Subgroups — #31797876

Order 17536.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578510	C₂	1	2	yes	maximal
2	C₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
3	#17984178	C₄	1	4	yes	complemented
4	#17984161	C₂²	1	1	yes	characteristic
5	C₈×#10587739	C₄	1	1	yes	characteristic
6	C₁₆×D₁₃₇	C₄	1	1	yes	characteristic
7	#14555518	C₈	1	1	yes	characteristic
8	C₄×#10587739	C₈	1	1	yes	characteristic
9	#14555513	C₂×C₄	1	2	yes	—
10	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
11	#14555485	C₈	1	1	yes	characteristic
12	C₁₆×C₁₃₇	C₈	1	1	yes	characteristic
13	C₂×#10587739	C₁₆	1	2	yes	—
14	#11764047	C₁₆	1	2	yes	—
15	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
16	#11764040	C₄²	1	1	yes	characteristic
17	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
18	C₂×D₁₃₇	C₂×C₁₆	1	1	yes	characteristic
19	#10587739	—	2	2	no	—
20	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
21	Dic₁₃₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
23	D₁₃₇	M₆₄	1	2	yes	—
24	C₁₃₇	#1397	1	1	yes	characteristic, complemented, Sylow
25	#1397	—	137	1	no	maximal, Sylow
26	C₂×C₃₂	—	137	2	no	—
27	C₄×C₁₆	—	137	1	no	—
28	C₂×C₁₆	—	137	1	no	—
29	C₂×C₁₆	—	137	1	no	—
30	C₄×C₈	—	137	1	no	—
31	C₃₂	—	137	4	no	—
32	C₂×C₈	—	137	1	no	—
33	C₂×C₈	—	137	1	no	—
34	C₄²	—	137	1	no	—
35	C₁₆	—	137	2	no	—
36	C₁₆	—	137	1	no	—
37	C₁₆	#11764049	1	1	yes	characteristic
38	C₂×C₄	—	137	2	no	—
39	C₂×C₄	—	137	1	no	—
40	C₈	—	137	2	no	—
41	C₈	—	137	1	no	—
42	C₈	C₂×#11764049	1	1	yes	characteristic
43	C₂²	—	137	1	no	—
44	C₄	—	274	2	no	—
45	C₄	—	137	1	no	—
46	C₄	C₄×#11764049	1	1	yes	characteristic
47	C₂	—	137	2	no	—
48	C₂	#10587739⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×#11764049
commutator, socle	C₂×C₁₃₇	C₄×C₁₆
Fitting	C₁₆×C₁₃₇	C₈
prime radical	C₂×D₁₃₇	C₂×C₁₆
cosocle	#17984161	C₂²
hypercenter	C₁₆	#11764049
hypocenter	C₁₃₇	#1397
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
3. #17984178	C₄	semidirect
24. C₁₃₇	#1397	semidirect

Maximal subgroups

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

subgroup	index
#22578510	2
C₁₆×#10587739	2
#1397	137