Subgroups — C₂²×#16810398 #27536717

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810398	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂²×#11719908	C₄	1	8	yes	complemented
8	C₂²×#11719908	C₂²	1	6	yes	—
9	C₂×C₈×Dic₅₃	C₄	1	6	yes	—
10	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
11	#16810398	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
15	C₂³×Dic₅₃	C₈	1	2	yes	—
16	C₂²×#88322	C₈	1	8	yes	—
17	C₂²×#88322	C₂×C₄	1	6	yes	—
18	C₂×C₄×Dic₅₃	C₈	1	6	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
20	C₂×#11719908	C₂×C₄	1	24	yes	complemented
21	C₂×#11719908	C₂³	1	4	yes	—
22	C₈×Dic₅₃	C₂×C₄	1	12	yes	—
23	C₈×Dic₅₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	24	yes	complemented
30	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
31	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₅₃	—	2	8	no	—
37	C₂²×Dic₅₃	C₂×C₈	1	12	yes	—
38	C₂×C₈×C₅₃	C₄²	1	12	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
42	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂×C₈	1	12	yes	—
44	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
45	C₂×#88322	C₂×C₈	1	24	yes	—
46	C₂×#88322	C₂²×C₄	1	4	yes	—
47	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
48	#11719908	C₂²×C₄	1	8	yes	complemented
49	C₂³×C₅₃	C₄×C₈	1	4	yes	—
50	C₂³×C₅₃	M₃₂	1	8	yes	—
51	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
52	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
54	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	24	no	—
58	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
59	C₈×C₅₃	C₂×C₄²	1	6	yes	—
60	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	#88322	C₂²×C₈	1	8	yes	—
63	C₂²×#346	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	#346	1	16	yes	complemented
65	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
66	C₂²×C₅₃	C₂×M₃₂	1	12	yes	—
67	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
69	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
70	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
71	Dic₅₃	—	2	8	no	—
72	C₂³×C₁₆	—	53	1	no	—
73	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
74	C₂²×C₄×C₈	—	53	1	no	—
75	C₂×#346	—	53	12	no	—
76	C₂×C₅₃	C₂×#346	1	12	yes	complemented
77	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
78	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
79	C₂³×C₈	—	53	1	no	—
80	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂²×C₄²	—	53	1	no	—
82	C₂²×C₁₆	—	53	8	no	—
83	C₂²×C₁₆	—	53	6	no	—
84	C₂²×C₁₆	Dic₅₃	1	8	yes	complemented
85	C₂²×C₁₆	C₂×D₅₃	1	6	yes	—
86	C₂×C₄×C₈	—	53	6	no	—
87	C₂×C₄×C₈	—	53	6	no	—
88	#346	—	53	16	no	—
89	C₅₃	C₂²×#346	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	53	2	no	—
91	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
92	C₂²×C₈	—	53	6	no	—
93	C₂²×C₈	—	53	8	no	—
94	C₂²×C₈	C₄×D₅₃	1	4	yes	—
95	C₂²×C₈	C₄×D₅₃	1	3	yes	—
96	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
97	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
98	C₂×C₄²	—	53	6	no	—
99	C₂×C₄²	—	53	6	no	—
100	C₂×C₁₆	—	53	24	no	—
101	C₂×C₁₆	—	53	4	no	—
102	C₂×C₁₆	C₂×Dic₅₃	1	24	yes	complemented
103	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
104	C₄×C₈	—	53	12	no	—
105	C₄×C₈	—	53	4	no	—
106	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
107	C₂²×C₄	—	53	12	no	—
108	C₂²×C₄	—	106	8	no	—
109	C₂²×C₄	C₈×D₅₃	1	3	yes	—
110	C₂²×C₄	C₈×D₅₃	1	4	yes	—
111	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
112	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
113	C₂×C₈	—	53	24	no	—
114	C₂×C₈	—	53	4	no	—
115	C₂×C₈	C₄×Dic₅₃	1	12	yes	—
116	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
117	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
118	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
119	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
120	C₄²	—	53	12	no	—
121	C₄²	—	53	4	no	—
122	C₁₆	—	53	8	no	—
123	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
124	C₂³	C₈×Dic₅₃	1	4	yes	—
125	C₂³	#13133156	1	8	yes	—
126	C₂³	C₂×C₈×D₅₃	1	3	yes	—
127	C₂×C₄	—	53	8	no	—
128	C₂×C₄	—	106	24	no	—
129	C₂×C₄	C₈×Dic₅₃	1	12	yes	—
130	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
131	C₂×C₄	C₂×C₈×D₅₃	1	3	yes	—
132	C₂×C₄	C₂×C₈×D₅₃	1	6	yes	—
133	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
134	C₈	—	53	8	no	—
135	C₈	C₂×C₄×Dic₅₃	1	6	yes	—
136	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
137	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
138	C₂²	#16810398	1	16	yes	direct product factor
139	C₂²	C₂×C₈×Dic₅₃	1	6	yes	—
140	C₂²	C₂×#13133156	1	12	yes	—
141	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
142	C₄	—	106	8	no	—
143	C₄	C₂×C₈×Dic₅₃	1	6	yes	—
144	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
145	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
146	C₂	C₂×#16810398	1	12	yes	direct product factor
147	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
148	C₂	C₂²×#13133156	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×#346
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16810398	C₂	direct (N × H)
7. C₂²×#11719908	C₄	semidirect
11. #16810398	C₂²	direct (N × H)
13. C₂²×C₁₆×C₅₃	C₄	semidirect
20. C₂×#11719908	C₂×C₄	semidirect
29. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
47. C₁₆×C₅₃	C₂²×C₄	semidirect
48. #11719908	C₂²×C₄	semidirect
64. C₂²×C₅₃	#346	semidirect
76. C₂×C₅₃	C₂×#346	semidirect
84. C₂²×C₁₆	Dic₅₃	semidirect
89. C₅₃	C₂²×#346	semidirect
102. C₂×C₁₆	C₂×Dic₅₃	semidirect
123. C₁₆	C₂²×Dic₅₃	semidirect
138. C₂²	#16810398	direct (N × H)
146. C₂	C₂×#16810398	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂²×C₈×Dic₅₃	2
C₂×#16810398	2
C₂³×C₁₆×C₅₃	2
C₂²×#346	53

Subgroups — C22×#16810398 #27536717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810398 #27536717