Subgroups — #355×C₁₀₉ #20294513

Order 6976.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#101×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
4	C₂×D₄×C₁₀₉	C₄	1	1	yes	characteristic
5	M₁₆×C₁₀₉	—	2	2	no	—
6	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
7	C₄²×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂³×C₁₀₉	—	2	1	no	—
9	Q₈×C₁₀₉	—	4	1	no	—
10	D₄×C₁₀₉	—	4	1	no	—
11	C₂×C₄×C₁₀₉	—	2	1	no	—
12	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
14	C₈×C₁₀₉	—	4	2	no	—
15	C₂²×C₁₀₉	—	4	2	no	—
16	C₂²×C₁₀₉	—	4	1	no	—
17	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₁₀₉	—	2	2	no	—
19	C₄×C₁₀₉	—	2	1	no	—
20	C₄×C₁₀₉	—	4	1	no	—
21	C₂×C₁₀₉	—	2	1	no	—
22	C₂×C₁₀₉	—	8	1	no	—
23	C₂×C₁₀₉	#99	1	1	yes	characteristic
24	C₁₀₉	#355	1	1	yes	characteristic, direct product factor, Sylow
25	#355	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	#124	C₂×C₁₀₉	1	1	yes	characteristic
27	#101	C₂×C₁₀₉	1	2	yes	complemented
28	C₂×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
29	C₂×D₄	C₄×C₁₀₉	1	1	yes	characteristic
30	M₁₆	—	2	2	no	—
31	C₂²⋊C₄	—	2	2	no	—
32	C₄²	C₄×C₁₀₉	1	1	yes	characteristic
33	C₂³	—	2	1	no	—
34	Q₈	—	4	1	no	—
35	D₄	—	4	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
38	C₂×C₄	D₄×C₁₀₉	1	2	yes	—
39	C₈	—	4	2	no	—
40	C₂²	—	4	2	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₂	—	2	1	no	—
47	C₂	—	8	1	no	—
48	C₂	#99×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₀₉	#99
commutator	C₂×C₄	C₂×C₄×C₁₀₉
Frattini, cosocle	C₂×Q₈	C₂²×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₁₀₉ ◁ C₂×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #101×C₁₀₉	C₂	semidirect
24. C₁₀₉	#355	direct (N × H)
25. #355	C₁₀₉	direct (N × H)
27. #101	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#124×C₁₀₉	2
#101×C₁₀₉	2
#355	109

Subgroups — #355×C109 #20294513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #355×C₁₀₉ #20294513