Subgroups — C₅₃×#33409 #27240921

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	2	yes	maximal
3	C₅₃×#2159	C₂	1	8	yes	maximal, complemented
4	C₅₃×#2158	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#117×C₅₃	—	2	2	no	—
7	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
8	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#427	—	2	2	no	—
11	C₅₃×#427	—	2	2	no	—
12	C₅₃×#427	C₂²	1	4	yes	—
13	C₅₃×#427	C₂²	1	4	yes	—
14	C₂×#107×C₅₃	C₂²	1	4	yes	—
15	C₂×#106×C₅₃	—	2	2	no	—
16	C₅₃×#367	C₂²	1	8	yes	complemented
17	C₅₃×#366	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
25	#117×C₅₃	—	2	4	no	—
26	#117×C₅₃	—	2	4	no	—
27	#117×C₅₃	—	4	2	no	—
28	#117×C₅₃	—	4	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
31	C₂×C₄²×C₅₃	—	4	2	no	—
32	M₃₂×C₅₃	C₂×C₄	1	16	yes	complemented
33	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
34	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
35	#107×C₅₃	—	2	4	no	—
36	#107×C₅₃	C₂³	1	8	yes	—
37	#106×C₅₃	—	2	4	no	—
38	#106×C₅₃	—	2	4	no	—
39	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	D₈	1	2	yes	—
44	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₂²⋊C₄×C₅₃	—	4	4	no	—
59	C₂²⋊C₄×C₅₃	—	4	4	no	—
60	C₄²×C₅₃	—	4	4	no	—
61	C₄²×C₅₃	—	4	4	no	—
62	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	#107	1	4	yes	—
65	C₂³×C₅₃	C₂×D₈	1	2	yes	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	4	8	no	—
68	C₂×C₄×C₅₃	—	8	8	no	—
69	C₂×C₄×C₅₃	#107	1	4	yes	—
70	C₂×C₄×C₅₃	#107	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	#33409	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₅₃	—	2	4	no	—
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×#107	1	4	yes	—
87	C₂²×C₅₃	C₂×#107	1	2	yes	—
88	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₅₃	—	8	8	no	—
90	C₄×C₅₃	C₂×#107	1	4	yes	—
91	C₄×C₅₃	C₂×#107	1	2	yes	—
92	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
94	C₂²×M₃₂	C₂×C₅₃	1	1	yes	characteristic
95	C₂×#427	C₂×C₅₃	1	2	yes	—
96	#2159	C₂×C₅₃	1	8	yes	complemented
97	#2158	C₂×C₅₃	1	4	yes	complemented
98	C₂×C₅₃	—	2	4	no	—
99	C₂×C₅₃	—	2	2	no	—
100	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
101	C₂×C₅₃	#3414	1	2	yes	—
102	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	C₂×#117	—	2	2	no	—
104	C₂×M₃₂	C₄×C₅₃	1	8	yes	complemented
105	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
106	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
107	#427	—	2	2	no	—
108	#427	—	2	2	no	—
109	#427	C₂²×C₅₃	1	4	yes	—
110	#427	C₂²×C₅₃	1	4	yes	—
111	C₂×#107	C₂²×C₅₃	1	4	yes	—
112	C₂×#106	—	2	2	no	—
113	#367	C₂²×C₅₃	1	8	yes	complemented
114	#366	C₂²×C₅₃	1	8	yes	complemented
115	C₅₃	#33409	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
118	C₂²×C₈	D₄×C₅₃	1	2	yes	—
119	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
120	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
121	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
122	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
123	#117	—	2	4	no	—
124	#117	—	2	4	no	—
125	#117	—	4	2	no	—
126	#117	—	4	2	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₂²⋊C₄	—	4	2	no	—
129	C₂×C₄²	—	4	2	no	—
130	M₃₂	C₂×C₄×C₅₃	1	16	yes	complemented
131	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
132	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
133	#107	—	2	4	no	—
134	#107	C₂³×C₅₃	1	8	yes	—
135	#106	—	2	4	no	—
136	#106	—	2	4	no	—
137	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
140	C₂²×C₄	D₈×C₅₃	1	2	yes	—
141	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
142	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
143	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
145	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
146	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
147	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
149	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
151	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
152	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
153	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₂²⋊C₄	—	4	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	4	no	—
160	C₁₆	C₂²×C₄×C₅₃	1	8	yes	—
161	C₂³	—	4	2	no	—
162	C₂³	#107×C₅₃	1	4	yes	—
163	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
164	C₂³	C₂×D₈×C₅₃	1	2	yes	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	8	8	no	—
167	C₂×C₄	#107×C₅₃	1	8	yes	—
168	C₂×C₄	#107×C₅₃	1	4	yes	—
169	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
171	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
172	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
174	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
175	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
176	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
177	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
178	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
179	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
180	C₂²	—	2	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×#107×C₅₃	1	2	yes	—
184	C₂²	C₂×#107×C₅₃	1	4	yes	—
185	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
186	C₄	—	8	8	no	—
187	C₄	C₂×#107×C₅₃	1	4	yes	—
188	C₄	C₂×#107×C₅₃	1	2	yes	—
189	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
190	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
191	C₂	—	2	4	no	—
192	C₂	—	2	2	no	—
193	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic
194	C₂	C₅₃×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2159	C₂	semidirect
4. C₅₃×#2158	C₂	semidirect
7. C₂×M₃₂×C₅₃	C₄	semidirect
16. C₅₃×#367	C₂²	semidirect
17. C₅₃×#366	C₂²	semidirect
32. M₃₂×C₅₃	C₂×C₄	semidirect
33. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
82. #33409	C₅₃	direct (N × H)
96. #2159	C₂×C₅₃	semidirect
97. #2158	C₂×C₅₃	semidirect
104. C₂×M₃₂	C₄×C₅₃	semidirect
113. #367	C₂²×C₅₃	semidirect
114. #366	C₂²×C₅₃	semidirect
115. C₅₃	#33409	direct (N × H)
130. M₃₂	C₂×C₄×C₅₃	semidirect
131. C₂×C₁₆	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₅₃	2
C₂×C₅₃×#427	2
C₅₃×#2159	2
C₅₃×#2158	2
#33409	53

Subgroups — C53×#33409 #27240921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33409 #27240921