	—
113	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
114	#391	—	2	2	no	—
115	#389	—	2	2	no	—
116	#389	C₂²×C₅₃	1	4	yes	—
117	#384	C₂²×C₅₃	1	2	yes	—
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
120	C₄³	C₂²×C₅₃	1	1	yes	characteristic
121	#340	C₂²×C₅₃	1	4	yes	—
122	#340	C₂²×C₅₃	1	4	yes	—
123	#336	C₂²×C₅₃	1	4	yes	—
124	C₅₃	#23837	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
127	C₄⋊Q₈	—	2	8	no	—
128	C₄×Q₈	—	4	8	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
134	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
135	C₂×C₄²	—	2	4	no	—
136	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
137	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
138	C₂×C₄²	D₄×C₅₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
140	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
141	#106	—	2	16	no	—
142	C₄×C₈	Q₈×C₅₃	1	8	yes	complemented
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	C₂×Q₈	—	2	8	no	—
146	C₂×Q₈	—	4	4	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	C₂×Q₈×C₅₃	1	8	yes	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	#41×C₅₃	1	8	yes	—
164	C₄²	—	2	4	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	—	4	8	no	—
168	C₄²	SD₁₆×C₅₃	1	8	yes	—
169	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
170	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
171	Q₈	—	4	16	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
185	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
186	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
187	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
188	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
189	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
191	C₈	—	2	8	no	—
192	C₂²	C₅₃×#525	1	2	yes	—
193	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
194	C₂²	C₂²×SD₁₆×C₅₃	1	2	yes	—
195	C₂²	C₅₃×#575	1	1	yes	characteristic
196	C₂²	C₅₃×#575	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	8	no	—
201	C₄	—	8	4	no	—
202	C₄	C₅₃×#498	1	4	yes	—
203	C₄	C₅₃×#533	1	4	yes	—
204	C₂	C₅₃×#2399	1	1	yes	characteristic
205	C₂	C₅₃×#3101	1	2	yes	—
206	C₂	C₅₃×#3103	1	2	yes	—
207	C₂	C₂×C₅₃×#498	1	1	yes	characteristic
208	C₂	C₅₃×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₄×C₈×C₅₃	Q₈	semidirect
87. #23837	C₅₃	direct (N × H)
124. C₅₃	#23837	direct (N × H)
142. 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
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#2084	2
C₅₃×#2058	2
C₄×#106×C₅₃	2
C₅₃×#1744	2
#23837	53

Subgroups — C53×#23837 #27231349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23837 #27231349