Subgroups — #44486801

Order 30592.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₉₁₂	C₂	1	4	yes	maximal
2	#29505152	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉₅₆	—	2	2	no	—
5	C₂×Dic₉₅₆	C₂²	1	4	yes	—
6	#21439960	—	2	2	no	—
7	Dic₁₉₁₂	—	2	8	no	—
8	C₂×C₁₆×C₂₃₉	C₂²	1	2	yes	—
9	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇₈	—	2	4	no	—
11	Dic₉₅₆	—	2	8	no	—
12	Dic₉₅₆	—	4	4	no	—
13	C₄⋊Dic₂₃₉	—	4	4	no	—
14	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
16	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
17	C₁₆×C₂₃₉	D₄	1	4	yes	—
18	Dic₄₇₈	—	4	4	no	—
19	Dic₄₇₈	—	4	4	no	—
20	C₂×Dic₂₃₉	—	4	4	no	—
21	C₂×C₄×C₂₃₉	D₈	1	2	yes	—
22	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
23	C₈×C₂₃₉	D₈	1	2	yes	—
24	C₈×C₂₃₉	C₂×D₄	1	2	yes	—
25	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
26	C₄×C₂₃₉	Q₃₂	1	4	yes	complemented
27	C₄×C₂₃₉	#127	1	1	yes	characteristic
28	C₄×C₂₃₉	C₂×D₈	1	1	yes	characteristic
29	Dic₂₃₉	—	8	4	no	—
30	C₂×C₂₃₉	#493	1	1	yes	characteristic
31	C₂×C₂₃₉	C₂×Q₃₂	1	2	yes	—
32	C₂₃₉	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	239	1	no	maximal, Sylow
34	C₂×Q₃₂	—	239	4	no	—
35	#494	—	239	2	no	—
36	C₄×C₁₆	D₂₃₉	1	1	yes	characteristic
37	C₂×Q₁₆	—	478	2	no	—
38	C₂×Q₁₆	—	239	4	no	—
39	C₄⋊Q₈	—	478	2	no	—
40	Q₃₂	—	478	8	no	—
41	C₂×C₁₆	C₂×D₂₃₉	1	2	yes	—
42	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
43	C₂×Q₈	—	478	4	no	—
44	Q₁₆	—	956	4	no	—
45	Q₁₆	—	478	8	no	—
46	C₂×C₈	D₉₅₆	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
48	C₄⋊C₄	—	956	4	no	—
49	C₄²	D₉₅₆	1	1	yes	characteristic
50	C₁₆	D₉₅₆	1	4	yes	—
51	Q₈	—	956	4	no	—
52	Q₈	—	956	4	no	—
53	C₂×C₄	—	956	4	no	—
54	C₂×C₄	D₁₉₁₂	1	2	yes	—
55	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
56	C₈	D₁₉₁₂	1	2	yes	—
57	C₈	C₂×D₉₅₆	1	2	yes	—
58	C₂²	C₂×D₁₉₁₂	1	1	yes	characteristic
59	C₄	—	1912	4	no	—
60	C₄	Dic₁₉₁₂	1	4	yes	complemented
61	C₄	#21439957	1	1	yes	characteristic
62	C₄	C₂×D₁₉₁₂	1	1	yes	characteristic
63	C₂	#29505149	1	1	yes	characteristic
64	C₂	C₂×Dic₁₉₁₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₉₁₂
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	C₄×C₁₆×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
hypercenter	C₄×C₁₆	D₂₃₉
hypocenter	C₂₃₉	#2247
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₂₃₉	Q₃₂	semidirect
32. C₂₃₉	#2247	semidirect
60. C₄	Dic₁₉₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₉₁₂	2
#29505152	2
C₄×C₁₆×C₂₃₉	2
#2247	239