Subgroups — C₂×#22578334 #31797208

Order 17536.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₁₆	#10587739
commutator, hypocenter	C₁₃₇	C₂²×C₃₂
Frattini	C₈	C₂²×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₁₆
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	#14555485	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22578334	C₂	direct (N × H)
2. C₂×#17984138	C₂	semidirect
4. #17984138	C₂²	semidirect
27. C₂×D₁₃₇	C₃₂	semidirect
32. C₂×C₁₃₇	C₂×C₃₂	semidirect
34. D₁₃₇	C₂×C₃₂	semidirect
35. C₁₃₇	C₂²×C₃₂	semidirect
69. C₂	#22578334	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578334	2
C₂×#17984138	2
C₂×C₁₆×D₁₃₇	2
C₂²×C₃₂	137

Subgroups — C2×#22578334 #31797208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578334 #31797208