Subgroups — #31783069

Order 17536.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577808	C₂	1	1	yes	maximal, characteristic
2	#22577230	C₂	1	4	yes	maximal
3	C₄×#14555485	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984088	—	2	1	no	—
6	C₂×#14555485	C₂²	1	2	yes	—
7	#17984022	—	2	1	no	—
8	#17984022	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
10	#107×C₁₃₇	—	2	1	no	—
11	#107×C₁₃₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₇₄	—	4	1	no	—
14	C₄⋊Dic₁₃₇	—	2	2	no	—
15	C₄⋊Dic₁₃₇	—	4	1	no	—
16	#14555485	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₃₇	—	4	1	no	—
18	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	—	2	2	no	—
21	C₄⋊C₄×C₁₃₇	—	4	1	no	—
22	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
23	Dic₂₇₄	—	8	2	no	—
24	C₂×Dic₁₃₇	—	4	2	no	—
25	Q₈×C₁₃₇	—	8	2	no	—
26	C₂×C₄×C₁₃₇	—	4	2	no	—
27	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₃₇	Q₁₆	1	2	yes	—
31	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₃₇	—	8	2	no	—
34	C₄×C₁₃₇	SD₃₂	1	2	yes	—
35	C₄×C₁₃₇	SD₃₂	1	2	yes	—
36	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₃₇	—	8	2	no	—
39	C₂×C₁₃₇	#500	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
42	C₁₃₇	#2254	1	1	yes	characteristic, complemented, Sylow
43	#2254	—	137	1	no	maximal, Sylow
44	#500	—	137	1	no	—
45	#500	D₁₃₇	1	1	yes	characteristic
46	#367	—	137	4	no	—
47	C₄×C₁₆	—	137	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	274	1	no	—
50	C₂×C₁₆	—	137	2	no	—
51	#107	—	2	1	no	—
52	#107	—	274	1	no	—
53	#107	—	137	2	no	—
54	#107	C₂×D₁₃₇	1	2	yes	—
55	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	548	1	no	—
58	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	274	2	no	—
63	C₄⋊C₄	—	548	1	no	—
64	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
65	C₁₆	—	137	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1096	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	548	2	no	—
70	C₂×C₄	#14555495	1	1	yes	characteristic
71	C₂×C₄	#14555495	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
73	C₈	#14555500	1	2	yes	—
74	C₈	Q₈×D₁₃₇	1	2	yes	—
75	C₂²	C₂×#14555495	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1096	2	no	—
78	C₄	#17984031	1	2	yes	—
79	C₄	#17984032	1	2	yes	—
80	C₄	#17984086	1	1	yes	characteristic
81	C₄	C₂×#14555500	1	1	yes	characteristic
82	C₂	#22577806	1	1	yes	characteristic
83	C₂	C₂×#17984031	1	1	yes	characteristic
84	C₂	C₂×#17984032	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14555495
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#500×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
hypercenter	#500	D₁₃₇
hypocenter	C₁₃₇	#2254
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #14555485	Q₈	semidirect
42. C₁₃₇	#2254	semidirect

Maximal subgroups

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

subgroup	index
#22577808	2
#22577230	2
C₄×#14555485	2
#500×C₁₃₇	2
#2254	137