Subgroups — #16819644

Order 3712.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6597	C₂	1	6	yes	maximal
2	C₄³×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₂₉	C₄	1	3	yes	—
4	C₂×C₄²×C₂₉	C₄	1	1	yes	characteristic
5	C₂×C₄²×C₂₉	C₂²	1	3	yes	—
6	C₂²×#6597	C₂²	1	4	yes	—
7	C₄⋊#6597	—	2	12	no	—
8	C₂²×C₄×C₂₉	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₂₉	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
11	C₄²×C₂₉	—	2	6	no	—
12	C₄²×C₂₉	C₈	1	4	yes	complemented
13	C₄²×C₂₉	D₄	1	6	yes	—
14	C₄²×C₂₉	Q₈	1	6	yes	—
15	C₂×#6597	—	2	12	no	—
16	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₉	—	2	6	no	—
18	C₂×C₄×C₂₉	C₄⋊C₄	1	6	yes	—
19	C₂×C₄×C₂₉	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₂₉	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₂₉	M₁₆	1	6	yes	—
22	C₂×C₄×C₂₉	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₂₉	C₂×Q₈	1	3	yes	—
24	#6597	—	4	4	no	—
25	#1842	—	29	1	no	maximal, Sylow
26	C₂²×C₂₉	C₂×C₄⋊C₄	1	3	yes	—
27	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
28	C₂²×C₂₉	C₂×M₁₆	1	3	yes	—
29	C₄×C₂₉	—	2	6	no	—
30	C₄×C₂₉	C₄⋊C₈	1	12	yes	complemented
31	C₄×C₂₉	#127	1	1	yes	characteristic
32	C₄×C₂₉	C₄⋊Q₈	1	3	yes	—
33	C₂×C₄⋊C₈	—	29	6	no	—
34	C₄³	D₂₉	1	1	yes	characteristic
35	C₂×C₂₉	#384	1	1	yes	characteristic
36	C₂×C₂₉	C₂×C₄⋊C₈	1	3	yes	—
37	C₂×C₂₉	#423	1	3	yes	—
38	C₂²×C₈	—	29	4	no	—
39	C₂×C₄²	Dic₂₉	1	3	yes	—
40	C₂×C₄²	Dic₂₉	1	1	yes	characteristic
41	C₂×C₄²	C₂×D₂₉	1	3	yes	—
42	C₄⋊C₈	—	58	12	no	—
43	C₂₉	#1842	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	C₂×Dic₂₉	1	3	yes	—
45	C₂²×C₄	C₂×Dic₂₉	1	3	yes	—
46	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
47	C₂×C₈	—	58	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	#6597	1	4	yes	complemented
50	C₄²	Dic₅₈	1	6	yes	—
51	C₄²	D₁₁₆	1	6	yes	—
52	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₂×#6597	1	6	yes	—
55	C₂×C₄	#90777	1	6	yes	—
56	C₂×C₄	C₄⋊Dic₂₉	1	6	yes	—
57	C₂×C₄	C₄⋊Dic₂₉	1	6	yes	—
58	C₂×C₄	C₂×Dic₅₈	1	3	yes	—
59	C₂×C₄	C₂×D₁₁₆	1	3	yes	—
60	C₈	—	116	4	no	—
61	C₂²	C₂²×#6597	1	1	yes	characteristic
62	C₂²	C₂×#90777	1	3	yes	—
63	C₂²	C₂×C₄⋊Dic₂₉	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊#6597	1	12	yes	complemented
66	C₄	#11745231	1	3	yes	—
67	C₄	#11745236	1	1	yes	characteristic
68	C₂	#14271033	1	1	yes	characteristic
69	C₂	C₂×C₄⋊#6597	1	3	yes	—
70	C₂	#14271187	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂²×C₈
Fitting	C₄³×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₄³	D₂₉
hypocenter	C₂₉	#1842
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
12. C₄²×C₂₉	C₈	semidirect
30. C₄×C₂₉	C₄⋊C₈	semidirect
43. C₂₉	#1842	semidirect
49. C₄²	#6597	semidirect
65. C₄	C₄⋊#6597	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6597	2
C₄³×C₂₉	2
#1842	29