Subgroups — C₅₃×#22746 #27230258

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1950	C₂	1	6	yes	maximal
4	C₅₃×#1837	C₂	1	3	yes	maximal
5	C₄×#108×C₅₃	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#446	—	2	6	no	—
8	C₈×Q₈×C₅₃	—	2	2	no	—
9	C₈×Q₈×C₅₃	C₄	1	6	yes	—
10	C₈×Q₈×C₅₃	C₄	1	2	yes	—
11	C₂×#108×C₅₃	C₂²	1	6	yes	—
12	C₅₃×#423	C₂²	1	6	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
14	C₄×M₁₆×C₅₃	C₂²	1	6	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
16	C₅₃×#341	C₂²	1	6	yes	—
17	C₅₃×#341	C₂²	1	1	yes	characteristic
18	C₅₃×#341	C₂²	1	3	yes	—
19	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
21	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
24	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₅₃	D₄	1	3	yes	—
28	C₂×C₄⋊C₄×C₅₃	Q₈	1	3	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
30	#108×C₅₃	—	2	12	no	—
31	C₄⋊C₈×C₅₃	—	2	6	no	—
32	C₄⋊C₈×C₅₃	—	2	6	no	—
33	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
34	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
35	#97×C₅₃	—	2	6	no	—
36	C₄×C₈×C₅₃	—	2	6	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	C₄⋊C₄	1	3	yes	—
41	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	M₁₆×C₅₃	—	2	2	no	—
46	M₁₆×C₅₃	—	2	6	no	—
47	M₁₆×C₅₃	—	2	6	no	—
48	M₁₆×C₅₃	#41	1	4	yes	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	6	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	6	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
55	C₄⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
56	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
57	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
60	C₄²×C₅₃	C₂×D₄	1	3	yes	—
61	C₄²×C₅₃	C₂×D₄	1	3	yes	—
62	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
63	Q₈×C₅₃	—	2	6	no	—
64	Q₈×C₅₃	#108	1	4	yes	complemented
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	4	3	no	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
74	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
75	C₈×C₅₃	—	2	6	no	—
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	—	2	4	no	—
78	C₈×C₅₃	—	4	6	no	—
79	#22746	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₅₃	#557	1	2	yes	—
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	—	2	3	no	—
86	C₄×C₅₃	—	2	3	no	—
87	C₄×C₅₃	C₂×#108	1	6	yes	—
88	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
89	C₄×C₅₃	#519	1	1	yes	characteristic
90	C₄×C₅₃	#548	1	2	yes	—
91	Q₈×M₁₆	C₂×C₅₃	1	2	yes	—
92	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
93	#1950	C₂×C₅₃	1	6	yes	—
94	#1837	C₂×C₅₃	1	3	yes	—
95	C₄×#108	C₂×C₅₃	1	3	yes	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	#2351	1	1	yes	characteristic
98	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
99	C₂×C₅₃	#2916	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
101	#446	—	2	6	no	—
102	C₈×Q₈	—	2	2	no	—
103	C₈×Q₈	C₄×C₅₃	1	6	yes	—
104	C₈×Q₈	C₄×C₅₃	1	2	yes	—
105	C₂×#108	C₂²×C₅₃	1	6	yes	—
106	#423	C₂²×C₅₃	1	6	yes	—
107	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	—
108	C₄×M₁₆	C₂²×C₅₃	1	6	yes	—
109	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
110	#341	C₂²×C₅₃	1	3	yes	—
111	#341	C₂²×C₅₃	1	6	yes	—
112	#341	C₂²×C₅₃	1	1	yes	characteristic
113	C₅₃	#22746	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
115	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
116	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
117	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
119	C₄×Q₈	C₂×C₄×C₅₃	1	3	yes	—
120	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
121	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
122	C₄×Q₈	D₄×C₅₃	1	3	yes	—
123	C₂×C₄⋊C₄	Q₈×C₅₃	1	3	yes	—
124	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
125	#108	—	2	12	no	—
126	C₄⋊C₈	—	2	6	no	—
127	C₄⋊C₈	—	2	6	no	—
128	C₄⋊C₈	C₂×C₄×C₅₃	1	6	yes	—
129	C₄⋊C₈	C₂×C₄×C₅₃	1	6	yes	—
130	#97	—	2	6	no	—
131	C₄×C₈	—	2	6	no	—
132	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
133	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₄⋊C₄×C₅₃	1	3	yes	—
136	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
138	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
139	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
140	M₁₆	—	2	2	no	—
141	M₁₆	—	2	6	no	—
142	M₁₆	—	2	6	no	—
143	M₁₆	#41×C₅₃	1	4	yes	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	6	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	6	no	—
148	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
150	C₄⋊C₄	C₄⋊C₄×C₅₃	1	6	yes	—
151	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
152	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
153	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
154	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
155	C₄²	C₂×D₄×C₅₃	1	3	yes	—
156	C₄²	C₂×D₄×C₅₃	1	3	yes	—
157	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
158	Q₈	—	2	6	no	—
159	Q₈	#108×C₅₃	1	4	yes	complemented
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
166	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
168	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
169	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
170	C₈	—	2	6	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	4	no	—
173	C₈	—	4	6	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂²	C₅₃×#557	1	2	yes	—
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#108×C₅₃	1	6	yes	—
182	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
183	C₄	C₅₃×#519	1	1	yes	characteristic
184	C₄	C₅₃×#548	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₅₃×#2351	1	1	yes	characteristic
187	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic
188	C₂	C₅₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
64. Q₈×C₅₃	#108	semidirect
79. #22746	C₅₃	direct (N × H)
113. C₅₃	#22746	direct (N × H)
159. Q₈	#108×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₅₃	2
C₂×C₈×Q₈×C₅₃	2
C₅₃×#1950	2
C₅₃×#1837	2
C₄×#108×C₅₃	2
#22746	53

Subgroups — C53×#22746 #27230258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22746 #27230258