Subgroups — C₂³×#14555522 #31798034

Order 17536.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555522	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#11764047	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555522	C₂²	1	28	yes	direct product factor
4	C₂²×#11764047	C₄	1	7	yes	—
5	C₂²×#11764047	C₂²	1	7	yes	—
6	C₂³×Dic₁₃₇	C₄	1	1	yes	characteristic
7	#14555522	C₂³	1	8	yes	direct product factor
8	C₂×#11764047	C₂×C₄	1	21	yes	—
9	C₂×#11764047	C₂³	1	7	yes	—
10	C₂²×Dic₁₃₇	C₈	1	7	yes	—
11	C₂²×Dic₁₃₇	C₂×C₄	1	7	yes	—
12	C₂⁴×C₁₃₇	C₈	1	1	yes	characteristic
13	#11764047	C₂²×C₄	1	7	yes	—
14	#11764047	C₂⁴	1	1	yes	characteristic
15	C₂×Dic₁₃₇	C₂×C₈	1	21	yes	—
16	C₂×Dic₁₃₇	C₂²×C₄	1	7	yes	—
17	C₂³×C₁₃₇	C₁₆	1	8	yes	complemented
18	C₂³×C₁₃₇	C₂×C₈	1	7	yes	—
19	C₂²×C₁₃₇	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₁₃₇	C₂²×C₈	1	7	yes	—
21	Dic₁₃₇	C₂²×C₈	1	7	yes	—
22	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₃₇	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
25	C₁₃₇	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₁₆	—	137	1	no	maximal, Sylow
27	C₂³×C₈	—	137	1	no	—
28	C₂²×C₁₆	—	137	14	no	—
29	C₂³×C₄	—	137	1	no	—
30	C₂²×C₈	—	137	7	no	—
31	C₂²×C₈	—	137	7	no	—
32	C₂×C₁₆	—	137	28	no	—
33	C₂⁴	#11764049	1	1	yes	characteristic
34	C₂²×C₄	—	137	7	no	—
35	C₂²×C₄	—	137	7	no	—
36	C₂×C₈	—	137	7	no	—
37	C₂×C₈	—	137	21	no	—
38	C₁₆	—	137	8	no	—
39	C₂³	#14555522	1	8	yes	direct product factor
40	C₂³	C₂×#11764049	1	7	yes	—
41	C₂×C₄	—	137	7	no	—
42	C₂×C₄	—	137	21	no	—
43	C₈	—	137	1	no	—
44	C₈	—	137	7	no	—
45	C₂²	C₂×#14555522	1	28	yes	direct product factor
46	C₂²	C₂²×#11764049	1	7	yes	—
47	C₄	—	137	1	no	—
48	C₄	—	137	7	no	—
49	C₂	C₂²×#14555522	1	14	yes	direct product factor
50	C₂	C₂³×#11764049	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	#11764049
commutator, hypocenter	C₁₃₇	C₂³×C₁₆
Frattini	C₂	C₂³×#11764049
Fitting, socle, prime radical	C₂⁴×C₁₃₇	C₈
cosocle	#11764047	C₂⁴
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555522	C₂	direct (N × H)
3. C₂×#14555522	C₂²	direct (N × H)
7. #14555522	C₂³	direct (N × H)
17. C₂³×C₁₃₇	C₁₆	semidirect
19. C₂²×C₁₃₇	C₂×C₁₆	semidirect
23. C₂×C₁₃₇	C₂²×C₁₆	semidirect
25. C₁₃₇	C₂³×C₁₆	semidirect
39. C₂³	#14555522	direct (N × H)
45. C₂²	C₂×#14555522	direct (N × H)
49. C₂	C₂²×#14555522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555522	2
C₂³×#11764047	2
C₂³×C₁₆	137

Subgroups — C23×#14555522 #31798034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14555522 #31798034