Subgroups — #31783042

Order 17536.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₉₆	C₂	1	4	yes	maximal
2	#22577783	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₄₈	—	2	2	no	—
5	C₂×Dic₅₄₈	C₂²	1	4	yes	—
6	#17984088	—	2	2	no	—
7	Dic₁₀₉₆	—	2	8	no	—
8	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₇₄	—	2	4	no	—
11	Dic₅₄₈	—	2	8	no	—
12	Dic₅₄₈	—	4	4	no	—
13	C₄⋊Dic₁₃₇	—	4	4	no	—
14	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₇	D₄	1	4	yes	—
18	Dic₂₇₄	—	4	4	no	—
19	Dic₂₇₄	—	4	4	no	—
20	C₂×Dic₁₃₇	—	4	4	no	—
21	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
22	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
23	C₈×C₁₃₇	D₈	1	2	yes	—
24	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
25	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
26	C₄×C₁₃₇	Q₃₂	1	4	yes	complemented
27	C₄×C₁₃₇	#127	1	1	yes	characteristic
28	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
29	Dic₁₃₇	—	8	4	no	—
30	C₂×C₁₃₇	#493	1	1	yes	characteristic
31	C₂×C₁₃₇	C₂×Q₃₂	1	2	yes	—
32	C₁₃₇	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	137	1	no	maximal, Sylow
34	C₂×Q₃₂	—	137	4	no	—
35	#494	—	137	2	no	—
36	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	274	2	no	—
38	C₂×Q₁₆	—	137	4	no	—
39	C₄⋊Q₈	—	274	2	no	—
40	Q₃₂	—	274	8	no	—
41	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
42	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
43	C₂×Q₈	—	274	4	no	—
44	Q₁₆	—	274	8	no	—
45	Q₁₆	—	548	4	no	—
46	C₂×C₈	D₅₄₈	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
48	C₄⋊C₄	—	548	4	no	—
49	C₄²	D₅₄₈	1	1	yes	characteristic
50	C₁₆	D₅₄₈	1	4	yes	—
51	Q₈	—	548	4	no	—
52	Q₈	—	548	4	no	—
53	C₂×C₄	—	548	4	no	—
54	C₂×C₄	D₁₀₉₆	1	2	yes	—
55	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
56	C₈	D₁₀₉₆	1	2	yes	—
57	C₈	C₂×D₅₄₈	1	2	yes	—
58	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
59	C₄	—	1096	4	no	—
60	C₄	Dic₁₀₉₆	1	4	yes	complemented
61	C₄	#17984085	1	1	yes	characteristic
62	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
63	C₂	#22577780	1	1	yes	characteristic
64	C₂	C₂×Dic₁₀₉₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₉₆
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2247
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₁₃₇	Q₃₂	semidirect
32. C₁₃₇	#2247	semidirect
60. C₄	Dic₁₀₉₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₉₆	2
#22577783	2
C₄×C₁₆×C₁₃₇	2
#2247	137