Subgroups — C₄×#41×C₁₃₇ #22577052

Order 8768.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₄×D₄×C₁₃₇	C₂	1	6	yes	maximal, complemented
4	#117×C₁₃₇	C₂	1	3	yes	maximal, complemented
5	C₂×C₄²×C₁₃₇	C₂	1	3	yes	maximal, complemented
6	#41×C₁₃₇	C₄	1	8	yes	direct product factor
7	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₁₃₇	C₂²	1	3	yes	—
9	C₂²×C₄×C₁₃₇	C₂²	1	6	yes	—
10	C₂²×C₄×C₁₃₇	C₂²	1	3	yes	—
11	C₄⋊C₄×C₁₃₇	C₂²	1	6	yes	—
12	C₂²⋊C₄×C₁₃₇	C₂²	1	6	yes	—
13	C₄²×C₁₃₇	C₂²	1	3	yes	—
14	C₄²×C₁₃₇	C₂²	1	6	yes	—
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂³×C₁₃₇	C₂³	1	3	yes	—
17	Q₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
18	D₄×C₁₃₇	C₂×C₄	1	12	yes	complemented
19	C₂×C₄×C₁₃₇	—	2	12	no	—
20	C₂×C₄×C₁₃₇	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₁₃₇	C₂³	1	6	yes	—
22	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₂³	1	3	yes	—
24	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
25	C₂²×C₁₃₇	—	2	6	no	—
26	C₂²×C₁₃₇	C₂²×C₄	1	6	yes	—
27	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
28	C₄×C₁₃₇	—	2	6	no	—
29	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
30	C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
31	C₄×C₁₃₇	#41	1	4	yes	direct product factor
32	C₂×C₁₃₇	—	2	6	no	—
33	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
34	C₂×C₁₃₇	C₂×#41	1	2	yes	—
35	C₁₃₇	C₄×#41	1	1	yes	characteristic, direct product factor, Sylow
36	C₄×#41	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×#41	C₂×C₁₃₇	1	1	yes	characteristic
38	C₄×Q₈	C₂×C₁₃₇	1	2	yes	complemented
39	C₄×D₄	C₂×C₁₃₇	1	6	yes	complemented
40	#117	C₂×C₁₃₇	1	3	yes	complemented
41	C₂×C₄²	C₂×C₁₃₇	1	3	yes	complemented
42	#41	C₄×C₁₃₇	1	8	yes	direct product factor
43	C₂×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
44	C₂×D₄	C₂²×C₁₃₇	1	3	yes	—
45	C₂²×C₄	C₂²×C₁₃₇	1	6	yes	—
46	C₂²×C₄	C₂²×C₁₃₇	1	3	yes	—
47	C₄⋊C₄	C₂²×C₁₃₇	1	6	yes	—
48	C₂²⋊C₄	C₂²×C₁₃₇	1	6	yes	—
49	C₄²	C₂²×C₁₃₇	1	3	yes	—
50	C₄²	C₂²×C₁₃₇	1	6	yes	—
51	C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
52	C₂³	C₂³×C₁₃₇	1	3	yes	—
53	Q₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
54	D₄	C₂×C₄×C₁₃₇	1	12	yes	complemented
55	C₂×C₄	—	2	12	no	—
56	C₂×C₄	C₂×C₄×C₁₃₇	1	12	yes	complemented
57	C₂×C₄	C₂³×C₁₃₇	1	6	yes	—
58	C₂×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
59	C₂×C₄	C₂³×C₁₃₇	1	3	yes	—
60	C₂×C₄	C₂³×C₁₃₇	1	2	yes	—
61	C₂²	—	2	6	no	—
62	C₂²	C₂²×C₄×C₁₃₇	1	6	yes	—
63	C₂²	C₂⁴×C₁₃₇	1	1	yes	characteristic
64	C₄	—	2	6	no	—
65	C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
66	C₄	C₂²×C₄×C₁₃₇	1	6	yes	—
67	C₄	#41×C₁₃₇	1	4	yes	direct product factor
68	C₂	—	2	6	no	—
69	C₂	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
70	C₂	C₂×#41×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	C₂²
commutator	C₂	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂⁴
prime radical	C₂×#41×C₁₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×C₁₃₇	C₂	semidirect
3. C₄×D₄×C₁₃₇	C₂	semidirect
4. #117×C₁₃₇	C₂	semidirect
5. C₂×C₄²×C₁₃₇	C₂	semidirect
6. #41×C₁₃₇	C₄	direct (N × H)
17. Q₈×C₁₃₇	C₂×C₄	semidirect
18. D₄×C₁₃₇	C₂×C₄	semidirect
20. C₂×C₄×C₁₃₇	C₂×C₄	semidirect
31. C₄×C₁₃₇	#41	direct (N × H)
35. C₁₃₇	C₄×#41	direct (N × H)
36. C₄×#41	C₁₃₇	direct (N × H)
38. C₄×Q₈	C₂×C₁₃₇	semidirect
39. C₄×D₄	C₂×C₁₃₇	semidirect
40. #117	C₂×C₁₃₇	semidirect
41. C₂×C₄²	C₂×C₁₃₇	semidirect
42. #41	C₄×C₁₃₇	direct (N × H)
53. Q₈	C₂×C₄×C₁₃₇	semidirect
54. D₄	C₂×C₄×C₁₃₇	semidirect
56. C₂×C₄	C₂×C₄×C₁₃₇	semidirect
67. C₄	#41×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×C₁₃₇	2
C₄×Q₈×C₁₃₇	2
C₄×D₄×C₁₃₇	2
#117×C₁₃₇	2
C₂×C₄²×C₁₃₇	2
C₄×#41	137

Subgroups — C4×#41×C137 #22577052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#41×C₁₃₇ #22577052