Subgroups — #336×C₃₀₇ #34087355

Order 19648.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₃₀₇	C₂	1	2	yes	maximal
2	C₂×C₄²×C₃₀₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₃₀₇	C₄	1	2	yes	—
4	C₂²×C₄×C₃₀₇	C₂²	1	1	yes	characteristic
5	C₂×C₈×C₃₀₇	—	2	2	no	—
6	C₂×C₈×C₃₀₇	—	2	2	no	—
7	C₂×C₈×C₃₀₇	C₄	1	4	yes	complemented
8	C₄²×C₃₀₇	—	2	2	no	—
9	C₂³×C₃₀₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₃₀₇	—	2	2	no	—
11	C₂×C₄×C₃₀₇	—	2	2	no	—
12	C₂×C₄×C₃₀₇	C₈	1	4	yes	complemented
13	C₂×C₄×C₃₀₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₃₀₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₃₀₇	D₄	1	2	yes	—
16	C₂×C₄×C₃₀₇	Q₈	1	1	yes	characteristic
17	C₈×C₃₀₇	—	2	4	no	—
18	C₂²×C₃₀₇	C₄²	1	1	yes	characteristic
19	C₂²×C₃₀₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₃₀₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₃₀₇	C₂×C₈	1	2	yes	—
22	C₂²×C₃₀₇	M₁₆	1	2	yes	—
23	C₄×C₃₀₇	—	2	4	no	—
24	C₄×C₃₀₇	C₂²⋊C₄	1	2	yes	—
25	C₄×C₃₀₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₃₀₇	#95	1	1	yes	characteristic
27	C₂×C₃₀₇	C₄×C₈	1	1	yes	characteristic
28	C₂×C₃₀₇	#97	1	1	yes	characteristic
29	C₂×C₃₀₇	C₂²⋊C₈	1	2	yes	—
30	C₂×C₃₀₇	C₄⋊C₈	1	2	yes	—
31	C₃₀₇	#336	1	1	yes	characteristic, direct product factor, Sylow
32	#336	C₃₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×C₈	C₂×C₃₀₇	1	2	yes	—
34	C₂×C₄²	C₂×C₃₀₇	1	1	yes	characteristic
35	C₂²×C₄	C₄×C₃₀₇	1	2	yes	—
36	C₂²×C₄	C₂²×C₃₀₇	1	1	yes	characteristic
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	C₄×C₃₀₇	1	4	yes	complemented
40	C₄²	—	2	2	no	—
41	C₂³	C₂×C₄×C₃₀₇	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₈×C₃₀₇	1	4	yes	complemented
45	C₂×C₄	C₂×C₄×C₃₀₇	1	2	yes	—
46	C₂×C₄	D₄×C₃₀₇	1	1	yes	characteristic
47	C₂×C₄	D₄×C₃₀₇	1	2	yes	—
48	C₂×C₄	Q₈×C₃₀₇	1	1	yes	characteristic
49	C₈	—	2	4	no	—
50	C₂²	C₄²×C₃₀₇	1	1	yes	characteristic
51	C₂²	C₂²⋊C₄×C₃₀₇	1	1	yes	characteristic
52	C₂²	C₄⋊C₄×C₃₀₇	1	1	yes	characteristic
53	C₂²	C₂×C₈×C₃₀₇	1	2	yes	—
54	C₂²	M₁₆×C₃₀₇	1	2	yes	—
55	C₄	—	2	4	no	—
56	C₄	C₂²⋊C₄×C₃₀₇	1	2	yes	—
57	C₄	C₄⋊C₄×C₃₀₇	1	2	yes	—
58	C₂	#95×C₃₀₇	1	1	yes	characteristic
59	C₂	C₄×C₈×C₃₀₇	1	1	yes	characteristic
60	C₂	#97×C₃₀₇	1	1	yes	characteristic
61	C₂	C₂²⋊C₈×C₃₀₇	1	2	yes	—
62	C₂	C₄⋊C₈×C₃₀₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₀₇	C₂²
commutator	C₂	C₄×C₈×C₃₀₇
Frattini, cosocle	C₂²×C₄	C₂²×C₃₀₇
socle, prime radical	C₂³×C₃₀₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂×C₈×C₃₀₇	C₄	semidirect
12. C₂×C₄×C₃₀₇	C₈	semidirect
31. C₃₀₇	#336	direct (N × H)
32. #336	C₃₀₇	direct (N × H)
39. C₂×C₈	C₄×C₃₀₇	semidirect
44. C₂×C₄	C₈×C₃₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₃₀₇	2
C₂×C₄²×C₃₀₇	2
#336	307

Subgroups — #336×C307 #34087355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #336×C₃₀₇ #34087355