Subgroups — C₄×C₈×#88324 #28300029

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#88324	C₂	1	2	yes	maximal
2	C₂×C₈×#88324	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88324	C₄	1	8	yes	—
5	C₂×C₄×#88324	C₂²	1	4	yes	—
6	C₈×#88324	C₄	1	16	yes	direct product factor
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	C₂×C₄	1	8	yes	—
13	C₄×#88324	C₈	1	32	yes	direct product factor
14	C₄×#88324	C₂×C₄	1	16	yes	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
19	C₄×#88322	C₈	1	4	yes	complemented
20	C₈×Dic₅₃	C₈	1	8	yes	complemented
21	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
25	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₅₃	C₄²	1	2	yes	—
28	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
29	C₂×#88324	C₄²	1	16	yes	—
30	C₂×#88324	C₂×C₈	1	32	yes	—
31	C₂×C₈×C₅₃	C₄²	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₄²×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	C₂×C₈	1	2	yes	—
36	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
37	C₄×Dic₅₃	C₂×C₈	1	16	yes	—
38	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
39	C₂×#88322	C₄²	1	4	yes	—
40	C₂×#88322	C₂×C₈	1	4	yes	—
41	C₂×#88322	C₂×C₈	1	4	yes	—
42	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
44	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
49	C₂×Dic₅₃	C₄×C₈	1	8	yes	—
50	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
51	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
52	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
53	#88324	C₄×C₈	1	32	yes	direct product factor
54	C₈×C₅₃	C₄×C₈	1	8	yes	complemented
55	#88322	C₄×C₈	1	8	yes	complemented
56	C₄×C₈²	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	C₄³	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
59	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₄×C₅₃	C₈²	1	8	yes	complemented
62	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
63	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
64	Dic₅₃	C₈²	1	8	yes	complemented
65	Dic₅₃	C₂×C₄×C₈	1	8	yes	—
66	C₄²×C₈	—	53	2	no	—
67	C₄²×C₈	—	53	1	no	—
68	C₂×C₈²	—	53	4	no	—
69	C₂×C₅₃	C₂×C₈²	1	2	yes	—
70	C₂×C₅₃	C₂×C₈²	1	2	yes	—
71	C₂×C₅₃	C₄²×C₈	1	2	yes	—
72	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₈	—	53	8	no	—
74	C₂×C₄×C₈	—	53	4	no	—
75	C₂×C₄×C₈	—	53	2	no	—
76	C₂×C₄×C₈	—	53	2	no	—
77	C₂×C₄×C₈	—	53	1	no	—
78	C₂×C₄×C₈	#6157	1	1	yes	characteristic
79	C₄³	—	53	1	no	—
80	C₈²	—	53	16	no	—
81	C₅₃	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	53	8	no	—
83	C₂²×C₈	—	53	2	no	—
84	C₂²×C₈	C₂×#6157	1	2	yes	—
85	C₂×C₄²	—	53	4	no	—
86	C₂×C₄²	—	53	1	no	—
87	C₂×C₄²	—	53	1	no	—
88	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
89	C₄×C₈	—	53	16	no	—
90	C₄×C₈	—	53	32	no	—
91	C₄×C₈	—	53	8	no	—
92	C₄×C₈	—	53	8	no	—
93	C₄×C₈	—	53	4	no	—
94	C₄×C₈	#88324	1	4	yes	direct product factor
95	C₂²×C₄	—	53	2	no	—
96	C₂²×C₄	—	53	2	no	—
97	C₂²×C₄	C₄×#6157	1	2	yes	—
98	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
99	C₂×C₈	—	53	16	no	—
100	C₂×C₈	—	53	32	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	—	53	4	no	—
103	C₂×C₈	—	53	4	no	—
104	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
105	C₂×C₈	C₄×#6157	1	4	yes	—
106	C₂×C₈	C₂×#88324	1	4	yes	—
107	C₄²	—	53	16	no	—
108	C₄²	—	53	4	no	—
109	C₄²	—	53	4	no	—
110	C₄²	D₅₃⋊C₈	1	2	yes	—
111	C₄²	C₂×#88324	1	2	yes	—
112	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
113	C₂×C₄	—	53	4	no	—
114	C₂×C₄	—	53	8	no	—
115	C₂×C₄	—	53	8	no	—
116	C₂×C₄	—	53	4	no	—
117	C₂×C₄	C₈×#6157	1	8	yes	—
118	C₂×C₄	C₄×#88324	1	4	yes	—
119	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
120	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
121	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
122	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
123	C₈	—	53	32	no	—
124	C₈	—	53	8	no	—
125	C₈	C₄×#88324	1	8	yes	direct product factor
126	C₂²	C₄²×#6157	1	1	yes	characteristic
127	C₂²	C₂×C₈×#6157	1	4	yes	—
128	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
129	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
130	C₄	—	53	8	no	—
131	C₄	—	53	8	no	—
132	C₄	C₈×#88324	1	8	yes	direct product factor
133	C₄	C₂×C₄×#88324	1	2	yes	—
134	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
135	C₂	C₂×C₈×#88324	1	2	yes	—
136	C₂	C₈×D₅₃⋊C₈	1	2	yes	—
137	C₂	C₄×C₈×#6157	1	2	yes	—
138	C₂	C₄²×#88324	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#6157
commutator, hypocenter	C₅₃	C₄×C₈²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#88324	C₄	direct (N × H)
13. C₄×#88324	C₈	direct (N × H)
19. C₄×#88322	C₈	semidirect
20. C₈×Dic₅₃	C₈	semidirect
24. C₄×C₈×C₅₃	C₈	semidirect
53. #88324	C₄×C₈	direct (N × H)
54. C₈×C₅₃	C₄×C₈	semidirect
55. #88322	C₄×C₈	semidirect
61. C₄×C₅₃	C₈²	semidirect
64. Dic₅₃	C₈²	semidirect
81. C₅₃	C₄×C₈²	semidirect
94. C₄×C₈	#88324	direct (N × H)
125. C₈	C₄×#88324	direct (N × H)
132. C₄	C₈×#88324	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×#88324	2
C₂×C₈×#88324	2
C₄×C₈×Dic₅₃	2
C₄×C₈²	53

Subgroups — C4×C8×#88324 #28300029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×#88324 #28300029