Subgroups — C₂²×C₅₃×#331 #27225205

Order 13568.

2022 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#331	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₄×C₅₃	—	2	2	no	—
5	C₂³×D₄×C₅₃	C₄	1	2	yes	—
6	C₂³×C₈×C₅₃	—	2	2	no	—
7	C₂×#127×C₅₃	—	2	6	no	—
8	C₂×#127×C₅₃	C₄	1	6	yes	—
9	C₂×#127×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	—	2	6	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	12	yes	complemented
13	C₅₃×#331	C₂²	1	16	yes	direct product factor
14	C₂⁵×C₅₃	—	2	2	no	—
15	C₂²×D₄×C₅₃	—	2	6	no	—
16	C₂²×D₄×C₅₃	—	2	12	no	—
17	C₂²×D₄×C₅₃	—	2	12	no	—
18	C₂²×D₄×C₅₃	—	4	8	no	—
19	C₂²×D₄×C₅₃	—	4	8	no	—
20	C₂²×D₄×C₅₃	—	4	8	no	—
21	C₂²×D₄×C₅₃	C₂×C₄	1	12	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	12	no	—
25	C₂²×C₈×C₅₃	—	4	8	no	—
26	#127×C₅₃	—	2	12	no	—
27	#127×C₅₃	—	2	12	no	—
28	#127×C₅₃	C₂×C₄	1	12	yes	—
29	#127×C₅₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₅₃	D₄	1	6	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	C₄⋊C₈×C₅₃	—	2	12	no	—
34	C₄⋊C₈×C₅₃	C₂³	1	8	yes	complemented
35	C₂⁴×C₅₃	—	2	12	no	—
36	C₂⁴×C₅₃	—	4	8	no	—
37	C₂⁴×C₅₃	—	4	16	no	—
38	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×D₄×C₅₃	—	2	12	no	—
40	C₂×D₄×C₅₃	—	2	24	no	—
41	C₂×D₄×C₅₃	—	2	8	no	—
42	C₂×D₄×C₅₃	—	4	24	no	—
43	C₂×D₄×C₅₃	—	4	24	no	—
44	C₂×D₄×C₅₃	—	4	48	no	—
45	C₂×D₄×C₅₃	—	4	24	no	—
46	C₂×D₄×C₅₃	—	4	24	no	—
47	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
51	C₂²×C₄×C₅₃	D₈	1	8	yes	—
52	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
55	C₂×C₈×C₅₃	—	2	8	no	—
56	C₂×C₈×C₅₃	—	4	24	no	—
57	C₄²×C₅₃	C₂²⋊C₄	1	6	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₅₃	C₂×D₄	1	6	yes	—
60	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂³×C₅₃	—	2	8	no	—
62	C₂³×C₅₃	—	4	24	no	—
63	C₂³×C₅₃	—	4	48	no	—
64	C₂³×C₅₃	—	8	32	no	—
65	C₂³×C₅₃	#100	1	4	yes	—
66	C₂³×C₅₃	#102	1	8	yes	—
67	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
68	D₄×C₅₃	—	4	8	no	—
69	D₄×C₅₃	—	4	24	no	—
70	D₄×C₅₃	—	4	8	no	—
71	D₄×C₅₃	—	4	8	no	—
72	D₄×C₅₃	—	4	48	no	—
73	C₂×C₄×C₅₃	—	2	6	no	—
74	C₂×C₄×C₅₃	—	2	6	no	—
75	C₂×C₄×C₅₃	#102	1	24	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	12	yes	—
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
82	C₈×C₅₃	—	4	8	no	—
83	C₂²×#331	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	—	4	8	no	—
85	C₂²×C₅₃	—	4	16	no	—
86	C₂²×C₅₃	—	8	48	no	—
87	C₂²×C₅₃	#331	1	16	yes	direct product factor
88	C₂²×C₅₃	C₂×#100	1	6	yes	—
89	C₂²×C₅₃	C₂×#102	1	12	yes	—
90	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₂×#102	1	12	yes	—
94	C₄×C₅₃	C₂²×D₈	1	2	yes	—
95	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
96	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
97	C₂²×C₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
98	C₂×#331	C₂×C₅₃	1	12	yes	direct product factor
99	C₂×C₅₃	—	8	8	no	—
100	C₂×C₅₃	C₂×#331	1	12	yes	direct product factor
101	C₂×C₅₃	C₂²×#100	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×#102	1	2	yes	—
103	C₂³×D₄	—	2	2	no	—
104	C₂³×D₄	C₄×C₅₃	1	2	yes	—
105	C₂³×C₈	—	2	2	no	—
106	C₂×#127	—	2	6	no	—
107	C₂×#127	C₄×C₅₃	1	6	yes	—
108	C₂×#127	C₂²×C₅₃	1	6	yes	—
109	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	2	6	no	—
111	C₂×C₄⋊C₈	C₂²×C₅₃	1	12	yes	complemented
112	#331	C₂²×C₅₃	1	16	yes	direct product factor
113	C₅₃	C₂²×#331	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁵	—	2	2	no	—
115	C₂²×D₄	—	2	12	no	—
116	C₂²×D₄	—	2	6	no	—
117	C₂²×D₄	—	2	12	no	—
118	C₂²×D₄	—	4	8	no	—
119	C₂²×D₄	—	4	8	no	—
120	C₂²×D₄	—	4	8	no	—
121	C₂²×D₄	C₂×C₄×C₅₃	1	12	yes	—
122	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
123	C₂³×C₄	D₄×C₅₃	1	2	yes	—
124	C₂²×C₈	—	2	12	no	—
125	C₂²×C₈	—	4	8	no	—
126	#127	—	2	12	no	—
127	#127	—	2	12	no	—
128	#127	C₂×C₄×C₅₃	1	12	yes	—
129	#127	C₂³×C₅₃	1	4	yes	—
130	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
131	C₂×C₄²	D₄×C₅₃	1	6	yes	—
132	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
133	C₄⋊C₈	—	2	12	no	—
134	C₄⋊C₈	C₂³×C₅₃	1	8	yes	complemented
135	C₂⁴	—	2	12	no	—
136	C₂⁴	—	4	8	no	—
137	C₂⁴	—	4	16	no	—
138	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂×D₄	—	2	12	no	—
140	C₂×D₄	—	2	24	no	—
141	C₂×D₄	—	2	8	no	—
142	C₂×D₄	—	4	24	no	—
143	C₂×D₄	—	4	24	no	—
144	C₂×D₄	—	4	48	no	—
145	C₂×D₄	—	4	24	no	—
146	C₂×D₄	—	4	24	no	—
147	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
150	C₂²×C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
151	C₂²×C₄	D₈×C₅₃	1	8	yes	—
152	C₂²×C₄	SD₁₆×C₅₃	1	8	yes	—
153	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	4	24	no	—
157	C₄²	C₂²⋊C₄×C₅₃	1	6	yes	—
158	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
159	C₄²	C₂×D₄×C₅₃	1	6	yes	—
160	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
161	C₂³	—	2	8	no	—
162	C₂³	—	4	24	no	—
163	C₂³	—	4	48	no	—
164	C₂³	—	8	32	no	—
165	C₂³	#100×C₅₃	1	4	yes	—
166	C₂³	#102×C₅₃	1	8	yes	—
167	C₂³	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
168	D₄	—	4	8	no	—
169	D₄	—	4	24	no	—
170	D₄	—	4	8	no	—
171	D₄	—	4	8	no	—
172	D₄	—	4	48	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	#102×C₅₃	1	24	yes	—
176	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
177	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
178	C₂×C₄	C₂×D₈×C₅₃	1	12	yes	—
179	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
180	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
182	C₈	—	4	8	no	—
183	C₂²	—	4	8	no	—
184	C₂²	—	4	16	no	—
185	C₂²	—	8	48	no	—
186	C₂²	C₅₃×#331	1	16	yes	direct product factor
187	C₂²	C₂×#100×C₅₃	1	6	yes	—
188	C₂²	C₂×#102×C₅₃	1	12	yes	—
189	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₄	—	2	3	no	—
191	C₄	—	2	1	no	—
192	C₄	C₂×#102×C₅₃	1	12	yes	—
193	C₄	C₂²×D₈×C₅₃	1	2	yes	—
194	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
195	C₂	—	8	8	no	—
196	C₂	C₂×C₅₃×#331	1	12	yes	direct product factor
197	C₂	C₂²×#100×C₅₃	1	1	yes	characteristic
198	C₂	C₂²×#102×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₈×C₅₃	C₂	semidirect
3. C₂×C₅₃×#331	C₂	direct (N × H)
12. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
13. C₅₃×#331	C₂²	direct (N × H)
34. C₄⋊C₈×C₅₃	C₂³	semidirect
83. C₂²×#331	C₅₃	direct (N × H)
87. C₂²×C₅₃	#331	direct (N × H)
97. C₂²×C₄⋊C₈	C₂×C₅₃	semidirect
98. C₂×#331	C₂×C₅₃	direct (N × H)
100. C₂×C₅₃	C₂×#331	direct (N × H)
111. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
112. #331	C₂²×C₅₃	direct (N × H)
113. C₅₃	C₂²×#331	direct (N × H)
134. C₄⋊C₈	C₂³×C₅₃	semidirect
186. C₂²	C₅₃×#331	direct (N × H)
196. C₂	C₂×C₅₃×#331	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#127×C₅₃	2
C₂²×C₄⋊C₈×C₅₃	2
C₂×C₅₃×#331	2
C₂²×#331	53

Subgroups — C22×C53×#331 #27225205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#331 #27225205