Subgroups — C₅₃×#9825 #27217337

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1990	C₂	1	4	yes	maximal
3	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₃	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#494	—	2	2	no	—
8	C₅₃×#494	C₄	1	4	yes	complemented
9	C₂×#107×C₅₃	—	2	2	no	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#384	C₂²	1	2	yes	—
13	C₅₃×#340	—	2	8	no	—
14	C₈²×C₅₃	—	2	2	no	—
15	C₂²×Q₈×C₅₃	—	2	2	no	—
16	C₂×Q₁₆×C₅₃	—	2	8	no	—
17	C₂×Q₁₆×C₅₃	—	4	4	no	—
18	C₂×Q₁₆×C₅₃	—	4	4	no	—
19	C₂²×C₈×C₅₃	D₄	1	4	yes	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₄⋊Q₈×C₅₃	—	2	2	no	—
22	C₄⋊Q₈×C₅₃	—	2	4	no	—
23	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#107×C₅₃	—	4	4	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×Q₈×C₅₃	—	2	4	no	—
34	C₂×Q₈×C₅₃	—	4	2	no	—
35	C₂×Q₈×C₅₃	—	4	4	no	—
36	C₂×Q₈×C₅₃	—	4	4	no	—
37	C₂²×C₄×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	Q₁₆×C₅₃	—	4	16	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₅₃	D₈	1	4	yes	—
46	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
47	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
48	C₄⋊C₄×C₅₃	—	2	8	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	8	no	—
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂³×C₅₃	#127	1	1	yes	characteristic
56	Q₈×C₅₃	—	4	8	no	—
57	Q₈×C₅₃	—	8	4	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	—	8	4	no	—
63	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₅₃	#121	1	2	yes	—
66	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
70	C₂×C₄×C₅₃	#135	1	4	yes	—
71	C₈×C₅₃	—	2	8	no	—
72	C₈×C₅₃	#102	1	8	yes	—
73	#9825	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	#390	1	1	yes	characteristic
75	C₂²×C₅₃	#492	1	1	yes	characteristic
76	C₂²×C₅₃	#493	1	1	yes	characteristic
77	C₂²×C₅₃	#494	1	2	yes	—
78	C₂²×C₅₃	#495	1	2	yes	—
79	C₄×C₅₃	—	8	4	no	—
80	C₄×C₅₃	—	16	2	no	—
81	C₄×C₅₃	C₂×#102	1	2	yes	—
82	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
83	C₄×C₅₃	#467	1	4	yes	—
84	C₄×C₅₃	#486	1	2	yes	—
85	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
86	#1990	C₂×C₅₃	1	4	yes	—
87	#1845	C₂×C₅₃	1	1	yes	characteristic
88	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
89	C₂×C₅₃	#1707	1	2	yes	—
90	C₂×C₅₃	#1709	1	1	yes	characteristic
91	C₂×C₅₃	#1713	1	1	yes	characteristic
92	C₂×C₅₃	#1957	1	1	yes	characteristic
93	C₂×C₅₃	#1969	1	2	yes	—
94	C₂²×Q₁₆	—	2	2	no	—
95	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
96	#494	—	2	2	no	—
97	#494	C₄×C₅₃	1	4	yes	complemented
98	C₂×#107	—	2	2	no	—
99	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
101	#384	C₂²×C₅₃	1	2	yes	—
102	#340	—	2	8	no	—
103	C₈²	—	2	2	no	—
104	C₅₃	#9825	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	8	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂²×C₈	D₄×C₅₃	1	4	yes	—
110	C₂²×C₈	D₄×C₅₃	1	2	yes	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
122	C₄×C₈	D₄×C₅₃	1	2	yes	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	4	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
131	Q₁₆	—	4	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
135	C₂×C₈	D₈×C₅₃	1	4	yes	—
136	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
137	C₂×C₈	Q₁₆×C₅₃	1	8	yes	complemented
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
143	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₄²	#41×C₅₃	1	2	yes	—
145	C₂³	#127×C₅₃	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
155	C₂×C₄	#121×C₅₃	1	2	yes	—
156	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
159	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
160	C₂×C₄	#135×C₅₃	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	#102×C₅₃	1	8	yes	—
163	C₂²	C₅₃×#390	1	1	yes	characteristic
164	C₂²	C₅₃×#492	1	1	yes	characteristic
165	C₂²	C₅₃×#493	1	1	yes	characteristic
166	C₂²	C₅₃×#494	1	2	yes	—
167	C₂²	C₅₃×#495	1	2	yes	—
168	C₄	—	8	4	no	—
169	C₄	—	16	2	no	—
170	C₄	C₂×#102×C₅₃	1	2	yes	—
171	C₄	C₄×Q₁₆×C₅₃	1	4	yes	—
172	C₄	C₅₃×#467	1	4	yes	—
173	C₄	C₅₃×#486	1	2	yes	—
174	C₂	C₅₃×#1707	1	2	yes	—
175	C₂	C₅₃×#1709	1	1	yes	characteristic
176	C₂	C₅₃×#1713	1	1	yes	characteristic
177	C₂	C₅₃×#1957	1	1	yes	characteristic
178	C₂	C₅₃×#1969	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₅₃×#494	C₄	semidirect
47. C₂×C₈×C₅₃	Q₁₆	semidirect
73. #9825	C₅₃	direct (N × H)
97. #494	C₄×C₅₃	semidirect
104. C₅₃	#9825	direct (N × H)
137. C₂×C₈	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#494	2
C₅₃×#1990	2
C₅₃×#1845	2
C₂×C₈²×C₅₃	2
#9825	53

Subgroups — C53×#9825 #27217337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9825 #27217337