Subgroups — C₅₃×#36608 #27244120

Order 13568.

1626 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2862	C₂	1	3	yes	maximal
2	C₅₃×#2581	C₂	1	6	yes	maximal
3	C₅₃×#2514	C₂	1	12	yes	maximal
4	C₅₃×#2356	C₂	1	6	yes	maximal
5	C₄×C₄⋊Q₈×C₅₃	C₂	1	3	yes	maximal
6	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
9	C₂×C₄×Q₈×C₅₃	C₂²	1	6	yes	—
10	C₅₃×#401	C₂²	1	8	yes	—
11	C₅₃×#400	C₂²	1	12	yes	—
12	C₅₃×#395	C₂²	1	4	yes	—
13	C₅₃×#391	C₂²	1	6	yes	—
14	C₅₃×#391	C₂²	1	12	yes	—
15	C₅₃×#391	C₂²	1	12	yes	—
16	C₅₃×#389	C₂²	1	12	yes	—
17	C₅₃×#387	C₂²	1	24	yes	—
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₅₃×#383	C₂²	1	12	yes	—
20	C₅₃×#382	C₂²	1	6	yes	—
21	C₅₃×#382	C₂²	1	6	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
25	C₅₃×#376	C₂²	1	4	yes	—
26	C₄³×C₅₃	C₂²	1	3	yes	—
27	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
29	C₄⋊Q₈×C₅₃	—	2	12	no	—
30	C₄×Q₈×C₅₃	—	2	12	no	—
31	C₄×Q₈×C₅₃	—	2	24	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#95×C₅₃	C₂³	1	1	yes	characteristic
44	#95×C₅₃	C₂³	1	12	yes	—
45	#95×C₅₃	C₂³	1	3	yes	—
46	#95×C₅₃	C₂³	1	2	yes	—
47	#95×C₅₃	C₂³	1	24	yes	—
48	#95×C₅₃	C₂³	1	24	yes	—
49	#95×C₅₃	C₂³	1	12	yes	—
50	C₂×Q₈×C₅₃	—	2	6	no	—
51	C₂×Q₈×C₅₃	—	2	12	no	—
52	C₂×Q₈×C₅₃	—	2	12	no	—
53	C₂×Q₈×C₅₃	—	2	12	no	—
54	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
60	C₄⋊C₄×C₅₃	—	2	12	no	—
61	C₄⋊C₄×C₅₃	—	2	24	no	—
62	C₄⋊C₄×C₅₃	—	2	6	no	—
63	C₄⋊C₄×C₅₃	—	2	12	no	—
64	C₄⋊C₄×C₅₃	—	2	24	no	—
65	C₄²×C₅₃	—	2	12	no	—
66	C₄²×C₅₃	—	2	24	no	—
67	C₄²×C₅₃	—	2	12	no	—
68	C₄²×C₅₃	—	2	6	no	—
69	C₄²×C₅₃	#41	1	12	yes	—
70	C₄²×C₅₃	#41	1	4	yes	—
71	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
72	Q₈×C₅₃	—	2	8	no	—
73	Q₈×C₅₃	—	4	24	no	—
74	C₂×C₄×C₅₃	—	2	6	no	—
75	C₂×C₄×C₅₃	—	2	6	no	—
76	C₂×C₄×C₅₃	—	2	12	no	—
77	C₂×C₄×C₅₃	—	2	12	no	—
78	C₂×C₄×C₅₃	—	2	12	no	—
79	C₂×C₄×C₅₃	—	2	6	no	—
80	C₂×C₄×C₅₃	—	2	12	no	—
81	C₂×C₄×C₅₃	—	2	6	no	—
82	C₂×C₄×C₅₃	—	2	6	no	—
83	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
84	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
86	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
87	C₂×C₄×C₅₃	2₋¹⁺⁴	1	6	yes	—
88	#36608	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	C₂²×#41	1	3	yes	—
90	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	3	yes	—
92	C₄×C₅₃	—	2	6	no	—
93	C₄×C₅₃	—	2	6	no	—
94	C₄×C₅₃	—	4	4	no	—
95	C₄×C₅₃	—	4	12	no	—
96	C₄×C₅₃	—	4	6	no	—
97	C₄×C₅₃	#556	1	12	yes	—
98	#2862	C₂×C₅₃	1	3	yes	—
99	#2581	C₂×C₅₃	1	6	yes	—
100	#2514	C₂×C₅₃	1	12	yes	—
101	#2356	C₂×C₅₃	1	6	yes	—
102	C₄×C₄⋊Q₈	C₂×C₅₃	1	3	yes	—
103	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
104	C₂×C₅₃	#3444	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#556	1	3	yes	—
106	C₂×C₅₃	#3509	1	3	yes	—
107	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
108	C₂×C₄×Q₈	C₂²×C₅₃	1	6	yes	—
109	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
110	#401	C₂²×C₅₃	1	8	yes	—
111	#400	C₂²×C₅₃	1	12	yes	—
112	#395	C₂²×C₅₃	1	4	yes	—
113	#391	C₂²×C₅₃	1	6	yes	—
114	#391	C₂²×C₅₃	1	12	yes	—
115	#391	C₂²×C₅₃	1	12	yes	—
116	#389	C₂²×C₅₃	1	12	yes	—
117	#387	C₂²×C₅₃	1	24	yes	—
118	#384	C₂²×C₅₃	1	1	yes	characteristic
119	#383	C₂²×C₅₃	1	12	yes	—
120	#382	C₂²×C₅₃	1	6	yes	—
121	#382	C₂²×C₅₃	1	6	yes	—
122	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
123	C₄×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
124	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
125	#376	C₂²×C₅₃	1	4	yes	—
126	C₄³	C₂²×C₅₃	1	3	yes	—
127	C₅₃	#36608	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
129	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
130	C₄⋊Q₈	—	2	12	no	—
131	C₄×Q₈	—	2	12	no	—
132	C₄×Q₈	—	2	24	no	—
133	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
134	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
135	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
136	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
137	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
139	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
140	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
141	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
142	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
143	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
144	#95	C₂³×C₅₃	1	1	yes	characteristic
145	#95	C₂³×C₅₃	1	12	yes	—
146	#95	C₂³×C₅₃	1	3	yes	—
147	#95	C₂³×C₅₃	1	2	yes	—
148	#95	C₂³×C₅₃	1	24	yes	—
149	#95	C₂³×C₅₃	1	24	yes	—
150	#95	C₂³×C₅₃	1	12	yes	—
151	C₂×Q₈	—	2	6	no	—
152	C₂×Q₈	—	2	12	no	—
153	C₂×Q₈	—	2	12	no	—
154	C₂×Q₈	—	2	12	no	—
155	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
156	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
157	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
158	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
159	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
160	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	—	2	6	no	—
163	C₄⋊C₄	—	2	12	no	—
164	C₄⋊C₄	—	2	24	no	—
165	C₄⋊C₄	—	2	24	no	—
166	C₄²	—	2	12	no	—
167	C₄²	—	2	24	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₄²	#41×C₅₃	1	12	yes	—
171	C₄²	#41×C₅₃	1	4	yes	—
172	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	4	24	no	—
175	C₂×C₄	—	2	6	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	12	no	—
184	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
185	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
186	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
187	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
188	C₂×C₄	2₋¹⁺⁴×C₅₃	1	6	yes	—
189	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
190	C₂²	C₂²×#41×C₅₃	1	3	yes	—
191	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	3	yes	—
192	C₄	—	2	6	no	—
193	C₄	—	2	6	no	—
194	C₄	—	4	12	no	—
195	C₄	—	4	6	no	—
196	C₄	—	4	4	no	—
197	C₄	C₅₃×#556	1	12	yes	—
198	C₂	C₅₃×#3444	1	1	yes	characteristic
199	C₂	C₂×C₅₃×#556	1	3	yes	—
200	C₂	C₅₃×#3509	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
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
88. #36608	C₅₃	direct (N × H)
127. C₅₃	#36608	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×#2862	2
C₅₃×#2581	2
C₅₃×#2514	2
C₅₃×#2356	2
C₄×C₄⋊Q₈×C₅₃	2
C₄²×Q₈×C₅₃	2
#36608	53

Subgroups — C53×#36608 #27244120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36608 #27244120