Subgroups — C₂²×#17984138 #31797207

Order 17536.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984138	C₂	1	6	yes	maximal, direct product factor
2	C₂²×#14555485	C₂	1	1	yes	maximal, characteristic
3	#17984138	C₂²	1	4	yes	direct product factor
4	C₂×#14555485	C₄	1	3	yes	—
5	C₂×#14555485	C₂²	1	3	yes	—
6	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	#14555485	C₂×C₄	1	3	yes	—
8	#14555485	C₂³	1	1	yes	characteristic
9	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
10	C₂×C₈×C₁₃₇	C₈	1	3	yes	—
11	C₂×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
12	C₂³×C₁₃₇	C₁₆	1	1	yes	characteristic
13	C₂×C₄×C₁₃₇	C₁₆	1	3	yes	—
14	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
15	C₈×C₁₃₇	C₂×C₈	1	3	yes	—
16	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
17	C₂²×C₁₃₇	C₃₂	1	4	yes	complemented
18	C₂²×C₁₃₇	C₂×C₁₆	1	3	yes	—
19	C₄×C₁₃₇	C₂×C₁₆	1	3	yes	—
20	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
21	C₂×C₁₃₇	C₂×C₃₂	1	6	yes	complemented
22	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
23	C₁₃₇	C₂²×C₃₂	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₃₂	—	137	1	no	maximal, Sylow
25	C₂²×C₁₆	—	137	1	no	—
26	C₂×C₃₂	—	137	6	no	—
27	C₂²×C₈	#10587739	1	1	yes	characteristic
28	C₂×C₁₆	—	137	3	no	—
29	C₂×C₁₆	—	137	3	no	—
30	C₃₂	—	137	4	no	—
31	C₂²×C₄	#11764047	1	1	yes	characteristic
32	C₂×C₈	#11764047	1	3	yes	—
33	C₂×C₈	C₂×#10587739	1	3	yes	—
34	C₁₆	—	137	3	no	—
35	C₁₆	—	137	1	no	—
36	C₂³	#14555513	1	1	yes	characteristic
37	C₂×C₄	#14555513	1	3	yes	—
38	C₂×C₄	C₂×#11764047	1	3	yes	—
39	C₈	C₂×#11764047	1	3	yes	—
40	C₈	C₂²×#10587739	1	1	yes	characteristic
41	C₂²	#17984138	1	4	yes	direct product factor
42	C₂²	C₂×#14555513	1	3	yes	—
43	C₄	C₂×#14555513	1	3	yes	—
44	C₄	C₂²×#11764047	1	1	yes	characteristic
45	C₂	C₂×#17984138	1	6	yes	direct product factor
46	C₂	C₂²×#14555513	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	#10587739
commutator, hypocenter	C₁₃₇	C₂²×C₃₂
Frattini	C₈	C₂²×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₁₆
cosocle	#14555485	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17984138	C₂	direct (N × H)
3. #17984138	C₂²	direct (N × H)
17. C₂²×C₁₃₇	C₃₂	semidirect
21. C₂×C₁₃₇	C₂×C₃₂	semidirect
23. C₁₃₇	C₂²×C₃₂	semidirect
41. C₂²	#17984138	direct (N × H)
45. C₂	C₂×#17984138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984138	2
C₂²×#14555485	2
C₂²×C₃₂	137

Subgroups — C22×#17984138 #31797207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17984138 #31797207