Subgroups — #14271566

Order 1856.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11745231	C₂	1	1	yes	maximal, characteristic
3	#11745155	C₂	1	4	yes	maximal
4	C₄×#6597	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₉	—	2	1	no	—
6	C₂×Dic₅₈	—	2	1	no	—
7	C₄⋊C₄×C₂₉	—	2	1	no	—
8	C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
9	C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₉	—	2	1	no	—
11	C₄⋊Dic₂₉	C₂²	1	2	yes	—
12	C₂×#6597	C₂²	1	2	yes	—
13	Q₈×C₂₉	—	4	2	no	—
14	C₂×C₄×C₂₉	—	2	2	no	—
15	C₂×C₄×C₂₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₉	—	2	2	no	—
19	Dic₅₈	—	4	2	no	—
20	#6597	Q₈	1	4	yes	complemented
21	C₂²×C₂₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₉	—	4	2	no	—
23	C₄×C₂₉	D₈	1	2	yes	—
24	C₄×C₂₉	Q₁₆	1	2	yes	—
25	C₄×C₂₉	C₂×Q₈	1	2	yes	—
26	Dic₂₉	—	4	2	no	—
27	#500	—	29	1	no	maximal, Sylow
28	C₂×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₂₉	C₂×D₈	1	1	yes	characteristic
30	C₂×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
31	C₄⋊Q₈	—	29	1	no	—
32	C₄⋊Q₈	D₂₉	1	1	yes	characteristic
33	#107	—	29	4	no	—
34	C₄×C₈	—	29	1	no	—
35	C₂₉	#500	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	58	1	no	—
38	C₂×C₈	—	29	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	29	2	no	—
41	C₄⋊C₄	—	58	1	no	—
42	C₄⋊C₄	C₂×D₂₉	1	2	yes	—
43	C₄²	C₂×D₂₉	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	116	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	58	2	no	—
48	C₂×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
51	C₈	—	29	4	no	—
52	C₂²	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	116	2	no	—
55	C₄	#90782	1	2	yes	—
56	C₄	#90785	1	2	yes	—
57	C₄	Q₈×D₂₉	1	2	yes	—
58	C₂	C₂×#90782	1	1	yes	characteristic
59	C₂	C₂×#90785	1	1	yes	characteristic
60	C₂	#11745306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₉
Fitting	C₂₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₂₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₉
hypocenter	C₂₉	#500
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
20. #6597	Q₈	semidirect
35. C₂₉	#500	semidirect

Maximal subgroups

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

subgroup	index
C₂₉×C₄⋊Q₈	2
#11745231	2
#11745155	2
C₄×#6597	2
#500	29