Subgroups — #31795938

Order 17536.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578334	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578329	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984157	—	2	1	no	—
5	#17984138	—	2	1	no	—
6	D₈×D₁₃₇	—	2	1	no	—
7	D₂₁₉₂	C₄	1	1	yes	characteristic, complemented
8	#17984029	—	2	1	no	—
9	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
10	D₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10587739	—	4	1	no	—
12	D₄×D₁₃₇	—	4	1	no	—
13	D₁₀₉₆	—	2	1	no	—
14	#14555495	—	4	1	no	—
15	C₈×D₁₃₇	D₄	1	1	yes	characteristic
16	#14555485	D₄	1	1	yes	characteristic
17	D₈×C₁₃₇	—	2	1	no	—
18	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#10587739	—	8	1	no	—
20	C₂²×D₁₃₇	—	8	1	no	—
21	D₅₄₈	—	4	1	no	—
22	C₂²⋊D₁₃₇	—	8	1	no	—
23	C₄×D₁₃₇	D₈	1	1	yes	characteristic
24	#11764040	SD₁₆	1	1	yes	characteristic
25	D₄×C₁₃₇	—	4	1	no	—
26	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	—	8	1	no	—
28	C₂×D₁₃₇	—	8	1	no	—
29	C₂×D₁₃₇	—	16	1	no	—
30	C₂×D₁₃₇	—	16	1	no	—
31	C₂×D₁₃₇	D₁₆	1	1	yes	characteristic
32	#10587739	—	16	1	no	—
33	C₄×C₁₃₇	#102	1	1	yes	characteristic
34	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
35	C₂×C₁₃₇	—	16	1	no	—
36	C₂×C₁₃₇	#357	1	1	yes	characteristic
37	D₁₃₇	—	16	1	no	—
38	D₁₃₇	D₃₂	1	1	yes	characteristic
39	D₁₃₇	SD₆₄	1	1	yes	characteristic
40	C₁₃₇	#1415	1	1	yes	characteristic, complemented, Sylow
41	#1415	—	137	1	no	maximal, Sylow
42	C₂×D₁₆	—	137	1	no	—
43	C₂×C₃₂	—	137	1	no	—
44	#366	—	137	1	no	—
45	C₂×D₈	—	274	1	no	—
46	D₁₆	—	274	1	no	—
47	D₁₆	—	137	1	no	—
48	D₁₆	#10587739	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	137	1	no	—
50	#107	—	274	1	no	—
51	C₃₂	—	274	1	no	—
52	C₂×D₄	—	548	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	274	1	no	—
55	D₈	—	548	1	no	—
56	C₂×C₈	—	137	1	no	—
57	C₄⋊C₄	—	548	1	no	—
58	C₁₆	—	137	1	no	—
59	C₁₆	C₂×#10587739	1	1	yes	characteristic
60	C₂³	—	1096	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	548	1	no	—
63	D₄	—	1096	1	no	—
64	C₂×C₄	—	1096	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₈	—	137	1	no	—
67	C₈	C₂²⋊#10587739	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	1096	1	no	—
70	C₂²	—	137	1	no	—
71	C₂²	—	2192	1	no	—
72	C₂²	—	2192	1	no	—
73	C₄	—	137	1	no	—
74	C₄	—	2192	1	no	—
75	C₄	#17984150	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	137	1	no	—
78	C₂	—	137	1	no	—
79	C₂	—	2192	1	no	—
80	C₂	#22578315	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578315
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₁₆	C₂×#10587739
Fitting	D₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	#357
prime radical	D₁₆×D₁₃₇	C₂
cosocle	C₁₆×D₁₃₇	C₂²
hypercenter	D₁₆	#10587739
hypocenter	C₁₃₇	#1415
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #22578334	C₂	semidirect
2. #22578329	C₂	semidirect
7. D₂₁₉₂	C₄	semidirect
10. D₁₆×C₁₃₇	C₄	semidirect
40. C₁₃₇	#1415	semidirect
48. D₁₆	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578334	2
#22578329	2
D₁₆×D₁₃₇	2
#1415	137