Subgroups — C₂×#22577231 #31782569

Order 17536.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984022	C₂	1	2	yes	maximal
2	#22577231	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
5	#17984022	—	2	2	no	—
6	#17984022	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₃₇	C₄	1	4	yes	complemented
9	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
10	C₂²×Dic₁₃₇	—	4	2	no	—
11	C₄⋊Dic₁₃₇	—	2	4	no	—
12	C₄⋊Dic₁₃₇	—	4	2	no	—
13	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
17	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
18	C₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
19	C₂×Dic₁₃₇	—	4	4	no	—
20	C₂×Dic₁₃₇	—	8	4	no	—
21	C₂³×C₁₃₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	Q₁₆	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
26	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
27	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₁₃₇	#107	1	2	yes	—
30	C₂²×C₁₃₇	SD₃₂	1	4	yes	—
31	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₃₇	#107	1	2	yes	—
33	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₃₇	—	8	4	no	—
36	C₂×C₁₃₇	#367	1	4	yes	complemented
37	C₂×C₁₃₇	C₂×#107	1	1	yes	characteristic
38	C₂×C₁₃₇	C₂×SD₃₂	1	2	yes	—
39	C₁₃₇	C₂×#367	1	1	yes	characteristic, complemented, Sylow
40	C₂×#367	—	137	1	no	maximal, Sylow
41	C₂²×C₁₆	D₁₃₇	1	1	yes	characteristic
42	C₂×#107	—	137	2	no	—
43	#367	—	137	4	no	—
44	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	274	2	no	—
46	C₂×C₁₆	Dic₁₃₇	1	4	yes	complemented
47	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
48	#107	—	274	2	no	—
49	#107	—	137	4	no	—
50	C₂²×C₄	—	548	2	no	—
51	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
52	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
53	C₂×C₈	D₅₄₈	1	1	yes	characteristic
54	C₂×C₈	Dic₂₇₄	1	2	yes	—
55	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
56	C₄⋊C₄	—	274	4	no	—
57	C₄⋊C₄	—	548	2	no	—
58	C₁₆	C₂×Dic₁₃₇	1	4	yes	complemented
59	C₂³	D₁₀₉₆	1	1	yes	characteristic
60	C₂×C₄	—	548	4	no	—
61	C₂×C₄	—	1096	4	no	—
62	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
63	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
64	C₂×C₄	Dic₅₄₈	1	2	yes	—
65	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
66	C₈	C₄⋊Dic₁₃₇	1	2	yes	—
67	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
68	C₈	C₂×Dic₂₇₄	1	1	yes	characteristic
69	C₂²	#17984022	1	2	yes	—
70	C₂²	#17984033	1	4	yes	—
71	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
72	C₄	—	1096	4	no	—
73	C₄	#17984022	1	2	yes	—
74	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
75	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
76	C₂	#22577231	1	4	yes	direct product factor
77	C₂	C₂×#17984022	1	1	yes	characteristic
78	C₂	C₂×#17984033	1	2	yes	—

	subgroup	G/H
center	C₂³	D₁₀₉₆
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂²×C₁₆×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	D₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂²×C₁₆	D₁₃₇
hypocenter	C₁₃₇	C₂×#367
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22577231	C₂	direct (N × H)
8. C₂×C₁₆×C₁₃₇	C₄	semidirect
18. C₁₆×C₁₃₇	C₂×C₄	semidirect
36. C₂×C₁₃₇	#367	semidirect
39. C₁₃₇	C₂×#367	semidirect
46. C₂×C₁₆	Dic₁₃₇	semidirect
58. C₁₆	C₂×Dic₁₃₇	semidirect
76. C₂	#22577231	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×#17984022	2
#22577231	2
C₂²×C₁₆×C₁₃₇	2
C₂×#367	137

Subgroups — C2×#22577231 #31782569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577231 #31782569