Subgroups — #31778982

Order 17536.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577237	C₂	1	1	yes	maximal, characteristic
2	#22577233	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984024	—	2	1	no	—
5	#17983994	—	2	1	no	—
6	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
7	Q₃₂×C₁₃₇	—	2	1	no	—
8	Q₃₂×C₁₃₇	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	#14555494	—	4	1	no	—
11	C₂×Q₈×C₁₃₇	—	4	1	no	—
12	Q₁₆×C₁₃₇	—	2	2	no	—
13	Q₁₆×C₁₃₇	—	4	1	no	—
14	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
17	#11764040	—	8	1	no	—
18	Q₈×C₁₃₇	—	4	2	no	—
19	Q₈×C₁₃₇	—	8	1	no	—
20	C₂×C₄×C₁₃₇	—	8	1	no	—
21	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
22	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₃₇	SD₃₂	1	1	yes	characteristic
25	C₄×C₁₃₇	—	8	2	no	—
26	C₄×C₁₃₇	#102	1	1	yes	characteristic
27	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
28	C₂×C₁₃₇	—	2	1	no	—
29	C₂×C₁₃₇	#357	1	1	yes	characteristic
30	C₁₃₇	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	137	1	no	maximal, Sylow
32	C₂×Q₃₂	D₁₃₇	1	1	yes	characteristic
33	M₆₄	—	137	1	no	—
34	#368	—	137	1	no	—
35	C₂×Q₁₆	—	2	1	no	—
36	Q₃₂	—	2	1	no	—
37	Q₃₂	Dic₁₃₇	1	2	yes	—
38	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
39	#108	—	274	1	no	—
40	C₃₂	—	274	1	no	—
41	C₂×Q₈	—	4	1	no	—
42	Q₁₆	—	2	2	no	—
43	Q₁₆	—	4	1	no	—
44	M₁₆	—	548	1	no	—
45	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
46	C₁₆	C₂×Dic₁₃₇	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₁₃₇	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	8	1	no	—
50	C₂×C₄	—	8	1	no	—
51	C₂×C₄	#14555495	1	1	yes	characteristic
52	C₈	—	1096	1	no	—
53	C₈	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
54	C₈	#14555498	1	1	yes	characteristic
55	C₂²	#17984032	1	1	yes	characteristic
56	C₄	—	8	2	no	—
57	C₄	#17984009	1	1	yes	characteristic
58	C₄	#17984029	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#22577204	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577204
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	C₂×Q₃₂×C₁₃₇	C₂
socle	C₂×C₁₃₇	#357
prime radical	C₂²×C₁₃₇	SD₃₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	C₂×Q₃₂	D₁₃₇
hypocenter	C₁₃₇	#1420
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
30. C₁₃₇	#1420	semidirect

Maximal subgroups

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

subgroup	index
#22577237	2
#22577233	2
C₂×Q₃₂×C₁₃₇	2
#1420	137