Subgroups — C₅₃×#33411 #27240923

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2913	C₂	1	2	yes	maximal
3	C₅₃×#2159	C₂	1	6	yes	maximal, complemented
4	C₅₃×#2158	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₃×#367	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#520	—	2	2	no	—
8	C₅₃×#504	C₄	1	8	yes	complemented
9	C₂×M₃₂×C₅₃	C₂²	1	3	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	3	yes	—
11	C₅₃×#428	—	2	6	no	—
12	C₅₃×#427	C₂²	1	6	yes	—
13	C₂×#107×C₅₃	C₂²	1	6	yes	—
14	C₅₃×#367	C₂²	1	3	yes	—
15	C₅₃×#367	C₂²	1	6	yes	—
16	C₅₃×#367	C₂²	1	1	yes	characteristic
17	C₅₃×#366	C₂²	1	6	yes	—
18	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
19	#131×C₅₃	C₂×C₄	1	4	yes	—
20	#131×C₅₃	Q₈	1	4	yes	—
21	C₂×M₁₆×C₅₃	D₄	1	3	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
23	C₄×Q₈×C₅₃	—	4	2	no	—
24	C₄×D₄×C₅₃	—	4	6	no	—
25	#117×C₅₃	—	2	6	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
27	M₃₂×C₅₃	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
31	#107×C₅₃	—	2	6	no	—
32	#107×C₅₃	C₂³	1	2	yes	—
33	#107×C₅₃	C₂³	1	6	yes	—
34	#106×C₅₃	—	2	6	no	—
35	#41×C₅₃	C₄⋊C₄	1	4	yes	—
36	#41×C₅₃	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
38	C₂×D₄×C₅₃	D₈	1	3	yes	—
39	C₂²×C₄×C₅₃	—	4	6	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
41	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₄⋊C₄×C₅₃	—	2	6	no	—
48	C₄⋊C₄×C₅₃	—	4	6	no	—
49	C₂²⋊C₄×C₅₃	—	4	6	no	—
50	C₄²×C₅₃	—	4	6	no	—
51	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
52	C₁₆×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂³×C₅₃	C₂×D₈	1	3	yes	—
54	Q₈×C₅₃	#107	1	4	yes	—
55	D₄×C₅₃	#107	1	12	yes	—
56	C₂×C₄×C₅₃	—	4	6	no	—
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	—	16	6	no	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
60	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
61	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
62	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
66	#33411	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	—	2	6	no	—
68	C₂²×C₅₃	C₂×#107	1	6	yes	—
69	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₅₃	—	8	2	no	—
71	C₄×C₅₃	—	8	6	no	—
72	C₄×C₅₃	C₂×#107	1	6	yes	—
73	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂×#504	C₂×C₅₃	1	1	yes	characteristic
76	#2913	C₂×C₅₃	1	2	yes	—
77	#2159	C₂×C₅₃	1	6	yes	complemented
78	#2158	C₂×C₅₃	1	3	yes	complemented
79	C₂×#367	C₂×C₅₃	1	3	yes	complemented
80	C₂×C₅₃	—	2	6	no	—
81	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
82	C₂×C₅₃	#3416	1	2	yes	—
83	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
84	#520	—	2	2	no	—
85	#504	C₄×C₅₃	1	8	yes	complemented
86	C₂×M₃₂	C₂²×C₅₃	1	3	yes	—
87	C₂²×C₁₆	C₂²×C₅₃	1	3	yes	—
88	#428	—	2	6	no	—
89	#427	C₂²×C₅₃	1	6	yes	—
90	C₂×#107	C₂²×C₅₃	1	6	yes	—
91	#367	C₂²×C₅₃	1	3	yes	—
92	#367	C₂²×C₅₃	1	6	yes	—
93	#367	C₂²×C₅₃	1	1	yes	characteristic
94	#366	C₂²×C₅₃	1	6	yes	—
95	C₅₃	#33411	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
97	#131	C₂×C₄×C₅₃	1	4	yes	—
98	#131	Q₈×C₅₃	1	4	yes	—
99	C₂×M₁₆	D₄×C₅₃	1	3	yes	—
100	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
101	C₄×Q₈	—	4	2	no	—
102	C₄×D₄	—	4	6	no	—
103	#117	—	2	6	no	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	M₃₂	C₂×C₄×C₅₃	1	12	yes	complemented
106	C₂×C₁₆	C₂×C₄×C₅₃	1	12	yes	complemented
107	C₂×C₁₆	C₂³×C₅₃	1	3	yes	—
108	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
109	#107	—	2	6	no	—
110	#107	C₂³×C₅₃	1	2	yes	—
111	#107	C₂³×C₅₃	1	6	yes	—
112	#106	—	2	6	no	—
113	#41	C₄⋊C₄×C₅₃	1	4	yes	—
114	#41	Q₁₆×C₅₃	1	4	yes	—
115	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
116	C₂×D₄	D₈×C₅₃	1	3	yes	—
117	C₂²×C₄	—	4	6	no	—
118	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
119	M₁₆	C₄⋊C₄×C₅₃	1	12	yes	—
120	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
121	C₂×C₈	C₂×D₄×C₅₃	1	3	yes	—
122	C₂×C₈	C₂×Q₈×C₅₃	1	6	yes	—
123	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	6	no	—
126	C₄⋊C₄	—	4	6	no	—
127	C₂²⋊C₄	—	4	6	no	—
128	C₄²	—	4	6	no	—
129	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
130	C₁₆	C₂²×C₄×C₅₃	1	6	yes	—
131	C₂³	C₂×D₈×C₅₃	1	3	yes	—
132	Q₈	#107×C₅₃	1	4	yes	—
133	D₄	#107×C₅₃	1	12	yes	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	6	no	—
136	C₂×C₄	—	16	6	no	—
137	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
138	C₂×C₄	C₂×D₈×C₅₃	1	3	yes	—
139	C₂×C₄	C₂×Q₁₆×C₅₃	1	6	yes	—
140	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
141	C₈	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
142	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
143	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
144	C₂²	—	2	6	no	—
145	C₂²	C₂×#107×C₅₃	1	6	yes	—
146	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
147	C₄	—	8	6	no	—
148	C₄	—	8	2	no	—
149	C₄	C₂×#107×C₅₃	1	6	yes	—
150	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
151	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
152	C₂	—	2	6	no	—
153	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic
154	C₂	C₅₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×#41×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2159	C₂	semidirect
4. C₅₃×#2158	C₂	semidirect
5. C₂×C₅₃×#367	C₂	semidirect
8. C₅₃×#504	C₄	semidirect
27. M₃₂×C₅₃	C₂×C₄	semidirect
28. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
66. #33411	C₅₃	direct (N × H)
77. #2159	C₂×C₅₃	semidirect
78. #2158	C₂×C₅₃	semidirect
79. C₂×#367	C₂×C₅₃	semidirect
85. #504	C₄×C₅₃	semidirect
95. C₅₃	#33411	direct (N × H)
105. M₃₂	C₂×C₄×C₅₃	semidirect
106. C₂×C₁₆	C₂×C₄×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×#504	2
C₅₃×#2913	2
C₅₃×#2159	2
C₅₃×#2158	2
C₂×C₅₃×#367	2
#33411	53

Subgroups — C53×#33411 #27240923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33411 #27240923