Subgroups — C₅₃×#12834 #27220346

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#366	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1808	C₂	1	2	yes	maximal
4	C₅₃×#1380	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₅₃	—	2	4	no	—
7	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#107×C₅₃	C₂²	1	4	yes	—
11	C₅₃×#380	—	2	2	no	—
12	C₅₃×#366	—	2	4	no	—
13	C₅₃×#340	—	2	2	no	—
14	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
23	C₂×C₁₆×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	8	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
28	#107×C₅₃	—	2	4	no	—
29	#107×C₅₃	—	2	4	no	—
30	#95×C₅₃	—	4	2	no	—
31	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	4	4	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₄⋊C₄×C₅₃	—	4	4	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₂²⋊C₄×C₅₃	—	8	4	no	—
51	C₁₆×C₅₃	—	2	2	no	—
52	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	#107	1	2	yes	—
56	C₂³×C₅₃	D₁₆	1	2	yes	—
57	C₂³×C₅₃	Q₃₂	1	2	yes	—
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	8	4	no	—
64	C₂×C₄×C₅₃	—	8	4	no	—
65	C₂×C₄×C₅₃	—	8	4	no	—
66	C₂×C₄×C₅₃	#107	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#121	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#135	1	2	yes	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₅₃	#117	1	1	yes	characteristic
74	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
75	#12834	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	—	2	4	no	—
79	C₂²×C₅₃	—	2	4	no	—
80	C₂²×C₅₃	#366	1	4	yes	complemented
81	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
82	C₂²×C₅₃	#466	1	2	yes	—
83	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
85	C₂²×C₅₃	#508	1	2	yes	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	16	4	no	—
88	C₄×C₅₃	#380	1	1	yes	characteristic
89	C₄×C₅₃	#427	1	1	yes	characteristic
90	C₄×C₅₃	#467	1	2	yes	—
91	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic
92	C₂×#366	C₂×C₅₃	1	2	yes	complemented
93	#1808	C₂×C₅₃	1	2	yes	—
94	#1380	C₂×C₅₃	1	2	yes	complemented
95	C₂×C₅₃	—	2	4	no	—
96	C₂×C₅₃	#1808	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
98	C₂×C₅₃	#2158	1	1	yes	characteristic
99	C₂×C₅₃	#2215	1	2	yes	—
100	C₂×C₅₃	#2216	1	2	yes	—
101	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
102	C₂²×C₁₆	—	2	4	no	—
103	C₂²×C₁₆	C₄×C₅₃	1	4	yes	complemented
104	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
106	C₂×#107	C₂²×C₅₃	1	4	yes	—
107	#380	—	2	2	no	—
108	#366	—	2	4	no	—
109	#340	—	2	2	no	—
110	C₅₃	#12834	1	1	yes	characteristic, direct product factor, Sylow
111	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
115	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
116	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
117	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₂²⋊C₄	—	4	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
123	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
124	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
125	#107	—	2	4	no	—
126	#107	—	2	4	no	—
127	#95	—	4	2	no	—
128	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
133	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
134	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
143	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
144	C₂×C₈	#41×C₅₃	1	2	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₂²⋊C₄	—	8	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊C₄×C₅₃	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	#107×C₅₃	1	2	yes	—
153	C₂³	D₁₆×C₅₃	1	2	yes	—
154	C₂³	Q₃₂×C₅₃	1	2	yes	—
155	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	#107×C₅₃	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄	#121×C₅₃	1	2	yes	—
166	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
167	C₂×C₄	#135×C₅₃	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
170	C₈	#117×C₅₃	1	1	yes	characteristic
171	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	C₅₃×#366	1	4	yes	complemented
177	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
178	C₂²	C₅₃×#466	1	2	yes	—
179	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
180	C₂²	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
181	C₂²	C₅₃×#508	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	16	4	no	—
184	C₄	C₅₃×#380	1	1	yes	characteristic
185	C₄	C₅₃×#427	1	1	yes	characteristic
186	C₄	C₅₃×#467	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₅₃×#1808	1	1	yes	characteristic
189	C₂	C₂×C₅₃×#366	1	1	yes	characteristic
190	C₂	C₅₃×#2158	1	1	yes	characteristic
191	C₂	C₅₃×#2215	1	2	yes	—
192	C₂	C₅₃×#2216	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#366	C₂	semidirect
4. C₅₃×#1380	C₂	semidirect
7. C₂²×C₁₆×C₅₃	C₄	semidirect
52. C₁₆×C₅₃	C₂²⋊C₄	semidirect
75. #12834	C₅₃	direct (N × H)
80. C₂²×C₅₃	#366	semidirect
92. C₂×#366	C₂×C₅₃	semidirect
94. #1380	C₂×C₅₃	semidirect
103. C₂²×C₁₆	C₄×C₅₃	semidirect
110. C₅₃	#12834	direct (N × H)
149. C₁₆	C₂²⋊C₄×C₅₃	semidirect
176. C₂²	C₅₃×#366	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₅₃	2
C₂×C₅₃×#366	2
C₅₃×#1808	2
C₅₃×#1380	2
#12834	53

Subgroups — C53×#12834 #27220346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12834 #27220346