Subgroups — #16821745

Order 3712.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₃₂	C₂	1	4	yes	maximal
2	#14271543	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₁₆	—	2	2	no	—
5	C₂×Dic₁₁₆	C₂²	1	4	yes	—
6	#11745231	—	2	2	no	—
7	C₂×C₁₆×C₂₉	C₂²	1	2	yes	—
8	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	Dic₂₃₂	—	2	8	no	—
10	C₂×Dic₅₈	—	2	4	no	—
11	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
12	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
13	C₄²×C₂₉	D₄	1	1	yes	characteristic
14	C₄⋊Dic₂₉	—	4	4	no	—
15	Dic₁₁₆	—	2	8	no	—
16	Dic₁₁₆	—	4	4	no	—
17	C₁₆×C₂₉	D₄	1	4	yes	—
18	C₂×C₄×C₂₉	D₈	1	2	yes	—
19	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₂₉	—	4	4	no	—
21	Dic₅₈	—	4	4	no	—
22	Dic₅₈	—	4	4	no	—
23	C₈×C₂₉	D₈	1	2	yes	—
24	C₈×C₂₉	C₂×D₄	1	2	yes	—
25	#2247	—	29	1	no	maximal, Sylow
26	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
27	C₄×C₂₉	Q₃₂	1	4	yes	complemented
28	C₄×C₂₉	#127	1	1	yes	characteristic
29	C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
30	Dic₂₉	—	8	4	no	—
31	C₂×Q₃₂	—	29	4	no	—
32	#494	—	29	2	no	—
33	C₄×C₁₆	D₂₉	1	1	yes	characteristic
34	C₂×C₂₉	#493	1	1	yes	characteristic
35	C₂×C₂₉	C₂×Q₃₂	1	2	yes	—
36	C₂×Q₁₆	—	29	4	no	—
37	C₂×Q₁₆	—	58	2	no	—
38	C₄⋊Q₈	—	58	2	no	—
39	Q₃₂	—	58	8	no	—
40	C₂×C₁₆	C₂×D₂₉	1	2	yes	—
41	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
42	C₂₉	#2247	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	—	58	4	no	—
44	Q₁₆	—	58	8	no	—
45	Q₁₆	—	116	4	no	—
46	C₂×C₈	D₁₁₆	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
48	C₄⋊C₄	—	116	4	no	—
49	C₄²	D₁₁₆	1	1	yes	characteristic
50	C₁₆	D₁₁₆	1	4	yes	—
51	Q₈	—	116	4	no	—
52	Q₈	—	116	4	no	—
53	C₂×C₄	—	116	4	no	—
54	C₂×C₄	D₂₃₂	1	2	yes	—
55	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
56	C₈	D₂₃₂	1	2	yes	—
57	C₈	C₂×D₁₁₆	1	2	yes	—
58	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
59	C₄	—	232	4	no	—
60	C₄	Dic₂₃₂	1	4	yes	complemented
61	C₄	#11745236	1	1	yes	characteristic
62	C₄	C₂×D₂₃₂	1	1	yes	characteristic
63	C₂	#14271540	1	1	yes	characteristic
64	C₂	C₂×Dic₂₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₃₂
commutator, cosocle	C₂×C₈×C₂₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₄×C₁₆×C₂₉	C₂
socle, prime radical	C₂²×C₂₉	C₂×D₈
hypercenter	C₄×C₁₆	D₂₉
hypocenter	C₂₉	#2247
LCS	G ▷ C₂×C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
27. C₄×C₂₉	Q₃₂	semidirect
42. C₂₉	#2247	semidirect
60. C₄	Dic₂₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₃₂	2
#14271543	2
C₄×C₁₆×C₂₉	2
#2247	29