Subgroups — C₄×#16810469 #27305164

Order 13568.

2302 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×#16810469	C₂	1	4	yes	maximal
2	C₂×C₄²×Dic₅₃	C₂	1	2	yes	maximal
3	C₄×#13133124	C₂	1	8	yes	maximal
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₅₃	C₂²	1	8	yes	—
6	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
7	#16810469	C₄	1	32	yes	direct product factor
8	C₂×#13133124	C₂²	1	16	yes	—
9	C₄²×Dic₅₃	C₄	1	16	yes	complemented
10	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₄	1	8	yes	complemented
14	C₂³×Dic₅₃	C₂³	1	8	yes	—
15	C₂×C₄×Dic₅₃	—	2	32	no	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	32	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	16	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	32	yes	—
19	#13133124	C₂×C₄	1	64	yes	—
20	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	16	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
29	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	—	2	16	no	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
38	C₂²×C₄×C₅₃	#41	1	16	yes	—
39	C₂²×Dic₅₃	—	2	32	no	—
40	C₂²×Dic₅₃	C₂²×C₄	1	32	yes	—
41	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
42	C₄²×C₅₃	—	2	32	no	—
43	C₄²×C₅₃	C₄²	1	16	yes	—
44	C₄²×C₅₃	C₄²	1	32	yes	complemented
45	C₄×Dic₅₃	—	2	64	no	—
46	C₄×Dic₅₃	C₄²	1	64	yes	complemented
47	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
48	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
49	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₂³×C₅₃	C₂×#41	1	8	yes	—
51	C₂×C₄×C₅₃	—	2	16	no	—
52	C₂×C₄×C₅₃	—	2	32	no	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
54	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
55	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
58	C₂×C₄×C₅₃	#117	1	32	yes	—
59	C₂×C₄×C₅₃	#117	1	16	yes	—
60	C₂×Dic₅₃	—	2	32	no	—
61	C₂×Dic₅₃	—	2	64	no	—
62	C₂×Dic₅₃	C₂×C₄²	1	32	yes	—
63	C₄×#376	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
65	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
66	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
67	C₂²×C₅₃	C₂×#117	1	8	yes	—
68	C₂²×C₅₃	C₂×#117	1	4	yes	—
69	C₂²×C₅₃	C₄×#41	1	16	yes	—
70	C₄×C₅₃	—	2	16	no	—
71	C₄×C₅₃	C₄³	1	8	yes	—
72	C₄×C₅₃	#376	1	16	yes	complemented
73	Dic₅₃	—	2	32	no	—
74	C₂×#376	—	53	4	no	—
75	C₂×C₄³	—	53	2	no	—
76	C₂×C₄³	D₅₃	1	1	yes	characteristic
77	C₄×#95	—	53	8	no	—
78	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#376	1	2	yes	—
80	C₂×C₅₃	C₄×#117	1	4	yes	—
81	C₂×C₅₃	C₄×#117	1	8	yes	—
82	C₂²×C₄²	—	53	4	no	—
83	C₂²×C₄²	—	53	8	no	—
84	C₂²×C₄²	C₂×D₅₃	1	4	yes	—
85	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
86	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
87	#376	—	53	32	no	—
88	C₂×#95	—	53	16	no	—
89	C₄³	—	53	16	no	—
90	C₄³	Dic₅₃	1	8	yes	complemented
91	C₅₃	C₄×#376	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	—	53	8	no	—
93	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
94	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
95	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
96	C₂×C₄²	—	2	16	no	—
97	C₂×C₄²	—	53	16	no	—
98	C₂×C₄²	—	53	32	no	—
99	C₂×C₄²	—	53	32	no	—
100	C₂×C₄²	—	106	32	no	—
101	C₂×C₄²	C₄×D₅₃	1	8	yes	—
102	C₂×C₄²	C₄×D₅₃	1	16	yes	—
103	C₂×C₄²	C₂×Dic₅₃	1	16	yes	—
104	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
105	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
106	#95	—	53	64	no	—
107	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
108	C₂²×C₄	—	2	16	no	—
109	C₂²×C₄	—	53	32	no	—
110	C₂²×C₄	—	53	16	no	—
111	C₂²×C₄	—	106	32	no	—
112	C₂²×C₄	C₂×C₄×D₅₃	1	16	yes	—
113	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
115	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
116	C₂²×C₄	#11719945	1	16	yes	—
117	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
119	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
120	C₄²	—	2	32	no	—
121	C₄²	—	53	64	no	—
122	C₄²	—	106	64	no	—
123	C₄²	C₄×Dic₅₃	1	16	yes	—
124	C₄²	C₄×Dic₅₃	1	32	yes	complemented
125	C₂³	C₂²×C₄×D₅₃	1	2	yes	—
126	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
127	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
128	C₂³	C₂×#11719945	1	8	yes	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	2	32	no	—
131	C₂×C₄	—	53	32	no	—
132	C₂×C₄	—	106	32	no	—
133	C₂×C₄	—	106	64	no	—
134	C₂×C₄	C₄²×D₅₃	1	16	yes	—
135	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
136	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
137	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
138	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
139	C₂×C₄	#13133174	1	32	yes	—
140	C₂×C₄	#13133176	1	16	yes	—
141	C₂²	C₂×C₄²×D₅₃	1	4	yes	—
142	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
143	C₂²	C₂²×C₄×Dic₅₃	1	2	yes	—
144	C₂²	C₂×#13133174	1	8	yes	—
145	C₂²	C₂×#13133176	1	4	yes	—
146	C₂²	C₄×#11719945	1	16	yes	—
147	C₄	—	2	16	no	—
148	C₄	—	106	32	no	—
149	C₄	C₄²×Dic₅₃	1	8	yes	—
150	C₄	#16810469	1	16	yes	direct product factor
151	C₂	C₂×C₄²×Dic₅₃	1	1	yes	characteristic
152	C₂	C₂×#16810469	1	2	yes	—
153	C₂	C₄×#13133176	1	4	yes	—
154	C₂	C₄×#13133174	1	8	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄³
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₄×#376
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. #16810469	C₄	direct (N × H)
9. C₄²×Dic₅₃	C₄	semidirect
13. C₄³×C₅₃	C₄	semidirect
44. C₄²×C₅₃	C₄²	semidirect
46. C₄×Dic₅₃	C₄²	semidirect
72. C₄×C₅₃	#376	semidirect
90. C₄³	Dic₅₃	semidirect
91. C₅₃	C₄×#376	semidirect
124. C₄²	C₄×Dic₅₃	semidirect
150. C₄	#16810469	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×#16810469	2
C₂×C₄²×Dic₅₃	2
C₄×#13133124	2
C₂×C₄³×C₅₃	2
C₄×#376	53

Subgroups — C4×#16810469 #27305164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810469 #27305164