Subgroups — C₄×#16810345 #27273441

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	4	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88322	C₄	1	4	yes	—
6	C₂×C₄×#88322	C₄	1	4	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810345	C₄	1	16	yes	direct product factor
10	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂×C₄	1	4	yes	—
14	C₂²×#88322	C₂×C₄	1	4	yes	—
15	C₄×#88322	—	2	8	no	—
16	C₄×#88322	C₈	1	16	yes	complemented
17	C₄×#88322	C₂×C₄	1	8	yes	—
18	C₄×#88322	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₃	—	2	8	no	—
25	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	—	2	8	no	—
31	C₂×C₈×C₅₃	C₄²	1	8	yes	—
32	C₄²×C₅₃	C₂×C₈	1	8	yes	—
33	C₄²×C₅₃	M₁₆	1	8	yes	—
34	C₄²×C₅₃	M₁₆	1	8	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂×#88322	—	2	8	no	—
37	C₂×#88322	C₄²	1	8	yes	—
38	C₂×#88322	C₄²	1	8	yes	—
39	C₂×#88322	C₂×C₈	1	16	yes	—
40	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
42	C₂×C₄×C₅₃	#97	1	8	yes	—
43	C₂×C₄×C₅₃	#97	1	8	yes	—
44	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
47	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
49	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
50	C₈×C₅₃	—	2	8	no	—
51	#88322	—	2	8	no	—
52	#88322	C₄×C₈	1	16	yes	complemented
53	C₄×#322	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	C₄³	1	1	yes	characteristic
55	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
56	C₂²×C₅₃	C₂×#97	1	2	yes	—
57	C₂²×C₅₃	C₂×#97	1	2	yes	—
58	C₄×C₅₃	#322	1	16	yes	complemented
59	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
60	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
61	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
62	C₄²×C₈	—	53	1	no	—
63	C₄²×C₈	—	53	1	no	—
64	C₄²×C₈	D₅₃	1	1	yes	characteristic
65	C₂×#322	—	53	4	no	—
66	C₂×C₅₃	C₂×#322	1	4	yes	—
67	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
68	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
69	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
70	C₂×C₄×C₈	—	53	4	no	—
71	C₂×C₄×C₈	—	53	2	no	—
72	C₂×C₄×C₈	—	53	2	no	—
73	C₂×C₄×C₈	—	53	4	no	—
74	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
75	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
76	C₄³	C₂×D₅₃	1	1	yes	characteristic
77	#322	—	53	16	no	—
78	C₅₃	C₄×#322	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	53	4	no	—
80	C₂²×C₈	—	53	4	no	—
81	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
82	C₂×C₄²	C₄×D₅₃	1	2	yes	—
83	C₂×C₄²	C₄×D₅₃	1	2	yes	—
84	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
85	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
86	C₄×C₈	—	2	8	no	—
87	C₄×C₈	—	53	8	no	—
88	C₄×C₈	—	53	8	no	—
89	C₄×C₈	—	53	16	no	—
90	C₄×C₈	—	106	8	no	—
91	C₄×C₈	C₄×D₅₃	1	8	yes	—
92	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
93	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	53	8	no	—
98	C₂×C₈	—	53	16	no	—
99	C₂×C₈	—	53	8	no	—
100	C₂×C₈	—	106	8	no	—
101	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
102	C₄²	C₈×D₅₃	1	8	yes	—
103	C₄²	#11719912	1	8	yes	—
104	C₄²	#11719917	1	8	yes	—
105	C₄²	C₂×C₄×D₅₃	1	4	yes	—
106	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	C₈×Dic₅₃	1	8	yes	—
108	C₂×C₄	#13133127	1	8	yes	—
109	C₂×C₄	#13133128	1	8	yes	—
110	C₂×C₄	C₄²×D₅₃	1	2	yes	—
111	C₂×C₄	C₄²×D₅₃	1	2	yes	—
112	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
113	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
114	C₂×C₄	C₂×#11719912	1	4	yes	—
115	C₂×C₄	C₂×#11719917	1	4	yes	—
116	C₈	—	2	8	no	—
117	C₈	—	53	16	no	—
118	C₈	—	106	8	no	—
119	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
120	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
121	C₂²	C₂×#13133127	1	2	yes	—
122	C₂²	C₂×#13133128	1	2	yes	—
123	C₄	#16810345	1	16	yes	direct product factor
124	C₄	C₄×C₈×D₅₃	1	4	yes	—
125	C₄	C₄×#11719912	1	4	yes	—
126	C₄	C₄×#11719917	1	4	yes	—
127	C₂	C₂×#16810345	1	4	yes	—
128	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
129	C₂	C₄×#13133127	1	1	yes	characteristic
130	C₂	C₄×#13133128	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	C₄×#322
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
9. #16810345	C₄	direct (N × H)
16. C₄×#88322	C₈	semidirect
52. #88322	C₄×C₈	semidirect
58. C₄×C₅₃	#322	semidirect
78. C₅₃	C₄×#322	semidirect
123. C₄	#16810345	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×#88322	2
C₂×#16810345	2
C₄²×C₈×C₅₃	2
C₄×#322	53

Subgroups — C4×#16810345 #27273441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810345 #27273441