Subgroups — #137×C₇₅₇ #37564285

Order 24224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₁₆×C₇₅₇	C₂	1	2	yes	maximal, complemented
4	SD₁₆×C₇₅₇	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₇₅₇	—	2	1	no	—
7	Q₈×C₇₅₇	C₂²	1	2	yes	—
8	Q₈×C₇₅₇	C₂²	1	1	yes	characteristic
9	D₄×C₇₅₇	—	2	1	no	—
10	D₄×C₇₅₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₅₇	—	2	1	no	—
12	C₂×C₄×C₇₅₇	—	2	1	no	—
13	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
14	C₈×C₇₅₇	C₂²	1	2	yes	—
15	C₂²×C₇₅₇	—	2	1	no	—
16	C₂²×C₇₅₇	D₄	1	1	yes	characteristic
17	C₄×C₇₅₇	—	2	1	no	—
18	C₄×C₇₅₇	—	2	2	no	—
19	C₄×C₇₅₇	D₄	1	1	yes	characteristic
20	C₄×C₇₅₇	C₂³	1	1	yes	characteristic
21	C₂×C₇₅₇	—	2	1	no	—
22	C₂×C₇₅₇	—	4	1	no	—
23	C₂×C₇₅₇	C₂×D₄	1	1	yes	characteristic
24	C₇₅₇	#137	1	1	yes	characteristic, direct product factor, Sylow
25	#137	C₇₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	#41	C₂×C₇₅₇	1	1	yes	characteristic
27	C₂×Q₈	C₂×C₇₅₇	1	1	yes	characteristic, complemented
28	Q₁₆	C₂×C₇₅₇	1	2	yes	complemented
29	SD₁₆	C₂×C₇₅₇	1	2	yes	complemented
30	M₁₆	C₂×C₇₅₇	1	1	yes	characteristic, complemented
31	Q₈	—	2	1	no	—
32	Q₈	C₂²×C₇₅₇	1	2	yes	—
33	Q₈	C₂²×C₇₅₇	1	1	yes	characteristic
34	D₄	—	2	1	no	—
35	D₄	C₂²×C₇₅₇	1	1	yes	characteristic
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	C₂²×C₇₅₇	1	1	yes	characteristic
39	C₈	C₂²×C₇₅₇	1	2	yes	—
40	C₂²	—	2	1	no	—
41	C₂²	D₄×C₇₅₇	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	2	2	no	—
44	C₄	D₄×C₇₅₇	1	1	yes	characteristic
45	C₄	C₂³×C₇₅₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂×D₄×C₇₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇₅₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇₅₇
prime radical	#41×C₇₅₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₅₇ ◁ C₂×C₄×C₇₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×C₇₅₇	C₂	semidirect
3. Q₁₆×C₇₅₇	C₂	semidirect
4. SD₁₆×C₇₅₇	C₂	semidirect
5. M₁₆×C₇₅₇	C₂	semidirect
24. C₇₅₇	#137	direct (N × H)
25. #137	C₇₅₇	direct (N × H)
27. C₂×Q₈	C₂×C₇₅₇	semidirect
28. Q₁₆	C₂×C₇₅₇	semidirect
29. SD₁₆	C₂×C₇₅₇	semidirect
30. M₁₆	C₂×C₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₅₇	2
C₂×Q₈×C₇₅₇	2
Q₁₆×C₇₅₇	2
SD₁₆×C₇₅₇	2
M₁₆×C₇₅₇	2
#137	757

Subgroups — #137×C757 #37564285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #137×C₇₅₇ #37564285