Subgroups — #343×C₃₀₇ #34087362

Order 19648.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₃₀₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₄×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	M₁₆×C₃₀₇	—	2	2	no	—
6	M₁₆×C₃₀₇	C₄	1	4	yes	complemented
7	C₂×C₈×C₃₀₇	—	2	2	no	—
8	C₄⋊C₄×C₃₀₇	—	2	2	no	—
9	C₂³×C₃₀₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₃₀₇	—	2	2	no	—
11	C₂×C₄×C₃₀₇	—	4	2	no	—
12	C₂×C₄×C₃₀₇	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₃₀₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₃₀₇	D₄	1	2	yes	—
15	C₂×C₄×C₃₀₇	Q₈	1	1	yes	characteristic
16	C₈×C₃₀₇	—	2	4	no	—
17	C₂²×C₃₀₇	—	2	1	no	—
18	C₂²×C₃₀₇	—	2	1	no	—
19	C₂²×C₃₀₇	C₄²	1	1	yes	characteristic
20	C₂²×C₃₀₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₃₀₇	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₃₀₇	—	4	2	no	—
23	C₄×C₃₀₇	C₂²⋊C₄	1	2	yes	—
24	C₄×C₃₀₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₃₀₇	—	2	1	no	—
26	C₂×C₃₀₇	—	2	1	no	—
27	C₂×C₃₀₇	#95	1	1	yes	characteristic
28	C₂×C₃₀₇	#100	1	1	yes	characteristic
29	C₂×C₃₀₇	#101	1	1	yes	characteristic
30	C₃₀₇	#343	1	1	yes	characteristic, direct product factor, Sylow
31	#343	C₃₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×M₁₆	C₂×C₃₀₇	1	2	yes	—
33	C₂×C₄⋊C₄	C₂×C₃₀₇	1	1	yes	characteristic
34	C₂²×C₄	C₄×C₃₀₇	1	2	yes	—
35	C₂²×C₄	C₂²×C₃₀₇	1	1	yes	characteristic
36	M₁₆	—	2	2	no	—
37	M₁₆	C₄×C₃₀₇	1	4	yes	complemented
38	C₂×C₈	—	2	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂³	C₂×C₄×C₃₀₇	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₂×C₄×C₃₀₇	1	2	yes	—
44	C₂×C₄	D₄×C₃₀₇	1	1	yes	characteristic
45	C₂×C₄	D₄×C₃₀₇	1	2	yes	—
46	C₂×C₄	Q₈×C₃₀₇	1	1	yes	characteristic
47	C₈	—	2	4	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	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	C₄	—	4	2	no	—
54	C₄	C₂²⋊C₄×C₃₀₇	1	2	yes	—
55	C₄	C₄⋊C₄×C₃₀₇	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#95×C₃₀₇	1	1	yes	characteristic
59	C₂	#100×C₃₀₇	1	1	yes	characteristic
60	C₂	#101×C₃₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₀₇	C₂²⋊C₄
commutator	C₂²	C₄²×C₃₀₇
Frattini, cosocle	C₂²×C₄	C₂²×C₃₀₇
prime radical	C₂³×C₃₀₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₀₇ ◁ C₂²×C₄×C₃₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. M₁₆×C₃₀₇	C₄	semidirect
30. C₃₀₇	#343	direct (N × H)
31. #343	C₃₀₇	direct (N × H)
37. M₁₆	C₄×C₃₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₃₀₇	2
C₂×C₄⋊C₄×C₃₀₇	2
#343	307

Subgroups — #343×C307 #34087362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #343×C₃₀₇ #34087362