Subgroups — C₄×#16810400 #27298919

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810400	C₂	1	4	yes	maximal
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133127	C₄	1	2	yes	—
7	C₂×#13133127	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	#16810400	C₄	1	16	yes	direct product factor
11	#16810399	C₄	1	2	yes	complemented
12	C₄×#11719908	C₄	1	2	yes	complemented
13	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#346	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	—	2	2	no	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	#13133157	C₂×C₄	1	8	yes	—
21	C₂×#11719908	C₂×C₄	1	4	yes	—
22	#13133127	—	2	4	no	—
23	#13133127	C₂×C₄	1	8	yes	—
24	C₄×#88322	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₄²	1	1	yes	characteristic
44	C₄²×C₅₃	C₄²	1	1	yes	characteristic
45	C₄²×C₅₃	M₁₆	1	2	yes	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	—	4	4	no	—
48	C₄×Dic₅₃	C₄²	1	4	yes	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	C₄²	1	4	yes	—
52	C₁₆×C₅₃	C₄²	1	8	yes	complemented
53	#11719908	C₄²	1	8	yes	complemented
54	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	#97	1	2	yes	—
57	C₂×C₄×C₅₃	#97	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×Dic₅₃	—	2	4	no	—
63	C₂×Dic₅₃	—	4	4	no	—
64	C₈×C₅₃	#97	1	2	yes	—
65	C₈×C₅₃	#97	1	2	yes	—
66	C₈×C₅₃	C₂×C₄²	1	2	yes	—
67	C₈×C₅₃	C₂×C₄²	1	2	yes	—
68	#88322	—	2	4	no	—
69	C₄×#347	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
72	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#347	1	4	yes	complemented
75	C₄×C₅₃	C₄³	1	1	yes	characteristic
76	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
77	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
78	Dic₅₃	—	4	4	no	—
79	C₂×#347	—	53	4	no	—
80	C₄×M₃₂	—	53	1	no	—
81	C₄×M₃₂	D₅₃	1	1	yes	characteristic
82	C₄×#97	—	53	1	no	—
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#347	1	2	yes	—
86	C₂×M₃₂	—	53	2	no	—
87	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
88	C₂×#97	—	53	2	no	—
89	C₂×#97	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄³	—	53	1	no	—
93	#347	—	53	16	no	—
94	#346	—	53	2	no	—
95	#346	Dic₅₃	1	2	yes	complemented
96	C₄×C₁₆	—	53	2	no	—
97	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
98	C₅₃	C₄×#347	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	2	no	—
100	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×C₄²	—	53	2	no	—
103	C₂×C₄²	—	106	2	no	—
104	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
105	M₃₂	—	53	8	no	—
106	M₃₂	C₄×D₅₃	1	8	yes	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
109	#97	—	53	8	no	—
110	#97	—	106	4	no	—
111	C₄×C₈	—	106	2	no	—
112	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂²×C₄	—	106	2	no	—
117	C₂²×C₄	#11719912	1	2	yes	—
118	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₈	—	53	4	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	#11719912	1	4	yes	—
123	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
124	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
127	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
128	C₄²	—	53	4	no	—
129	C₄²	—	106	2	no	—
130	C₄²	—	212	4	no	—
131	C₄²	#11719912	1	2	yes	—
132	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
133	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
134	C₁₆	—	53	8	no	—
135	C₁₆	C₄×Dic₅₃	1	8	yes	complemented
136	C₂³	C₂×#11719912	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	—	212	4	no	—
140	C₂×C₄	#13133127	1	2	yes	—
141	C₂×C₄	#13133127	1	2	yes	—
142	C₂×C₄	C₄²×D₅₃	1	2	yes	—
143	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂×#11719912	1	2	yes	—
145	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
146	C₈	—	106	4	no	—
147	C₈	#13133127	1	2	yes	—
148	C₈	#13133127	1	2	yes	—
149	C₈	C₄²×D₅₃	1	2	yes	—
150	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂×#13133127	1	1	yes	characteristic
153	C₂²	C₄×#11719912	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	212	4	no	—
156	C₄	#16810400	1	4	yes	direct product factor
157	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
158	C₄	C₂×#13133127	1	1	yes	characteristic
159	C₄	C₄×#11719912	1	2	yes	—
160	C₂	—	2	2	no	—
161	C₂	C₄×#13133127	1	1	yes	characteristic
162	C₂	C₂×#16810400	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16810400	C₄	direct (N × H)
11. #16810399	C₄	semidirect
12. C₄×#11719908	C₄	semidirect
15. C₅₃×#346	C₄	semidirect
16. C₄×C₁₆×C₅₃	C₄	semidirect
52. C₁₆×C₅₃	C₄²	semidirect
53. #11719908	C₄²	semidirect
74. C₄×C₅₃	#347	semidirect
95. #346	Dic₅₃	semidirect
97. C₄×C₁₆	Dic₅₃	semidirect
98. C₅₃	C₄×#347	semidirect
135. C₁₆	C₄×Dic₅₃	semidirect
156. C₄	#16810400	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×#16810400	2
C₄×#13133157	2
C₄×#13133127	2
C₄×M₃₂×C₅₃	2
C₄×#347	53

Subgroups — C4×#16810400 #27298919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810400 #27298919