Subgroups — C₅₃×#3177 #20272114

Order 6784.

338 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#541	C₂	1	2	yes	maximal
2	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#491	C₂	1	2	yes	maximal, complemented
5	C₅₃×#470	C₂	1	2	yes	maximal
6	C₅₃×#467	C₂	1	2	yes	maximal
7	C₅₃×#441	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	#126×C₅₃	—	2	4	no	—
16	#125×C₅₃	—	2	4	no	—
17	#125×C₅₃	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
19	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
20	#117×C₅₃	C₂²	1	1	yes	characteristic
21	#107×C₅₃	C₂²	1	2	yes	—
22	#106×C₅₃	C₂²	1	2	yes	—
23	#103×C₅₃	C₂²	1	4	yes	—
24	#103×C₅₃	C₂²	1	4	yes	—
25	#97×C₅₃	C₂²	1	1	yes	characteristic
26	#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	Q₁₆×C₅₃	—	2	4	no	—
32	Q₁₆×C₅₃	—	2	4	no	—
33	M₁₆×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₅₃	—	2	4	no	—
46	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
47	C₄²×C₅₃	—	2	4	no	—
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	#41	1	2	yes	—
64	C₈×C₅₃	#41	1	2	yes	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	4	4	no	—
72	C₄×C₅₃	—	4	4	no	—
73	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
74	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
75	#3177	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#532	1	1	yes	characteristic
78	C₂×C₅₃	#578	1	2	yes	—
79	#541	C₂×C₅₃	1	2	yes	—
80	#499	C₂×C₅₃	1	1	yes	characteristic, complemented
81	#494	C₂×C₅₃	1	1	yes	characteristic, complemented
82	#491	C₂×C₅₃	1	2	yes	complemented
83	#470	C₂×C₅₃	1	2	yes	—
84	#467	C₂×C₅₃	1	2	yes	—
85	#441	C₂×C₅₃	1	2	yes	complemented
86	C₄×Q₁₆	C₂×C₅₃	1	2	yes	complemented
87	#405	C₂×C₅₃	1	1	yes	characteristic
88	C₅₃	#3177	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×Q₁₆	C₂²×C₅₃	1	2	yes	—
90	C₂×Q₁₆	C₂²×C₅₃	1	2	yes	—
91	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
93	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
94	#126	—	2	4	no	—
95	#125	—	2	4	no	—
96	#125	C₂²×C₅₃	1	2	yes	—
97	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
98	C₄×Q₈	C₂²×C₅₃	1	4	yes	—
99	#117	C₂²×C₅₃	1	1	yes	characteristic
100	#107	C₂²×C₅₃	1	2	yes	—
101	#106	C₂²×C₅₃	1	2	yes	—
102	#103	C₂²×C₅₃	1	4	yes	—
103	#103	C₂²×C₅₃	1	4	yes	—
104	#97	C₂²×C₅₃	1	1	yes	characteristic
105	#97	C₂²×C₅₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
108	C₂×Q₈	C₂³×C₅₃	1	4	yes	—
109	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
117	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
123	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
126	C₄²	—	2	4	no	—
127	C₄²	C₂³×C₅₃	1	1	yes	characteristic
128	C₄²	C₂³×C₅₃	1	1	yes	characteristic
129	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
137	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#41×C₅₃	1	2	yes	—
143	C₈	#41×C₅₃	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	4	4	no	—
152	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
153	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	C₅₃×#532	1	1	yes	characteristic
156	C₂	C₅₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₂³×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#499	C₂	semidirect
3. C₅₃×#494	C₂	semidirect
4. C₅₃×#491	C₂	semidirect
7. C₅₃×#441	C₂	semidirect
8. C₄×Q₁₆×C₅₃	C₂	semidirect
75. #3177	C₅₃	direct (N × H)
80. #499	C₂×C₅₃	semidirect
81. #494	C₂×C₅₃	semidirect
82. #491	C₂×C₅₃	semidirect
85. #441	C₂×C₅₃	semidirect
86. C₄×Q₁₆	C₂×C₅₃	semidirect
88. C₅₃	#3177	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×#541	2
C₅₃×#499	2
C₅₃×#494	2
C₅₃×#491	2
C₅₃×#470	2
C₅₃×#467	2
C₅₃×#441	2
C₄×Q₁₆×C₅₃	2
C₅₃×#405	2
#3177	53

Subgroups — C53×#3177 #20272114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3177 #20272114