Subgroups — #34094500

Order 20032.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756494	C₂	1	1	yes	maximal, characteristic
2	#23756417	C₂	1	4	yes	maximal, complemented
3	C₄×#14578809	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₅₂	—	2	1	no	—
6	C₂×D₁₂₅₂	—	2	1	no	—
7	C₂×D₁₂₅₂	C₂²	1	2	yes	—
8	C₂×#14578809	C₂²	1	2	yes	—
9	C₂×Q₈×C₃₁₃	—	2	1	no	—
10	C₄⋊C₄×C₃₁₃	—	2	1	no	—
11	C₄⋊C₄×C₃₁₃	C₂²	1	2	yes	—
12	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₃₁₃	—	2	2	no	—
14	D₁₂₅₂	—	2	4	no	—
15	D₁₂₅₂	—	4	2	no	—
16	D₁₂₅₂	—	4	2	no	—
17	#14578809	—	2	2	no	—
18	Q₈×C₃₁₃	—	4	2	no	—
19	C₂×C₄×C₃₁₃	—	2	2	no	—
20	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₁₃	—	4	2	no	—
25	C₂×D₃₁₃	—	4	2	no	—
26	C₂×D₃₁₃	—	4	2	no	—
27	C₄×C₃₁₃	—	4	2	no	—
28	C₄×C₃₁₃	D₈	1	2	yes	—
29	C₄×C₃₁₃	SD₁₆	1	2	yes	—
30	C₄×C₃₁₃	#41	1	2	yes	—
31	C₂×C₃₁₃	#124	1	1	yes	characteristic
32	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
33	C₂×C₃₁₃	C₂×SD₁₆	1	1	yes	characteristic
34	D₃₁₃	—	8	2	no	—
35	C₃₁₃	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	313	1	no	maximal, Sylow
37	C₄⋊Q₈	D₃₁₃	1	1	yes	characteristic, complemented
38	#127	—	313	1	no	—
39	#102	—	313	4	no	—
40	C₄×C₈	—	313	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	313	2	no	—
43	C₂×D₄	—	626	1	no	—
44	C₂×D₄	—	626	1	no	—
45	C₂×C₈	—	313	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₃₁₃	1	2	yes	—
48	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
49	C₂³	—	626	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	626	4	no	—
52	D₄	—	1252	2	no	—
53	D₄	—	1252	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
58	C₈	—	626	2	no	—
59	C₂²	—	1252	2	no	—
60	C₂²	—	1252	2	no	—
61	C₂²	—	1252	2	no	—
62	C₂²	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#19097546	1	2	yes	—
65	C₄	#19097548	1	2	yes	—
66	C₄	#19097562	1	2	yes	—
67	C₂	—	2504	2	no	—
68	C₂	#23756486	1	1	yes	characteristic
69	C₂	C₂×#19097546	1	1	yes	characteristic
70	C₂	C₂×#19097548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁₃
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₄⋊Q₈×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
prime radical	#23756494	C₂
hypercenter	C₄⋊Q₈	D₃₁₃
hypocenter	C₃₁₃	#486
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #23756417	C₂	semidirect
3. C₄×#14578809	C₂	semidirect
4. C₄⋊Q₈×C₃₁₃	C₂	semidirect
35. C₃₁₃	#486	semidirect
37. C₄⋊Q₈	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756494	2
#23756417	2
C₄×#14578809	2
C₄⋊Q₈×C₃₁₃	2
#486	313