Subgroups — #4881

Order 192.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	#892	C₂	1	1	yes	maximal, characteristic
3	D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#817	C₂	1	1	yes	maximal, characteristic, complemented
6	#808	C₂	1	1	yes	maximal, characteristic, complemented
7	#807	C₂	1	1	yes	maximal, characteristic, complemented
8	#428	—	3	1	no	maximal, Sylow
9	C₂×C₄×D₃	—	2	1	no	—
10	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	Dic₃⋊C₄	—	2	1	no	—
18	C₄×Dic₃	—	2	1	no	—
19	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
21	C₂×M₁₆	—	3	1	no	—
22	#117	—	3	1	no	—
23	C₂×C₄⋊C₄	—	3	1	no	—
24	#107	—	3	1	no	—
25	#107	D₃	1	1	yes	characteristic, complemented
26	#106	—	3	1	no	—
27	#106	—	3	1	no	—
28	C₂²×D₃	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	—	2	1	no	—
33	C₂×Dic₃	—	2	1	no	—
34	C₂×Dic₃	D₄	1	1	yes	characteristic
35	C₄×D₃	—	4	2	no	—
36	C₄×D₃	C₂×C₄	1	2	yes	—
37	C₄×D₃	Q₈	1	2	yes	—
38	C₃×C₈	C₂×C₄	1	2	yes	complemented
39	C₃⋊C₈	C₂×C₄	1	2	yes	complemented
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	6	1	no	—
42	M₁₆	—	3	4	no	—
43	C₂×C₈	—	3	1	no	—
44	C₂×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄⋊C₄	—	3	1	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₄⋊C₄	—	6	1	no	—
48	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
50	C₂²⋊C₄	—	6	1	no	—
51	C₄²	—	6	1	no	—
52	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
53	C₂×D₃	—	2	1	no	—
54	C₂×D₃	—	2	1	no	—
55	C₂×D₃	C₄⋊C₄	1	2	yes	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	—	4	1	no	—
58	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
60	Dic₃	—	4	1	no	—
61	Dic₃	—	4	1	no	—
62	Dic₃	C₄⋊C₄	1	2	yes	—
63	C₂³	—	3	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	2	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	—	6	1	no	—
71	C₂×C₄	—	12	2	no	—
72	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₈	—	3	2	no	—
74	C₈	C₄×D₃	1	2	yes	complemented
75	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×C₃	#137	1	1	yes	characteristic
78	D₃	—	2	2	no	—
79	C₂²	—	3	2	no	—
80	C₂²	—	6	1	no	—
81	C₂²	—	6	1	no	—
82	C₂²	D₃×D₄	1	1	yes	characteristic
83	C₄	—	3	2	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	12	1	no	—
87	C₄	—	12	1	no	—
88	C₄	C₂×C₄×D₃	1	1	yes	characteristic
89	C₄	D₃×Q₈	1	1	yes	characteristic
90	C₃	#428	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	6	2	no	—
92	C₂	D₃×C₄⋊C₄	1	1	yes	characteristic
93	C₂	#911	1	1	yes	characteristic
94	C₂	#918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#107	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂²×D₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#107	D₃
hypocenter	C₃	#428
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
4. C₃×#107	C₂	semidirect
5. #817	C₂	semidirect
6. #808	C₂	semidirect
7. #807	C₂	semidirect
20. ΓL(1,𝔽₂₅)	C₄	semidirect
25. #107	D₃	semidirect
38. C₃×C₈	C₂×C₄	semidirect
39. C₃⋊C₈	C₂×C₄	semidirect
74. C₈	C₄×D₃	semidirect
90. C₃	#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×ΓL(1,𝔽₂₅)	2
#892	2
D₃×C₄⋊C₄	2
C₃×#107	2
#817	2
#808	2
#807	2
#428	3