Subgroups — C₄×#16810449 #27299491

Order 13568.

366 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×#16810449	C₂	1	2	yes	maximal
2	C₄×#13133151	C₂	1	2	yes	maximal
3	#20272807	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	C₂²	1	4	yes	—
6	C₄×C₄⋊Dic₅₃	—	2	2	no	—
7	#16810449	C₄	1	8	yes	direct product factor
8	#16810386	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₅₃	—	2	4	no	—
14	C₂×C₄×Dic₅₃	—	4	2	no	—
15	#13133151	—	2	4	no	—
16	#13133151	C₂×C₄	1	8	yes	—
17	#13133124	—	4	2	no	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	4	4	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	#41	1	2	yes	—
37	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₄²×C₅₃	Q₁₆	1	2	yes	—
39	C₄⋊Dic₅₃	—	2	8	no	—
40	C₄⋊Dic₅₃	—	4	4	no	—
41	C₄×Dic₅₃	—	8	4	no	—
42	C₁₆×C₅₃	—	2	2	no	—
43	C₁₆×C₅₃	C₄²	1	4	yes	complemented
44	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	#107	1	2	yes	—
47	C₂×C₄×C₅₃	#107	1	2	yes	—
48	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#135	1	2	yes	—
54	C₂×Dic₅₃	—	4	8	no	—
55	C₂×Dic₅₃	—	8	4	no	—
56	C₂×Dic₅₃	—	8	4	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₅₃	#117	1	1	yes	characteristic
60	C₈×C₅₃	C₄×Q₈	1	2	yes	—
61	C₄×#366	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
63	C₂²×C₅₃	C₄×D₈	1	2	yes	—
64	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
66	C₂²×C₅₃	#508	1	2	yes	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	#366	1	4	yes	complemented
69	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₅₃	#427	1	1	yes	characteristic
71	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
72	Dic₅₃	—	8	8	no	—
73	C₂×#366	—	53	2	no	—
74	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
75	C₄×#107	—	53	2	no	—
76	#1380	—	53	2	no	—
77	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
79	C₂×C₅₃	#2158	1	1	yes	characteristic
80	C₂×C₅₃	C₄×D₁₆	1	2	yes	—
81	C₂×C₅₃	C₄×Q₃₂	1	2	yes	—
82	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂×#107	—	53	4	no	—
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	106	2	no	—
87	#366	—	53	8	no	—
88	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
89	#340	—	106	2	no	—
90	C₅₃	C₄×#366	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	106	4	no	—
94	C₂×C₄²	—	212	2	no	—
95	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
98	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
100	#107	—	53	8	no	—
101	#107	—	106	4	no	—
102	C₄×C₈	Dic₁₀₆	1	2	yes	—
103	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
104	#95	—	212	2	no	—
105	C₂²×C₄	—	212	4	no	—
106	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
107	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
111	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
113	C₂×C₈	#11719945	1	2	yes	—
114	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
115	C₄⋊C₄	—	106	8	no	—
116	C₄⋊C₄	—	212	4	no	—
117	C₄²	—	424	4	no	—
118	C₄²	Dic₂₁₂	1	2	yes	—
119	C₄²	C₄⋊Dic₅₃	1	2	yes	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₄×Dic₅₃	1	4	yes	complemented
122	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	424	4	no	—
125	C₂×C₄	—	424	4	no	—
126	C₂×C₄	—	212	8	no	—
127	C₂×C₄	#13133151	1	2	yes	—
128	C₂×C₄	#13133151	1	2	yes	—
129	C₂×C₄	D₈₄₈	1	2	yes	—
130	C₂×C₄	Dic₄₂₄	1	2	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
133	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
134	C₂×C₄	#13133239	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
137	C₈	#13133176	1	1	yes	characteristic
138	C₈	C₄×Dic₁₀₆	1	2	yes	—
139	C₂²	C₂×#13133151	1	1	yes	characteristic
140	C₂²	C₄×D₄₂₄	1	2	yes	—
141	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
142	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
143	C₂²	#16811055	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	424	8	no	—
146	C₄	#16810449	1	4	yes	direct product factor
147	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
148	C₄	#16810661	1	1	yes	characteristic
149	C₄	C₄×Dic₂₁₂	1	2	yes	—
150	C₂	C₄×#13133151	1	1	yes	characteristic
151	C₂	C₂×#16810449	1	1	yes	characteristic
152	C₂	#20276499	1	1	yes	characteristic
153	C₂	C₄×D₈₄₈	1	2	yes	—
154	C₂	C₄×Dic₄₂₄	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₄×#366
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. #16810449	C₄	direct (N × H)
12. C₄×C₁₆×C₅₃	C₄	semidirect
43. C₁₆×C₅₃	C₄²	semidirect
68. C₄×C₅₃	#366	semidirect
88. C₄×C₁₆	Dic₅₃	semidirect
90. C₅₃	C₄×#366	semidirect
121. C₁₆	C₄×Dic₅₃	semidirect
146. C₄	#16810449	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810449	2
C₄×#13133151	2
#20272807	2
C₂×C₄×C₁₆×C₅₃	2
C₄×#366	53

Subgroups — C4×#16810449 #27299491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810449 #27299491