Subgroups — #4837

Order 192.

126 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×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#423	—	3	1	no	maximal, Sylow
7	C₂×C₄×D₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₄×Dic₃	C₄	1	1	yes	characteristic
13	C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	ΓL(1,𝔽₂₅)	—	2	4	no	—
17	C₂×M₁₆	—	3	2	no	—
18	C₂×C₄²	—	3	1	no	—
19	C₄⋊C₈	—	3	2	no	—
20	C₄⋊C₈	—	3	1	no	—
21	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
22	C₂²×D₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₄×D₃	—	2	2	no	—
30	C₄×D₃	—	2	2	no	—
31	C₄×D₃	D₄	1	2	yes	—
32	C₄×D₃	Q₈	1	2	yes	—
33	C₃×C₈	—	2	2	no	—
34	C₃⋊C₈	—	2	2	no	—
35	C₂²×C₄	—	3	1	no	—
36	C₂²×C₄	—	3	1	no	—
37	C₂²×C₄	—	3	1	no	—
38	M₁₆	—	6	4	no	—
39	C₂×C₈	—	3	2	no	—
40	C₂×C₈	C₂×D₃	1	2	yes	—
41	C₄²	—	3	1	no	—
42	C₄²	—	3	1	no	—
43	C₄²	—	3	1	no	—
44	C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×D₃	—	2	1	no	—
47	C₂×D₃	—	2	1	no	—
48	C₂×D₃	C₄⋊C₄	1	2	yes	—
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	M₁₆	1	2	yes	complemented
51	C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
53	Dic₃	—	2	1	no	—
54	Dic₃	C₄⋊C₄	1	2	yes	—
55	Dic₃	M₁₆	1	2	yes	complemented
56	C₂³	—	3	1	no	—
57	C₂×C₄	—	3	1	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₄	—	3	2	no	—
60	C₂×C₄	—	3	2	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	6	2	no	—
63	C₂×C₄	—	6	2	no	—
64	C₂×C₄	C₄×D₃	1	1	yes	characteristic
65	C₂×C₄	C₄×D₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	6	2	no	—
69	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
72	D₃	—	2	2	no	—
73	C₂²	—	3	2	no	—
74	C₂²	—	6	1	no	—
75	C₂²	—	6	1	no	—
76	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	3	2	no	—
79	C₄	—	3	2	no	—
80	C₄	—	6	1	no	—
81	C₄	ΓL(1,𝔽₂₅)	1	2	yes	complemented
82	C₄	D₃×D₄	1	1	yes	characteristic
83	C₄	D₃×Q₈	1	1	yes	characteristic
84	C₃	#423	1	1	yes	characteristic, complemented, Sylow
85	C₂	—	6	2	no	—
86	C₂	D₃×C₄⋊C₄	1	1	yes	characteristic
87	C₂	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
88	C₂	D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₄⋊C₈	C₂
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊C₈	C₂	semidirect
4. Dic₃⋊C₈	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
21. C₄⋊C₈	D₃	semidirect
50. C₃×C₄	M₁₆	semidirect
55. Dic₃	M₁₆	semidirect
81. C₄	ΓL(1,𝔽₂₅)	semidirect
84. C₃	#423	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×C₄⋊C₈	2
Dic₃⋊C₈	2
C₄⋊(C₃⋊C₈)	2
#423	3