Subgroups — #14271353

Order 1856.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#11745297	C₂	1	3	yes	maximal, complemented
3	#11745182	C₂	1	3	yes	maximal, complemented
4	#41×C₂₉	—	2	3	no	—
5	C₂×D₄×C₂₉	—	2	3	no	—
6	C₂×D₄×C₂₉	C₂²	1	3	yes	—
7	C₂²×Dic₂₉	C₂²	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₂₉	—	2	3	no	—
9	C₂²⋊Dic₂₉	C₂²	1	3	yes	complemented
10	#90779	—	2	3	no	—
11	C₂³×C₂₉	—	2	1	no	—
12	C₂³×C₂₉	D₄	1	3	yes	—
13	C₂³×C₂₉	C₂³	1	1	yes	characteristic
14	Q₈×C₂₉	—	2	1	no	—
15	D₄×C₂₉	—	2	3	no	—
16	D₄×C₂₉	—	2	6	no	—
17	C₂×C₄×C₂₉	—	2	3	no	—
18	C₂×C₄×C₂₉	D₄	1	3	yes	—
19	C₂×Dic₂₉	—	2	3	no	—
20	C₂×Dic₂₉	—	2	3	no	—
21	C₂²×C₂₉	—	2	3	no	—
22	C₂²×C₂₉	—	4	1	no	—
23	C₂²×C₂₉	—	4	3	no	—
24	C₂²×C₂₉	—	8	1	no	—
25	C₂²×C₂₉	C₂×D₄	1	3	yes	—
26	C₄×C₂₉	—	2	3	no	—
27	Dic₂₉	—	4	3	no	—
28	Dic₂₉	—	4	1	no	—
29	#458	—	29	1	no	maximal, Sylow
30	C₂×C₂₉	—	2	3	no	—
31	C₂×C₂₉	—	4	3	no	—
32	C₂×C₂₉	#120	1	1	yes	characteristic
33	H(5,𝔽₂)	D₂₉	1	1	yes	characteristic
34	#123	—	29	3	no	—
35	#99	—	29	3	no	—
36	C₂₉	#458	1	1	yes	characteristic, complemented, Sylow
37	#41	—	2	3	no	—
38	C₂×D₄	—	2	3	no	—
39	C₂×D₄	C₂×D₂₉	1	3	yes	—
40	C₂²×C₄	—	29	1	no	—
41	C₄⋊C₄	—	58	3	no	—
42	C₂²⋊C₄	—	29	3	no	—
43	C₂²⋊C₄	—	58	3	no	—
44	C₂³	—	2	1	no	—
45	C₂³	C₂²⋊D₂₉	1	3	yes	—
46	C₂³	C₂²×D₂₉	1	1	yes	characteristic
47	Q₈	—	2	1	no	—
48	D₄	—	2	3	no	—
49	D₄	—	2	6	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	58	3	no	—
52	C₂×C₄	—	58	3	no	—
53	C₂×C₄	C₂²⋊D₂₉	1	3	yes	—
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×C₂²⋊D₂₉	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	116	3	no	—
61	C₄	—	116	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#11745315	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11745315
commutator, cosocle	C₂³×C₂₉	C₂³
Frattini	C₂³	C₂²×D₂₉
Fitting, prime radical	C₂₉×H(5,𝔽₂)	C₂
socle	C₂×C₂₉	#120
hypercenter	H(5,𝔽₂)	D₂₉
hypocenter	C₂₉	#458
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #11745297	C₂	semidirect
3. #11745182	C₂	semidirect
7. C₂²×Dic₂₉	C₂²	semidirect
9. C₂²⋊Dic₂₉	C₂²	semidirect
36. C₂₉	#458	semidirect

Maximal subgroups

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

subgroup	index
C₂₉×H(5,𝔽₂)	2
#11745297	2
#11745182	2
#458	29