Subgroups — C₅₃×#19603 #27227115

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	4	yes	maximal, complemented
2	C₅₃×#3147	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1709	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1706	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	—	2	4	no	—
9	C₅₃×#532	C₂²	1	4	yes	—
10	C₅₃×#500	C₂²	1	4	yes	—
11	C₅₃×#495	—	2	4	no	—
12	C₅₃×#494	C₂²	1	2	yes	—
13	C₅₃×#493	C₂²	1	4	yes	—
14	C₅₃×#493	C₂²	1	2	yes	—
15	C₅₃×#492	C₂²	1	4	yes	—
16	C₅₃×#486	C₂²	1	8	yes	—
17	C₅₃×#466	—	2	8	no	—
18	C₄×D₈×C₅₃	—	2	8	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₈²×C₅₃	C₂²	1	2	yes	—
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	C₈²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#41×C₅₃	—	2	4	no	—
25	#135×C₅₃	—	4	16	no	—
26	C₂×Q₁₆×C₅₃	—	2	4	no	—
27	C₂×SD₁₆×C₅₃	—	2	8	no	—
28	C₂×D₈×C₅₃	—	2	4	no	—
29	C₂×D₈×C₅₃	—	2	8	no	—
30	C₂²×C₈×C₅₃	D₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
33	#127×C₅₃	C₂³	1	4	yes	—
34	#124×C₅₃	—	2	4	no	—
35	#121×C₅₃	—	2	8	no	—
36	C₄×D₄×C₅₃	—	2	8	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	8	no	—
39	#102×C₅₃	—	2	16	no	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₄×C₈×C₅₃	D₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	2	yes	—
46	#41×C₅₃	—	4	16	no	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×D₄×C₅₃	—	2	8	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₅₃	—	4	8	no	—
55	SD₁₆×C₅₃	—	4	16	no	—
56	D₈×C₅₃	—	4	8	no	—
57	D₈×C₅₃	—	4	16	no	—
58	C₂×C₈×C₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	D₈	1	8	yes	complemented
60	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₂²⋊C₄×C₅₃	—	4	8	no	—
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂³×C₅₃	—	4	4	no	—
70	C₂³×C₅₃	#127	1	1	yes	characteristic
71	Q₈×C₅₃	—	4	8	no	—
72	D₄×C₅₃	—	4	8	no	—
73	D₄×C₅₃	—	4	16	no	—
74	D₄×C₅₃	—	8	8	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	4	4	no	—
77	C₂×C₄×C₅₃	—	8	8	no	—
78	C₂×C₄×C₅₃	#127	1	2	yes	—
79	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	C₂×D₈	1	4	yes	—
85	C₈×C₅₃	C₂×D₈	1	4	yes	—
86	C₈×C₅₃	#135	1	8	yes	complemented
87	#19603	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	8	4	no	—
90	C₂²×C₅₃	—	8	8	no	—
91	C₂²×C₅₃	#493	1	2	yes	—
92	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	8	4	no	—
95	C₄×C₅₃	#493	1	2	yes	—
96	C₄×C₅₃	C₂²×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×#135	1	4	yes	—
98	#3169	C₂×C₅₃	1	4	yes	complemented
99	#3147	C₂×C₅₃	1	2	yes	complemented
100	#1713	C₂×C₅₃	1	1	yes	characteristic, complemented
101	#1712	C₂×C₅₃	1	1	yes	characteristic, complemented
102	#1709	C₂×C₅₃	1	2	yes	complemented
103	#1706	C₂×C₅₃	1	4	yes	complemented
104	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	—	16	4	no	—
107	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
108	C₂×C₅₃	#3147	1	2	yes	—
109	C₂×#135	—	2	4	no	—
110	#532	C₂²×C₅₃	1	4	yes	—
111	#500	C₂²×C₅₃	1	4	yes	—
112	#495	—	2	4	no	—
113	#494	C₂²×C₅₃	1	2	yes	—
114	#493	C₂²×C₅₃	1	2	yes	—
115	#493	C₂²×C₅₃	1	4	yes	—
116	#492	C₂²×C₅₃	1	4	yes	—
117	#486	C₂²×C₅₃	1	8	yes	—
118	#466	—	2	8	no	—
119	C₄×D₈	—	2	8	no	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
122	C₈²	C₂²×C₅₃	1	2	yes	—
123	C₈²	C₂²×C₅₃	1	1	yes	characteristic
124	C₈²	C₂²×C₅₃	1	1	yes	characteristic
125	C₅₃	#19603	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×#41	—	2	4	no	—
127	#135	—	4	16	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×SD₁₆	—	2	8	no	—
130	C₂×D₈	—	2	4	no	—
131	C₂×D₈	—	2	8	no	—
132	C₂²×C₈	D₄×C₅₃	1	4	yes	—
133	C₂²×C₈	D₄×C₅₃	1	2	yes	—
134	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
135	#127	C₂³×C₅₃	1	4	yes	—
136	#124	—	2	4	no	—
137	#121	—	2	8	no	—
138	C₄×D₄	—	2	8	no	—
139	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
140	#107	—	2	8	no	—
141	#102	—	2	16	no	—
142	C₄×C₈	D₄×C₅₃	1	4	yes	—
143	C₄×C₈	D₄×C₅₃	1	2	yes	—
144	C₄×C₈	C₂³×C₅₃	1	2	yes	—
145	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂³×C₅₃	1	2	yes	—
148	#41	—	4	16	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×D₄	—	2	8	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
156	Q₁₆	—	4	8	no	—
157	SD₁₆	—	4	16	no	—
158	D₈	—	4	8	no	—
159	D₈	—	4	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₈×C₅₃	1	8	yes	complemented
162	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
163	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
166	C₄⋊C₄	—	2	8	no	—
167	C₂²⋊C₄	—	4	8	no	—
168	C₄²	C₂×D₄×C₅₃	1	2	yes	—
169	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
170	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
171	C₂³	—	4	4	no	—
172	C₂³	#127×C₅₃	1	1	yes	characteristic
173	Q₈	—	4	8	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	8	no	—
180	C₂×C₄	#127×C₅₃	1	2	yes	—
181	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
183	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₈×C₅₃	1	4	yes	—
187	C₈	C₂×D₈×C₅₃	1	4	yes	—
188	C₈	#135×C₅₃	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	8	no	—
192	C₂²	C₅₃×#493	1	2	yes	—
193	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	C₅₃×#493	1	2	yes	—
197	C₄	C₂²×D₈×C₅₃	1	2	yes	—
198	C₄	C₂×#135×C₅₃	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	4	no	—
201	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
202	C₂	C₅₃×#3147	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3169	C₂	semidirect
2. C₅₃×#3147	C₂	semidirect
3. C₅₃×#1713	C₂	semidirect
4. C₅₃×#1712	C₂	semidirect
5. C₅₃×#1709	C₂	semidirect
6. C₅₃×#1706	C₂	semidirect
7. C₂×C₈²×C₅₃	C₂	semidirect
59. C₂×C₈×C₅₃	D₈	semidirect
86. C₈×C₅₃	#135	semidirect
87. #19603	C₅₃	direct (N × H)
98. #3169	C₂×C₅₃	semidirect
99. #3147	C₂×C₅₃	semidirect
100. #1713	C₂×C₅₃	semidirect
101. #1712	C₂×C₅₃	semidirect
102. #1709	C₂×C₅₃	semidirect
103. #1706	C₂×C₅₃	semidirect
104. C₂×C₈²	C₂×C₅₃	semidirect
125. C₅₃	#19603	direct (N × H)
161. C₂×C₈	D₈×C₅₃	semidirect
188. C₈	#135×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3169	2
C₅₃×#3147	2
C₅₃×#1713	2
C₅₃×#1712	2
C₅₃×#1709	2
C₅₃×#1706	2
C₂×C₈²×C₅₃	2
#19603	53

Subgroups — C53×#19603 #27227115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19603 #27227115