Subgroups — C₂²×#87812 #13107340

Order 1600.

1032 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6045	C₂	1	1	yes	maximal, characteristic
3	C₂×#87812	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#6029	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	—	2	3	no	—
6	C₂×C₅²×Q₈	C₂²	1	6	yes	—
7	C₂²×C₄×C₅²	—	2	1	no	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×#1039	—	2	1	no	—
10	C₂×#6045	—	2	3	no	—
11	C₂×#6045	C₂²	1	6	yes	—
12	#87812	C₂²	1	16	yes	direct product factor
13	C₂×#6029	C₂²	1	6	yes	—
14	C₂²×#689	—	5	6	no	maximal
15	C₂³×C₅²	D₄	1	1	yes	characteristic
16	C₅²×Q₈	—	2	6	no	—
17	C₅²×Q₈	C₂³	1	4	yes	—
18	C₂×C₄×C₅²	—	2	6	no	—
19	C₂×C₄×C₅²	D₄	1	3	yes	—
20	C₂×C₄×C₅²	C₂³	1	3	yes	—
21	C₂×#1039	—	2	6	no	—
22	#6045	—	2	6	no	—
23	#6045	C₂³	1	4	yes	—
24	#6029	C₂³	1	4	yes	—
25	C₂²×C₅×Q₈	D₅	1	6	yes	—
26	C₂²×Dic₁₀	—	5	6	no	—
27	C₂×#689	—	5	72	no	—
28	C₂²×#168	—	5	6	no	—
29	C₂²×C₅²	Q₁₆	1	4	yes	complemented
30	C₂²×C₅²	C₂×D₄	1	3	yes	—
31	C₄×C₅²	—	2	4	no	—
32	C₄×C₅²	C₂×D₄	1	3	yes	—
33	C₄×C₅²	C₂⁴	1	1	yes	characteristic
34	#1039	—	2	4	no	—
35	C₂×C₅×Q₈	—	2	18	no	—
36	C₂×C₅×Q₈	C₂×D₅	1	36	yes	—
37	C₂²×C₄×C₅	—	2	6	no	—
38	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
39	C₂²×Dic₅	—	10	6	no	—
40	C₂×Dic₁₀	—	5	36	no	—
41	C₂×Dic₁₀	—	10	18	no	—
42	#689	—	5	96	no	—
43	C₂×#168	—	5	36	no	—
44	C₂²×Q₁₆	—	25	1	no	Sylow
45	C₂×C₅²	C₂×Q₁₆	1	6	yes	complemented
46	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
47	C₂³×C₅	C₂²⋊D₅	1	6	yes	—
48	C₅×Q₈	—	2	36	no	—
49	C₅×Q₈	C₂²×D₅	1	24	yes	—
50	C₂×C₄×C₅	—	2	36	no	—
51	C₂×C₄×C₅	C₂²⋊D₅	1	18	yes	—
52	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
53	C₂×Dic₅	—	10	36	no	—
54	Dic₁₀	—	5	24	no	—
55	Dic₁₀	—	10	36	no	—
56	#168	—	5	24	no	—
57	C₂²×Q₈	—	25	1	no	—
58	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic
59	C₂×Q₁₆	—	25	12	no	—
60	C₂²×C₈	—	25	1	no	—
61	C₅²	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₅	#689	1	24	yes	complemented
63	C₂²×C₅	C₂×C₂²⋊D₅	1	18	yes	—
64	C₄×C₅	—	2	24	no	—
65	C₄×C₅	C₂×C₂²⋊D₅	1	18	yes	—
66	C₄×C₅	C₂³×D₅	1	6	yes	—
67	Dic₅	—	10	24	no	—
68	C₂×Q₈	—	2	3	no	—
69	C₂×Q₈	—	25	6	no	—
70	C₂×Q₈	—	50	3	no	—
71	C₂×Q₈	C₂×C₅⋊D₅	1	6	yes	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	50	1	no	—
74	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
75	Q₁₆	—	25	16	no	—
76	C₂×C₈	—	25	6	no	—
77	C₂×C₅	C₂×#689	1	36	yes	complemented
78	C₂×C₅	C₂²×C₂²⋊D₅	1	6	yes	—
79	C₂³	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
80	Q₈	—	2	6	no	—
81	Q₈	—	25	4	no	—
82	Q₈	—	50	6	no	—
83	Q₈	C₂²×C₅⋊D₅	1	4	yes	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	50	6	no	—
86	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	3	yes	—
87	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
88	C₈	—	25	4	no	—
89	C₅	C₂²×#689	1	6	yes	complemented
90	C₂²	#87812	1	4	yes	direct product factor
91	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	3	yes	—
92	C₄	—	2	4	no	—
93	C₄	—	50	4	no	—
94	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	3	yes	—
95	C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
96	C₂	C₂×#87812	1	6	yes	direct product factor
97	C₂	C₂²×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂³×C₅⋊D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle, prime radical	C₂³×C₅²	D₄
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	C₂²×Q₁₆
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#87812	C₂	direct (N × H)
12. #87812	C₂²	direct (N × H)
29. C₂²×C₅²	Q₁₆	semidirect
45. C₂×C₅²	C₂×Q₁₆	semidirect
61. C₅²	C₂²×Q₁₆	semidirect
62. C₂²×C₅	#689	semidirect
77. C₂×C₅	C₂×#689	semidirect
89. C₅	C₂²×#689	semidirect
90. C₂²	#87812	direct (N × H)
96. C₂	C₂×#87812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂²×#6045	2
C₂×#87812	2
C₂²×#6029	2
C₂²×#689	5

Subgroups — C22×#87812 #13107340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#87812 #13107340