Subgroups — #31795896

Order 17536.

68 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	C₄×#14555485	C₂	1	1	yes	maximal, characteristic
3	#17984138	C₄	1	4	yes	complemented
4	C₂×#14555485	C₄	1	1	yes	characteristic
5	C₂×#14555485	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	#14555485	C₈	1	2	yes	—
8	#14555485	C₂×C₄	1	2	yes	—
9	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
10	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
12	C₂×C₄×C₁₃₇	C₁₆	1	2	yes	—
13	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
14	C₈×C₁₃₇	C₁₆	1	2	yes	—
15	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
16	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₁₃₇	—	2	2	no	—
19	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
21	C₂×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₁₃₇	M₆₄	1	2	yes	—
23	C₁₃₇	#1397	1	1	yes	characteristic, complemented, Sylow
24	#1397	—	137	1	no	maximal, Sylow
25	C₂×C₃₂	—	137	2	no	—
26	C₄×C₁₆	—	137	1	no	—
27	C₂×C₁₆	—	137	1	no	—
28	C₂×C₁₆	—	137	1	no	—
29	C₄×C₈	#10587739	1	1	yes	characteristic
30	C₃₂	—	137	4	no	—
31	C₂×C₈	#11764047	1	1	yes	characteristic
32	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
33	C₄²	#11764047	1	1	yes	characteristic
34	C₁₆	—	137	2	no	—
35	C₁₆	—	137	2	no	—
36	C₂×C₄	#14555513	1	2	yes	—
37	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
38	C₈	#14555513	1	2	yes	—
39	C₈	C₄×#10587739	1	1	yes	characteristic
40	C₈	#14555518	1	1	yes	characteristic
41	C₂²	C₂×#14555513	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	C₄×#11764047	1	1	yes	characteristic
44	C₄	C₂×#14555513	1	1	yes	characteristic
45	C₂	C₄×#14555513	1	1	yes	characteristic
46	C₂	#22578335	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₄×C₁₆
Fitting	C₄×C₈×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₂×C₁₆
cosocle	C₂×#14555485	C₂²
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1397
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #17984138	C₄	semidirect
23. C₁₃₇	#1397	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
C₄×#14555485	2
#1397	137