Subgroups — #31795947

Order 17536.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984138	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984138	—	2	2	no	—
4	C₂×#14555485	C₂²	1	1	yes	characteristic
5	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764040	C₈	1	1	yes	characteristic
8	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic
9	#14555485	D₄	1	1	yes	characteristic
10	#14555485	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	—	2	1	no	—
13	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
14	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
15	#11764040	—	2	1	no	—
16	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
17	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
19	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
20	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
22	Dic₁₃₇	—	2	1	no	—
23	Dic₁₃₇	C₃₂	1	2	yes	complemented
24	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
25	C₂×C₁₃₇	C₂×C₃₂	1	1	yes	characteristic
26	C₂×C₁₃₇	M₆₄	1	1	yes	characteristic
27	C₁₃₇	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₄⋊C₃₂	—	137	1	no	maximal, Sylow
29	C₂×C₃₂	—	137	2	no	—
30	C₄×C₁₆	—	137	1	no	—
31	C₂×C₁₆	—	137	1	no	—
32	C₂×C₁₆	#10587739	1	1	yes	characteristic
33	C₄×C₈	—	137	1	no	—
34	C₃₂	—	274	2	no	—
35	C₂×C₈	—	137	1	no	—
36	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
37	C₄²	—	137	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	—	137	1	no	—
40	C₁₆	—	137	1	no	—
41	C₂×C₄	—	137	1	no	—
42	C₂×C₄	—	137	1	no	—
43	C₂×C₄	#14555518	1	1	yes	characteristic
44	C₈	—	274	1	no	—
45	C₈	C₄⋊#10587739	1	1	yes	characteristic
46	C₈	#14555519	1	1	yes	characteristic
47	C₂²	#17984161	1	1	yes	characteristic
48	C₄	—	274	1	no	—
49	C₄	—	137	2	no	—
50	C₄	#17984155	1	1	yes	characteristic
51	C₄	#17984163	1	1	yes	characteristic
52	C₂	#22578324	1	1	yes	characteristic
53	C₂	#22578334	1	1	yes	characteristic
54	C₂	#22578336	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₃₂
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₂×C₁₆
cosocle	C₂×#14555485	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	C₄⋊C₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
23. Dic₁₃₇	C₃₂	semidirect
27. C₁₃₇	C₄⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984138	2
C₁₆×Dic₁₃₇	2
C₄⋊C₃₂	137