Subgroups — #16821394

Order 3712.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₅₈	C₂	1	1	yes	maximal, characteristic
2	#14270981	C₂	1	2	yes	maximal
3	C₄×#90768	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₂₉	C₂	1	2	yes	maximal
5	C₂₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₉	C₂²	1	2	yes	—
8	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	#11745162	C₄	1	2	yes	—
10	C₈×Dic₂₉	C₂²	1	2	yes	—
11	C₂×#90768	C₂²	1	1	yes	characteristic
12	C₂×#90768	C₂²	1	1	yes	characteristic
13	C₄⋊#6597	C₄	1	1	yes	characteristic
14	C₂×Dic₅₈	C₈	1	1	yes	characteristic
15	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
17	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₂₉	C₈	1	1	yes	characteristic
19	#90779	C₈	1	2	yes	—
20	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
21	C₂×#6597	C₂×C₄	1	2	yes	—
22	C₁₆×C₂₉	—	2	2	no	—
23	#90768	—	2	1	no	—
24	#90768	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₂₉	C₂×C₈	1	2	yes	—
29	C₂×Dic₂₉	C₂×C₈	1	2	yes	—
30	Dic₅₈	C₁₆	1	4	yes	complemented
31	C₈×C₂₉	—	2	1	no	—
32	C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₂₉	#41	1	1	yes	characteristic
34	#6597	—	2	2	no	—
35	Q₈×C₁₆	—	29	1	no	maximal, Sylow
36	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₂₉	—	2	1	no	—
38	C₄×C₂₉	C₂×C₁₆	1	2	yes	—
39	C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
40	C₄×C₂₉	#131	1	1	yes	characteristic
41	Dic₂₉	—	2	2	no	—
42	Dic₂₉	C₂×C₁₆	1	4	yes	—
43	C₈×Q₈	—	29	1	no	—
44	C₄⋊C₁₆	—	29	2	no	—
45	C₄⋊C₁₆	D₂₉	1	1	yes	characteristic
46	C₄×C₁₆	—	29	2	no	—
47	C₄×C₁₆	—	29	1	no	—
48	C₂×C₂₉	C₈×Q₈	1	1	yes	characteristic
49	C₂×C₂₉	C₂²×C₁₆	1	1	yes	characteristic
50	C₂×C₂₉	#504	1	1	yes	characteristic
51	C₄×Q₈	—	29	1	no	—
52	C₂×C₁₆	—	29	1	no	—
53	C₂×C₁₆	—	29	1	no	—
54	C₂×C₁₆	C₂×D₂₉	1	2	yes	—
55	C₄⋊C₈	—	29	2	no	—
56	C₄⋊C₈	—	29	1	no	—
57	C₄×C₈	—	29	2	no	—
58	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
59	C₂₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	29	1	no	—
61	C₂×C₈	—	29	2	no	—
62	C₂×C₈	C₄×D₂₉	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
64	C₄⋊C₄	—	29	2	no	—
65	C₄⋊C₄	—	29	1	no	—
66	C₄²	—	29	2	no	—
67	C₄²	C₄×D₂₉	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	29	2	no	—
70	C₁₆	—	58	1	no	—
71	Q₈	—	29	4	no	—
72	C₂×C₄	—	29	2	no	—
73	C₂×C₄	—	29	2	no	—
74	C₂×C₄	C₈×D₂₉	1	1	yes	characteristic
75	C₂×C₄	C₈×D₂₉	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	58	2	no	—
79	C₈	#90807	1	1	yes	characteristic
80	C₈	Q₈×D₂₉	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₂₉	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	29	4	no	—
84	C₄	—	58	2	no	—
85	C₄	C₁₆×D₂₉	1	2	yes	—
86	C₄	#11745249	1	1	yes	characteristic
87	C₄	#11745269	1	1	yes	characteristic
88	C₂	#14271287	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₂₉	1	1	yes	characteristic
90	C₂	#14271581	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₉
commutator	C₂×C₂₉	C₂²×C₁₆
Fitting	C₂₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₂₉	C₂²×C₈
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	C₄⋊C₁₆	D₂₉
hypocenter	C₂₉	Q₈×C₁₆
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. #90768	Q₈	semidirect
30. Dic₅₈	C₁₆	semidirect
59. C₂₉	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₅₈	2
#14270981	2
C₄×#90768	2
C₁₆×Dic₂₉	2
C₂₉×C₄⋊C₁₆	2
Q₈×C₁₆	29