Subgroups — #4707

Order 192.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal
2	C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₃	C₂	1	2	yes	maximal, complemented
5	#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×M₁₆	—	3	1	no	maximal, Sylow
7	C₂×C₄×D₃	C₄	1	2	yes	—
8	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈	C₂²	1	2	yes	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₄×Dic₃	C₄	1	2	yes	—
12	C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₈	C₂²	1	2	yes	—
14	ΓL(1,𝔽₂₅)	C₄	1	8	yes	complemented
15	C₂×M₁₆	—	3	2	no	—
16	C₂×C₄²	—	3	1	no	—
17	#97	—	3	1	no	—
18	#97	D₃	1	1	yes	characteristic, complemented
19	C₄×C₈	—	3	2	no	—
20	C₂²×D₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₃	C₂×C₄	1	2	yes	—
25	C₄×D₃	—	2	4	no	—
26	C₄×D₃	C₂×C₄	1	4	yes	—
27	C₃×C₈	C₂×C₄	1	4	yes	complemented
28	C₃⋊C₈	C₂×C₄	1	4	yes	complemented
29	C₂²×C₄	—	3	1	no	—
30	C₂²×C₄	—	3	2	no	—
31	M₁₆	—	3	8	no	—
32	C₂×C₈	—	3	2	no	—
33	C₂×C₈	C₂×D₃	1	2	yes	—
34	C₄²	—	3	2	no	—
35	C₄²	—	3	1	no	—
36	C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×D₃	—	2	2	no	—
39	C₂×D₃	C₄²	1	2	yes	—
40	C₃×C₄	—	2	2	no	—
41	C₃×C₄	C₂²×C₄	1	2	yes	—
42	Dic₃	C₄²	1	2	yes	—
43	Dic₃	M₁₆	1	4	yes	complemented
44	C₂³	—	3	1	no	—
45	C₂×C₄	—	3	1	no	—
46	C₂×C₄	—	3	2	no	—
47	C₂×C₄	—	3	4	no	—
48	C₂×C₄	—	6	4	no	—
49	C₂×C₄	C₄×D₃	1	2	yes	—
50	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
51	C₈	—	3	4	no	—
52	C₈	C₄×D₃	1	4	yes	complemented
53	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₃	C₂×M₁₆	1	2	yes	—
55	D₃	—	2	2	no	—
56	C₂²	—	3	2	no	—
57	C₂²	—	6	2	no	—
58	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	3	2	no	—
61	C₄	—	3	4	no	—
62	C₄	C₂×C₄×D₃	1	2	yes	—
63	C₃	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
64	C₂	—	6	2	no	—
65	C₂	C₄²×D₃	1	1	yes	characteristic
66	C₂	D₃×M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²
Fitting	C₃×#97	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#97	D₃
hypocenter	C₃	C₄×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. C₃×#97	C₂	semidirect
4. C₈×Dic₃	C₂	semidirect
5. #803	C₂	semidirect
14. ΓL(1,𝔽₂₅)	C₄	semidirect
18. #97	D₃	semidirect
27. C₃×C₈	C₂×C₄	semidirect
28. C₃⋊C₈	C₂×C₄	semidirect
43. Dic₃	M₁₆	semidirect
52. C₈	C₄×D₃	semidirect
63. C₃	C₄×M₁₆	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
C₄²×D₃	2
C₃×#97	2
C₈×Dic₃	2
#803	2
C₄×M₁₆	3