Subgroups — #31783072

Order 17536.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577805	C₂	1	1	yes	maximal, characteristic
2	#22577230	C₂	1	2	yes	maximal
3	#22577228	C₂	1	2	yes	maximal
4	#22577179	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984080	—	2	1	no	—
7	C₂×#14555485	C₂²	1	2	yes	—
8	#17984022	C₂²	1	1	yes	characteristic
9	#17984022	C₂²	1	1	yes	characteristic
10	#17984020	—	2	1	no	—
11	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
12	#107×C₁₃₇	—	2	1	no	—
13	#107×C₁₃₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	1	no	—
16	C₄⋊Dic₁₃₇	—	2	1	no	—
17	#14555489	—	4	1	no	—
18	#14555489	—	4	1	no	—
19	#14555485	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₁₃₇	—	4	1	no	—
21	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₄⋊C₄×C₁₃₇	—	4	1	no	—
25	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×Dic₁₃₇	—	4	1	no	—
27	C₂×Dic₁₃₇	—	4	1	no	—
28	Q₈×C₁₃₇	—	8	2	no	—
29	C₂×C₄×C₁₃₇	—	4	2	no	—
30	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₃₇	Q₁₆	1	2	yes	—
34	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	—	2	1	no	—
38	C₄×C₁₃₇	—	8	2	no	—
39	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₁₃₇	—	8	1	no	—
42	Dic₁₃₇	—	8	1	no	—
43	C₂×C₁₃₇	#500	1	1	yes	characteristic
44	C₂×C₁₃₇	#509	1	1	yes	characteristic
45	C₂×C₁₃₇	#510	1	1	yes	characteristic
46	C₁₃₇	#2255	1	1	yes	characteristic, complemented, Sylow
47	#2255	—	137	1	no	maximal, Sylow
48	#500	D₁₃₇	1	1	yes	characteristic
49	#499	—	137	1	no	—
50	#367	—	137	2	no	—
51	#366	—	137	2	no	—
52	#346	—	137	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	274	1	no	—
55	C₂×C₁₆	—	137	2	no	—
56	#107	—	2	1	no	—
57	#107	—	137	1	no	—
58	#107	—	137	1	no	—
59	#107	C₂×D₁₃₇	1	2	yes	—
60	#106	—	274	1	no	—
61	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	274	1	no	—
68	C₄⋊C₄	—	274	1	no	—
69	C₄⋊C₄	—	548	1	no	—
70	C₄⋊C₄	—	548	1	no	—
71	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
72	C₁₆	—	137	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	548	1	no	—
76	C₂×C₄	—	548	1	no	—
77	C₂×C₄	#14555495	1	1	yes	characteristic
78	C₂×C₄	#14555495	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
80	C₈	#14555500	1	2	yes	—
81	C₈	Q₈×D₁₃₇	1	2	yes	—
82	C₂²	C₂×#14555495	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	1096	1	no	—
87	C₄	—	1096	1	no	—
88	C₄	#17984086	1	1	yes	characteristic
89	C₄	C₂×#14555500	1	1	yes	characteristic
90	C₂	#22577806	1	1	yes	characteristic
91	C₂	#22577839	1	1	yes	characteristic
92	C₂	#22577844	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	#2255
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
19. #14555485	Q₈	semidirect
46. C₁₃₇	#2255	semidirect

Maximal subgroups

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

subgroup	index
#22577805	2
#22577230	2
#22577228	2
#22577179	2
#500×C₁₃₇	2
#2255	137