Subgroups — C₂×#22578348 #31797235

Order 17536.

624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578348	C₂	1	8	yes	maximal, direct product factor
2	C₂×(C₂×#10587739)⋊C₄	C₂	1	6	yes	maximal
3	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10587739	C₂²	1	4	yes	—
5	(C₂×#10587739)⋊C₄	C₂²	1	24	yes	—
6	C₂²×C₄×D₁₃₇	C₂²	1	3	yes	—
7	C₂×C₄×Dic₁₃₇	C₄	1	3	yes	complemented
8	C₄²×D₁₃₇	C₄	1	4	yes	complemented
9	C₄²×D₁₃₇	C₂²	1	4	yes	—
10	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂²×#10587739	—	2	24	no	—
12	C₂²×#10587739	C₂³	1	8	yes	—
13	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
14	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
15	C₂×C₄×D₁₃₇	—	2	12	no	—
16	C₂×C₄×D₁₃₇	C₂×C₄	1	6	yes	—
17	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
18	C₄×Dic₁₃₇	C₂×C₄	1	12	yes	complemented
19	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
20	C₄²×C₁₃₇	C₂×C₄	1	4	yes	complemented
21	C₂×#10587739	—	2	24	no	—
22	C₂×#10587739	—	4	16	no	—
23	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×D₁₃₇	#41	1	12	yes	—
25	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
26	C₂×Dic₁₃₇	—	2	6	no	—
27	C₂×Dic₁₃₇	C₂²×C₄	1	6	yes	—
28	C₄×D₁₃₇	—	2	12	no	—
29	C₄×D₁₃₇	—	2	12	no	—
30	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	—	2	6	no	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
33	C₂²×C₁₃₇	#117	1	6	yes	—
34	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
35	C₂×D₁₃₇	#117	1	6	yes	—
36	C₂×D₁₃₇	#126	1	16	yes	—
37	C₂×D₁₃₇	C₂×#41	1	6	yes	—
38	#10587739	—	4	8	no	—
39	C₄×C₁₃₇	—	2	6	no	—
40	Dic₁₃₇	—	2	6	no	—
41	C₂×C₁₃₇	#383	1	4	yes	complemented
42	C₂×C₁₃₇	C₂×#117	1	3	yes	—
43	D₁₃₇	#383	1	4	yes	complemented
44	D₁₃₇	C₂×#126	1	4	yes	—
45	C₁₃₇	C₂×#383	1	1	yes	characteristic, complemented, Sylow
46	C₂×#383	—	137	1	no	maximal, Sylow
47	C₂²×C₄²	—	137	1	no	—
48	#383	—	137	8	no	—
49	C₂×#95	—	137	6	no	—
50	C₂³×C₄	—	137	4	no	—
51	C₂³×C₄	—	137	3	no	—
52	C₂×C₄²	—	137	3	no	—
53	C₂×C₄²	—	137	4	no	—
54	C₂×C₄²	—	137	4	no	—
55	C₂×C₄²	#10587739	1	1	yes	characteristic, complemented
56	#95	—	137	24	no	—
57	C₂⁴	—	137	1	no	—
58	C₂²×C₄	—	274	12	no	—
59	C₂²×C₄	—	274	24	no	—
60	C₂²×C₄	—	137	8	no	—
61	C₂²×C₄	—	137	3	no	—
62	C₂²×C₄	—	137	6	no	—
63	C₂²×C₄	—	137	6	no	—
64	C₂²×C₄	C₂×#10587739	1	3	yes	—
65	C₄²	—	137	12	no	—
66	C₄²	C₂×#10587739	1	4	yes	complemented
67	C₂³	—	137	12	no	—
68	C₂³	—	137	1	no	—
69	C₂³	—	137	1	no	—
70	C₂³	C₂²×#10587739	1	1	yes	characteristic
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	274	12	no	—
73	C₂×C₄	—	274	12	no	—
74	C₂×C₄	—	274	6	no	—
75	C₂×C₄	—	274	24	no	—
76	C₂×C₄	—	548	16	no	—
77	C₂×C₄	—	137	6	no	—
78	C₂×C₄	C₂²×#10587739	1	6	yes	—
79	C₂²	—	137	6	no	—
80	C₂²	—	137	6	no	—
81	C₂²	—	137	16	no	—
82	C₂²	#17984167	1	6	yes	—
83	C₂²	C₂³×#10587739	1	1	yes	characteristic
84	C₄	—	2	6	no	—
85	C₄	—	274	6	no	—
86	C₄	—	548	8	no	—
87	C₂	—	137	4	no	—
88	C₂	—	137	4	no	—
89	C₂	#22578348	1	4	yes	direct product factor
90	C₂	C₂×#17984167	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×#10587739
commutator	C₂²×C₁₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂³×D₁₃₇	C₂³
cosocle	C₂²×D₁₃₇	C₂⁴
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	C₂×#383
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #22578348	C₂	direct (N × H)
7. C₂×C₄×Dic₁₃₇	C₄	semidirect
8. C₄²×D₁₃₇	C₄	semidirect
10. C₂×C₄²×C₁₃₇	C₄	semidirect
18. C₄×Dic₁₃₇	C₂×C₄	semidirect
20. C₄²×C₁₃₇	C₂×C₄	semidirect
41. C₂×C₁₃₇	#383	semidirect
43. D₁₃₇	#383	semidirect
45. C₁₃₇	C₂×#383	semidirect
55. C₂×C₄²	#10587739	semidirect
66. C₄²	C₂×#10587739	semidirect
89. C₂	#22578348	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578348	2
C₂×(C₂×#10587739)⋊C₄	2
C₂×C₄²×D₁₃₇	2
C₂×#383	137

Subgroups — C2×#22578348 #31797235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578348 #31797235