Subgroups — C₄×C₈×#88322 #27273438

Order 13568.

750 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#88322	C₂	1	2	yes	maximal
2	C₂×C₈×#88322	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₄	1	8	yes	—
5	C₂×C₄×#88322	C₂²	1	4	yes	—
6	C₈×#88322	C₄	1	16	yes	direct product factor
7	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂×C₄	1	8	yes	—
11	C₄×#88322	C₈	1	32	yes	direct product factor
12	C₄×#88322	C₂×C₄	1	16	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₅₃	C₈	1	16	yes	complemented
18	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
19	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₄²	1	8	yes	—
23	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
24	C₄²×C₅₃	C₂×C₈	1	16	yes	—
25	C₄²×C₅₃	C₂×C₈	1	8	yes	—
26	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
27	C₂×#88322	C₄²	1	16	yes	—
28	C₂×#88322	C₂×C₈	1	32	yes	—
29	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	—
31	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
32	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
33	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
35	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
36	C₈×C₅₃	C₄×C₈	1	16	yes	complemented
37	#88322	C₄×C₈	1	32	yes	direct product factor
38	C₄×C₈²	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₄³	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
41	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
42	C₄×C₅₃	C₈²	1	16	yes	complemented
43	C₄×C₅₃	C₂×C₄×C₈	1	8	yes	—
44	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
45	C₄²×C₈	—	53	2	no	—
46	C₄²×C₈	D₅₃	1	1	yes	characteristic
47	C₂×C₈²	—	53	4	no	—
48	C₂×C₅₃	C₂×C₈²	1	4	yes	—
49	C₂×C₅₃	C₄²×C₈	1	2	yes	—
50	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₈	—	53	8	no	—
52	C₂×C₄×C₈	—	53	4	no	—
53	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
54	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
55	C₄³	C₂×D₅₃	1	1	yes	characteristic
56	C₈²	—	53	16	no	—
57	C₅₃	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	53	8	no	—
59	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
60	C₂×C₄²	C₄×D₅₃	1	4	yes	—
61	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
62	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
63	C₄×C₈	—	53	16	no	—
64	C₄×C₈	—	53	32	no	—
65	C₄×C₈	#88322	1	16	yes	direct product factor
66	C₄×C₈	C₄×D₅₃	1	8	yes	—
67	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
68	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
69	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
70	C₂×C₈	—	53	16	no	—
71	C₂×C₈	—	53	32	no	—
72	C₂×C₈	C₂×#88322	1	16	yes	—
73	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
74	C₄²	C₈×D₅₃	1	16	yes	—
75	C₄²	C₂×#88322	1	8	yes	—
76	C₄²	C₂×C₄×D₅₃	1	4	yes	—
77	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
78	C₂×C₄	C₈×Dic₅₃	1	16	yes	—
79	C₂×C₄	C₄×#88322	1	8	yes	—
80	C₂×C₄	C₄²×D₅₃	1	4	yes	—
81	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
82	C₂×C₄	C₂×C₈×D₅₃	1	8	yes	—
83	C₂×C₄	C₂²×#88322	1	4	yes	—
84	C₈	—	53	32	no	—
85	C₈	C₄×#88322	1	16	yes	direct product factor
86	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
87	C₂²	C₂×C₈×Dic₅₃	1	4	yes	—
88	C₂²	C₂×C₄×#88322	1	2	yes	—
89	C₄	C₈×#88322	1	16	yes	direct product factor
90	C₄	C₄×C₈×D₅₃	1	8	yes	—
91	C₄	C₂×C₄×#88322	1	4	yes	—
92	C₂	C₂×C₈×#88322	1	4	yes	—
93	C₂	C₄×C₈×Dic₅₃	1	2	yes	—
94	C₂	C₄²×#88322	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄²×C₈	D₅₃
commutator, hypocenter	C₅₃	C₄×C₈²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#88322	C₄	direct (N × H)
11. C₄×#88322	C₈	direct (N × H)
17. C₄×C₈×C₅₃	C₈	semidirect
36. C₈×C₅₃	C₄×C₈	semidirect
37. #88322	C₄×C₈	direct (N × H)
42. C₄×C₅₃	C₈²	semidirect
57. C₅₃	C₄×C₈²	semidirect
65. C₄×C₈	#88322	direct (N × H)
85. C₈	C₄×#88322	direct (N × H)
89. C₄	C₈×#88322	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×C₈×#88322	2
C₄²×C₈×C₅₃	2
C₄×C₈²	53

Subgroups — C4×C8×#88322 #27273438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×#88322 #27273438