Subgroups — #31797848

Order 17536.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555522	C₂	1	2	yes	maximal
2	C₈×#11764047	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555522	—	2	2	no	—
4	C₄×#11764047	C₄	1	1	yes	characteristic
5	C₄×#11764047	C₂²	1	1	yes	characteristic
6	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
7	#14555522	—	4	2	no	—
8	C₂×#11764047	—	2	1	no	—
9	C₂×#11764047	D₄	1	1	yes	characteristic
10	C₂×#11764047	Q₈	1	1	yes	characteristic
11	C₂×#11764040	C₈	1	1	yes	characteristic
12	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
14	#11764047	—	2	2	no	—
15	#11764047	—	2	1	no	—
16	#11764047	SD₁₆	1	2	yes	—
17	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
19	#11764040	C₁₆	1	2	yes	complemented
20	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₈×C₁₃₇	—	2	1	no	—
22	C₂²×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
24	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
25	Dic₁₃₇	—	2	1	no	—
26	Dic₁₃₇	#106	1	1	yes	characteristic
27	Dic₁₃₇	#108	1	1	yes	characteristic
28	C₂×C₁₃₇	#334	1	1	yes	characteristic
29	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₁₃₇	#364	1	1	yes	characteristic
31	C₁₃₇	#1367	1	1	yes	characteristic, complemented, Sylow
32	#1367	—	137	1	no	maximal, Sylow
33	C₄⋊C₁₆	—	137	2	no	—
34	C₈²	—	137	1	no	—
35	C₂×C₁₆	—	274	2	no	—
36	C₄×C₈	—	137	1	no	—
37	C₄×C₈	—	137	1	no	—
38	C₄×C₈	—	137	1	no	—
39	C₂×C₈	—	274	1	no	—
40	C₂×C₈	—	137	1	no	—
41	C₂×C₈	—	137	1	no	—
42	C₂×C₈	—	137	1	no	—
43	C₂×C₈	#11764049	1	1	yes	characteristic
44	C₄²	—	137	1	no	—
45	C₁₆	—	548	2	no	—
46	C₂×C₄	—	137	1	no	—
47	C₂×C₄	—	137	1	no	—
48	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	—	274	2	no	—
51	C₈	—	274	1	no	—
52	C₈	—	137	2	no	—
53	C₈	—	137	2	no	—
54	C₂²	C₄⋊#11764049	1	1	yes	characteristic
55	C₄	—	274	1	no	—
56	C₄	—	137	1	no	—
57	C₄	—	137	1	no	—
58	C₄	#17984183	1	1	yes	characteristic
59	C₄	#17984184	1	1	yes	characteristic
60	C₂	#22578492	1	1	yes	characteristic
61	C₂	#22578506	1	1	yes	characteristic
62	C₂	#22578508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11764049
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₂×C₈×C₁₃₇	C₈
socle, prime radical	C₂²×C₁₃₇	C₄⋊C₈
cosocle	C₄×#11764047	C₂²
hypercenter	C₂×C₈	#11764049
hypocenter	C₁₃₇	#1367
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
19. #11764040	C₁₆	semidirect
31. C₁₃₇	#1367	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14555522	2
C₈×#11764047	2
#1367	137