Subgroups — #16821401

Order 3712.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#90768	C₂	1	2	yes	maximal
2	C₄⋊#90768	C₂	1	1	yes	maximal, characteristic
3	#14270939	C₂	1	2	yes	maximal
4	C₄×#90768	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
7	C₂₉×C₄⋊C₈	C₄	1	2	yes	—
8	C₂₉×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₂₉	C₂²	1	2	yes	—
11	C₂×#90768	C₂²	1	1	yes	characteristic
12	C₂×#90768	C₂²	1	2	yes	—
13	C₂×#90768	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₂₉	C₈	1	1	yes	characteristic
15	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₂₉	C₈	1	2	yes	—
19	C₄⋊C₄×C₂₉	C₈	1	1	yes	characteristic
20	C₄²×C₂₉	C₂×C₄	1	2	yes	—
21	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
22	#90768	—	2	1	no	—
23	#90768	—	2	2	no	—
24	#90768	Q₈	1	2	yes	complemented
25	Q₈×C₂₉	—	2	2	no	—
26	C₂×C₄×C₂₉	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₂₉	—	2	1	no	—
32	C₈×C₂₉	—	2	2	no	—
33	C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₂₉	#41	1	1	yes	characteristic
35	#2183	—	29	1	no	maximal, Sylow
36	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₂₉	—	2	2	no	—
38	C₄×C₂₉	—	2	2	no	—
39	C₄×C₂₉	—	2	1	no	—
40	C₄×C₂₉	M₃₂	1	2	yes	—
41	C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
42	C₄×C₂₉	#131	1	1	yes	characteristic
43	C₈×Q₈	D₂₉	1	1	yes	characteristic
44	C₄⋊C₁₆	—	29	1	no	—
45	C₄⋊C₁₆	—	29	2	no	—
46	#346	—	29	2	no	—
47	C₄×C₁₆	—	29	1	no	—
48	C₂×C₂₉	C₈×Q₈	1	1	yes	characteristic
49	C₂×C₂₉	C₂×M₃₂	1	1	yes	characteristic
50	C₂×C₂₉	#504	1	1	yes	characteristic
51	C₄×Q₈	Dic₂₉	1	1	yes	characteristic
52	C₂×C₁₆	—	29	1	no	—
53	C₂×C₁₆	—	29	1	no	—
54	C₂×C₁₆	—	29	2	no	—
55	C₄⋊C₈	Dic₂₉	1	2	yes	—
56	C₄⋊C₈	Dic₂₉	1	1	yes	characteristic
57	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
58	C₄×C₈	C₂×D₂₉	1	2	yes	—
59	C₂₉	#2183	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	#6597	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₂₉	1	1	yes	characteristic
62	C₂×C₈	C₂×Dic₂₉	1	2	yes	—
63	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
64	C₄⋊C₄	#6597	1	2	yes	—
65	C₄⋊C₄	#6597	1	1	yes	characteristic
66	C₄²	C₂×Dic₂₉	1	2	yes	—
67	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
68	C₁₆	—	29	2	no	—
69	C₁₆	—	58	1	no	—
70	C₁₆	—	58	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×#6597	1	2	yes	—
73	C₂×C₄	C₂×#6597	1	1	yes	characteristic
74	C₂×C₄	C₂×#6597	1	2	yes	—
75	C₂×C₄	C₂×#6597	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₂₉	1	1	yes	characteristic
80	C₈	#90809	1	1	yes	characteristic
81	C₂²	C₂²×#6597	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#11745160	1	2	yes	—
86	C₄	Q₈×Dic₂₉	1	1	yes	characteristic
87	C₄	#11745310	1	1	yes	characteristic
88	C₂	Q₈×#6597	1	1	yes	characteristic
89	C₂	C₂×#11745160	1	1	yes	characteristic
90	C₂	#14271580	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₉
commutator, socle, prime radical	C₂²×C₂₉	C₂²×C₈
Fitting	C₈×Q₈×C₂₉	C₂
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	C₈×Q₈	D₂₉
hypocenter	C₂₉	#2183
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. #90768	Q₈	semidirect
59. C₂₉	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#90768	2
#14270939	2
C₄×#90768	2
C₈×Q₈×C₂₉	2
#2183	29