Subgroups — C₅₃×#2434 #20271371

Order 6784.

800 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#395	C₂	1	4	yes	maximal, complemented
5	C₅₃×#391	C₂	1	2	yes	maximal, complemented
6	C₅₃×#385	C₂	1	4	yes	maximal
7	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₅₃	—	2	8	no	—
9	C₂²×Q₈×C₅₃	—	2	4	no	—
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₅₃	—	2	8	no	—
16	C₂²⋊Q₈×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
23	#95×C₅₃	C₂²	1	2	yes	—
24	#95×C₅₃	C₂²	1	8	yes	—
25	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₅₃	—	2	32	no	—
27	C₂×Q₈×C₅₃	—	2	8	no	—
28	C₂×Q₈×C₅₃	—	2	4	no	—
29	C₂×Q₈×C₅₃	—	2	4	no	—
30	C₂×Q₈×C₅₃	—	2	4	no	—
31	C₂×Q₈×C₅₃	—	2	2	no	—
32	C₂×Q₈×C₅₃	—	2	4	no	—
33	C₂×Q₈×C₅₃	D₄	1	8	yes	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	D₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	2	16	no	—
47	C₂²⋊C₄×C₅₃	—	2	4	no	—
48	C₂²⋊C₄×C₅₃	Q₈	1	8	yes	complemented
49	C₄²×C₅₃	—	2	4	no	—
50	C₂³×C₅₃	—	2	2	no	—
51	C₂³×C₅₃	—	2	2	no	—
52	C₂³×C₅₃	C₂×D₄	1	2	yes	—
53	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
55	Q₈×C₅₃	—	2	16	no	—
56	Q₈×C₅₃	—	2	16	no	—
57	Q₈×C₅₃	—	4	4	no	—
58	C₂×C₄×C₅₃	—	2	16	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	C₂×D₄	1	8	yes	—
70	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	complemented
74	C₂²×C₅₃	—	2	2	no	—
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	—	2	8	no	—
77	C₂²×C₅₃	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
80	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
81	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₅₃	—	2	8	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	8	no	—
85	C₄×C₅₃	—	4	2	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	#120	1	4	yes	—
88	#2434	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₅₃	—	2	4	no	—
90	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
91	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₅₃	#536	1	1	yes	characteristic
93	C₂×C₅₃	D₄×Q₈	1	4	yes	—
94	C₂³×Q₈	C₂×C₅₃	1	1	yes	characteristic
95	C₂×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
96	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
97	#395	C₂×C₅₃	1	4	yes	complemented
98	#391	C₂×C₅₃	1	2	yes	complemented
99	#385	C₂×C₅₃	1	4	yes	—
100	C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
101	C₅₃	#2434	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×Q₈	—	2	8	no	—
103	C₂²×Q₈	—	2	4	no	—
104	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
105	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
106	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
107	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
108	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
109	C₄⋊Q₈	—	2	8	no	—
110	C₂²⋊Q₈	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
116	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
117	#95	C₂²×C₅₃	1	2	yes	—
118	#95	C₂²×C₅₃	1	8	yes	—
119	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
120	C₂×Q₈	—	2	32	no	—
121	C₂×Q₈	—	2	8	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	2	2	no	—
126	C₂×Q₈	—	2	4	no	—
127	C₂×Q₈	D₄×C₅₃	1	8	yes	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	D₄×C₅₃	1	4	yes	—
132	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
135	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
137	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	16	no	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	Q₈×C₅₃	1	8	yes	complemented
143	C₄²	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂×D₄×C₅₃	1	2	yes	—
147	C₂³	C₂×Q₈×C₅₃	1	4	yes	—
148	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
149	Q₈	—	2	16	no	—
150	Q₈	—	2	16	no	—
151	Q₈	—	4	4	no	—
152	C₂×C₄	—	2	16	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	C₂×D₄×C₅₃	1	8	yes	—
164	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
166	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
167	C₂×C₄	C₂×Q₈×C₅₃	1	8	yes	complemented
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	8	no	—
171	C₂²	C₄⋊Q₈×C₅₃	1	4	yes	complemented
172	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
173	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
174	C₂²	C₂²×Q₈×C₅₃	1	2	yes	—
175	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
176	C₄	—	2	8	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	#120×C₅₃	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	C₂×#120×C₅₃	1	1	yes	characteristic
184	C₂	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
185	C₂	C₅₃×#536	1	1	yes	characteristic
186	C₂	D₄×Q₈×C₅₃	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂³
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
2. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#395	C₂	semidirect
5. C₅₃×#391	C₂	semidirect
48. C₂²⋊C₄×C₅₃	Q₈	semidirect
73. C₂×C₄×C₅₃	C₂×Q₈	semidirect
77. C₂²×C₅₃	C₄⋊Q₈	semidirect
88. #2434	C₅₃	direct (N × H)
95. C₂×C₄⋊Q₈	C₂×C₅₃	semidirect
97. #395	C₂×C₅₃	semidirect
98. #391	C₂×C₅₃	semidirect
101. C₅₃	#2434	direct (N × H)
142. C₂²⋊C₄	Q₈×C₅₃	semidirect
167. C₂×C₄	C₂×Q₈×C₅₃	semidirect
171. C₂²	C₄⋊Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₅₃	2
C₂×C₄⋊Q₈×C₅₃	2
C₂×C₂²⋊Q₈×C₅₃	2
C₅₃×#395	2
C₅₃×#391	2
C₅₃×#385	2
C₄×C₂²⋊C₄×C₅₃	2
#2434	53

Subgroups — C53×#2434 #20271371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2434 #20271371