Subgroups — C₅₃×#3341 #20272278

Order 6784.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#555	C₂	1	2	yes	maximal, complemented
2	C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#482	C₂	1	4	yes	maximal, complemented
4	C₅₃×#476	C₂	1	2	yes	maximal, complemented
5	C₅₃×#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#427	C₂	1	2	yes	maximal, complemented
9	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
12	#124×C₅₃	—	2	4	no	—
13	#121×C₅₃	—	2	2	no	—
14	#121×C₅₃	C₂²	1	4	yes	—
15	C₄×D₄×C₅₃	C₂²	1	2	yes	—
16	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	#117×C₅₃	—	2	2	no	—
18	#117×C₅₃	C₂²	1	4	yes	—
19	#107×C₅₃	C₂²	1	2	yes	—
20	#107×C₅₃	C₂²	1	2	yes	—
21	#107×C₅₃	C₂²	1	1	yes	characteristic
22	#106×C₅₃	C₂²	1	4	yes	—
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#102×C₅₃	C₂²	1	2	yes	—
25	#102×C₅₃	C₂²	1	4	yes	—
26	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	—	2	4	no	—
30	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×D₄×C₅₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	D₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₅₃	—	2	4	no	—
44	C₂²⋊C₄×C₅₃	—	2	4	no	—
45	C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
47	C₄²×C₅₃	—	2	4	no	—
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₅₃	—	2	2	no	—
50	C₂³×C₅₃	C₂×D₄	1	2	yes	—
51	Q₈×C₅₃	—	8	2	no	—
52	D₄×C₅₃	—	2	2	no	—
53	D₄×C₅₃	—	2	2	no	—
54	D₄×C₅₃	—	2	2	no	—
55	D₄×C₅₃	—	8	4	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	#41	1	2	yes	—
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	2	2	no	—
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	—	4	2	no	—
72	C₂²×C₅₃	—	4	2	no	—
73	C₂²×C₅₃	—	4	2	no	—
74	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	4	2	no	—
78	C₄×C₅₃	—	4	4	no	—
79	C₄×C₅₃	#135	1	2	yes	—
80	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
81	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
82	#3341	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₅₃	—	4	2	no	—
84	C₂×C₅₃	—	8	2	no	—
85	C₂×C₅₃	#547	1	1	yes	characteristic
86	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
87	C₂×C₅₃	#576	1	1	yes	characteristic
88	#555	C₂×C₅₃	1	2	yes	complemented
89	#500	C₂×C₅₃	1	1	yes	characteristic, complemented
90	#482	C₂×C₅₃	1	4	yes	complemented
91	#476	C₂×C₅₃	1	2	yes	complemented
92	#466	C₂×C₅₃	1	2	yes	complemented
93	C₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
94	C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
95	#427	C₂×C₅₃	1	2	yes	complemented
96	C₅₃	#3341	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×D₈	C₂²×C₅₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
99	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
100	#124	—	2	4	no	—
101	#121	—	2	2	no	—
102	#121	C₂²×C₅₃	1	4	yes	—
103	C₄×D₄	C₂²×C₅₃	1	2	yes	—
104	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	C₂²×C₅₃	1	4	yes	—
107	#107	C₂²×C₅₃	1	1	yes	characteristic
108	#107	C₂²×C₅₃	1	2	yes	—
109	#107	C₂²×C₅₃	1	2	yes	—
110	#106	C₂²×C₅₃	1	4	yes	—
111	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
112	#102	C₂²×C₅₃	1	2	yes	—
113	#102	C₂²×C₅₃	1	4	yes	—
114	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
115	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	D₄×C₅₃	1	1	yes	characteristic
119	C₂×D₄	C₂³×C₅₃	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
122	D₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₅₃	1	2	yes	—
126	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
129	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
130	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	C₂³×C₅₃	1	1	yes	characteristic
137	C₂³	—	2	2	no	—
138	C₂³	C₂×D₄×C₅₃	1	2	yes	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	#41×C₅₃	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	#135×C₅₃	1	2	yes	—
168	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
169	C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	C₅₃×#547	1	1	yes	characteristic
173	C₂	C₂×#135×C₅₃	1	1	yes	characteristic
174	C₂	C₅₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#555	C₂	semidirect
2. C₅₃×#500	C₂	semidirect
3. C₅₃×#482	C₂	semidirect
4. C₅₃×#476	C₂	semidirect
5. C₅₃×#466	C₂	semidirect
6. C₄×D₈×C₅₃	C₂	semidirect
7. C₈×D₄×C₅₃	C₂	semidirect
8. C₅₃×#427	C₂	semidirect
82. #3341	C₅₃	direct (N × H)
88. #555	C₂×C₅₃	semidirect
89. #500	C₂×C₅₃	semidirect
90. #482	C₂×C₅₃	semidirect
91. #476	C₂×C₅₃	semidirect
92. #466	C₂×C₅₃	semidirect
93. C₄×D₈	C₂×C₅₃	semidirect
94. C₈×D₄	C₂×C₅₃	semidirect
95. #427	C₂×C₅₃	semidirect
96. C₅₃	#3341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#555	2
C₅₃×#500	2
C₅₃×#482	2
C₅₃×#476	2
C₅₃×#466	2
C₄×D₈×C₅₃	2
C₈×D₄×C₅₃	2
C₅₃×#427	2
#3341	53

Subgroups — C53×#3341 #20272278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3341 #20272278