Subgroups — #31783083

Order 17536.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577237	C₂	1	3	yes	maximal, complemented
2	C₂×#17983994	C₂	1	3	yes	maximal, complemented
3	#504×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17983994	C₂²	1	1	yes	characteristic
5	#17983994	C₂²	1	3	yes	—
6	#131×C₁₃₇	C₄	1	1	yes	characteristic
7	M₃₂×C₁₃₇	C₄	1	3	yes	—
8	C₂×C₁₆×C₁₃₇	C₂²	1	3	yes	—
9	#41×C₁₃₇	C₈	1	1	yes	characteristic
10	M₁₆×C₁₃₇	C₈	1	3	yes	—
11	C₂×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
12	C₁₆×C₁₃₇	C₂×C₄	1	3	yes	—
13	C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
14	Q₈×C₁₃₇	C₁₆	1	1	yes	characteristic
15	D₄×C₁₃₇	C₁₆	1	3	yes	—
16	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
17	C₈×C₁₃₇	C₂×C₈	1	3	yes	—
18	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₁₃₇	C₂×C₁₆	1	3	yes	—
20	C₄×C₁₃₇	C₂×C₁₆	1	3	yes	—
21	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₁₃₇	—	2	3	no	—
23	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
24	C₁₃₇	#2258	1	1	yes	characteristic, complemented, Sylow
25	#2258	—	137	1	no	maximal, Sylow
26	#504	D₁₃₇	1	1	yes	characteristic
27	M₆₄	—	137	3	no	—
28	C₂×C₃₂	—	137	3	no	—
29	#131	Dic₁₃₇	1	1	yes	characteristic
30	M₃₂	Dic₁₃₇	1	3	yes	—
31	C₂×C₁₆	C₂×D₁₃₇	1	3	yes	—
32	C₃₂	—	137	1	no	—
33	C₃₂	—	137	3	no	—
34	#41	#11764040	1	1	yes	characteristic
35	M₁₆	#11764040	1	3	yes	—
36	C₂×C₈	C₂×Dic₁₃₇	1	3	yes	—
37	C₁₆	C₂×Dic₁₃₇	1	3	yes	—
38	C₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
39	Q₈	#14555485	1	1	yes	characteristic
40	D₄	#14555485	1	3	yes	—
41	C₂×C₄	C₂×#11764040	1	3	yes	—
42	C₈	C₂×#11764040	1	3	yes	—
43	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
44	C₂²	C₂×#14555485	1	3	yes	—
45	C₄	C₂×#14555485	1	3	yes	—
46	C₄	C₂²×#11764040	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×#14555485	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₃₇
commutator, socle	C₂×C₁₃₇	C₂²×C₁₆
Fitting	#504×C₁₃₇	C₂
prime radical	#41×C₁₃₇	C₈
cosocle	C₁₆×C₁₃₇	C₂³
hypercenter	#504	D₁₃₇
hypocenter	C₁₃₇	#2258
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22577237	C₂	semidirect
2. C₂×#17983994	C₂	semidirect
24. C₁₃₇	#2258	semidirect

Maximal subgroups

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

subgroup	index
#22577237	2
C₂×#17983994	2
#504×C₁₃₇	2
#2258	137