Subgroups — #31795906

Order 17536.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578335	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984138	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577820	C₂	1	1	yes	maximal, characteristic
4	#17984138	—	2	1	no	—
5	#17984138	—	2	1	no	—
6	#17984028	—	2	1	no	—
7	#17984028	C₄	1	1	yes	characteristic
8	C₂×#14555485	—	2	1	no	—
9	C₂×#14555485	C₂²	1	1	yes	characteristic
10	#131×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555485	—	2	1	no	—
12	#14555485	D₄	1	1	yes	characteristic
13	#14555485	D₄	1	1	yes	characteristic
14	#41×C₁₃₇	C₈	1	1	yes	characteristic
15	M₁₆×C₁₃₇	—	2	1	no	—
16	M₁₆×C₁₃₇	C₈	1	1	yes	characteristic
17	C₂×C₈×C₁₃₇	—	2	1	no	—
18	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₁₃₇	C₁₆	1	1	yes	characteristic
20	D₄×C₁₃₇	—	2	1	no	—
21	D₄×C₁₃₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₃₇	—	2	1	no	—
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	—	2	1	no	—
28	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
29	C₄×C₁₃₇	—	2	1	no	—
30	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	—	4	1	no	—
34	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	C₁₃₇	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	137	1	no	maximal, Sylow
37	#504	—	137	1	no	—
38	M₆₄	—	137	1	no	—
39	C₂×C₃₂	—	137	1	no	—
40	#131	#10587739	1	1	yes	characteristic
41	M₃₂	—	274	1	no	—
42	M₃₂	—	137	1	no	—
43	C₂×C₁₆	—	274	1	no	—
44	C₂×C₁₆	—	137	1	no	—
45	C₃₂	—	274	1	no	—
46	C₃₂	—	274	1	no	—
47	#41	#11764047	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	#11764047	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
52	C₁₆	—	274	1	no	—
53	C₁₆	—	137	1	no	—
54	C₁₆	—	137	1	no	—
55	Q₈	#14555513	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	#14555513	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊#10587739	1	1	yes	characteristic
62	C₈	#14555519	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#17984162	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊#11764047	1	1	yes	characteristic
67	C₄	C₂×#14555513	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊#14555513	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10587739
Fitting	#131×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	#41×C₁₃₇	C₈
cosocle	C₂×#14555485	C₂²
hypercenter	#131	#10587739
hypocenter	C₁₃₇	#1401
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #22578335	C₂	semidirect
2. C₂×#17984138	C₂	semidirect
35. C₁₃₇	#1401	semidirect

Maximal subgroups

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

subgroup	index
#22578335	2
C₂×#17984138	2
#22577820	2
#1401	137