Subgroups — #31795903

Order 17536.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984138	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984138	—	2	2	no	—
4	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
5	C₂×#14555485	C₂²	1	1	yes	characteristic
6	C₁₆×D₁₃₇	—	2	2	no	—
7	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₃₇	C₈	1	1	yes	characteristic
9	C₂×#11764040	C₈	1	1	yes	characteristic
10	C₈×D₁₃₇	—	2	1	no	—
11	C₈×D₁₃₇	—	2	1	no	—
12	#14555485	D₄	1	2	yes	—
13	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₁₃₇	—	2	1	no	—
15	C₂²×D₁₃₇	C₁₆	1	1	yes	characteristic
16	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
17	C₄×D₁₃₇	—	2	1	no	—
18	C₄×D₁₃₇	—	2	1	no	—
19	#11764040	—	2	1	no	—
20	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
23	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
24	C₂×D₁₃₇	—	2	1	no	—
25	C₂×D₁₃₇	—	2	1	no	—
26	C₂×D₁₃₇	C₃₂	1	2	yes	complemented
27	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
29	Dic₁₃₇	—	2	1	no	—
30	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₃₇	C₂×C₃₂	1	1	yes	characteristic
32	C₂×C₁₃₇	M₆₄	1	1	yes	characteristic
33	D₁₃₇	—	2	2	no	—
34	C₁₃₇	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
35	C₂²⋊C₃₂	—	137	1	no	maximal, Sylow
36	C₂²×C₁₆	—	137	1	no	—
37	C₂×C₃₂	—	137	2	no	—
38	C₂²×C₈	—	137	1	no	—
39	C₂×C₁₆	—	274	2	no	—
40	C₂×C₁₆	—	137	1	no	—
41	C₂×C₁₆	#10587739	1	1	yes	characteristic
42	C₃₂	—	274	2	no	—
43	C₂²×C₄	—	137	1	no	—
44	C₂×C₈	—	274	1	no	—
45	C₂×C₈	—	274	1	no	—
46	C₂×C₈	—	137	1	no	—
47	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	—	137	2	no	—
50	C₂³	—	137	1	no	—
51	C₂×C₄	—	274	1	no	—
52	C₂×C₄	—	274	1	no	—
53	C₂×C₄	—	137	1	no	—
54	C₂×C₄	#14555518	1	1	yes	characteristic
55	C₈	—	274	1	no	—
56	C₈	C₂²⋊#10587739	1	1	yes	characteristic
57	C₈	#14555520	1	1	yes	characteristic
58	C₂²	—	274	1	no	—
59	C₂²	—	274	1	no	—
60	C₂²	—	137	2	no	—
61	C₂²	#17984161	1	1	yes	characteristic
62	C₄	—	274	1	no	—
63	C₄	#17984145	1	1	yes	characteristic
64	C₄	#17984163	1	1	yes	characteristic
65	C₂	—	274	2	no	—
66	C₂	#22578298	1	1	yes	characteristic
67	C₂	#22578334	1	1	yes	characteristic
68	C₂	#22578336	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₃₂
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₁₆
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	C₂×#14555485	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	C₂²⋊C₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17984138	C₂	semidirect
26. C₂×D₁₃₇	C₃₂	semidirect
34. C₁₃₇	C₂²⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984138	2
C₂×C₁₆×D₁₃₇	2
C₂²⋊C₃₂	137