Subgroups — #31797836

Order 17536.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587739⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22578493	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
4	#17984183	—	2	1	no	—
5	C₄⋊#11764049	—	2	1	no	—
6	Q₈×#10587739	—	2	1	no	—
7	#17984157	C₄	1	1	yes	characteristic
8	#17984151	—	2	1	no	—
9	C₈×#10587739	C₂²	1	1	yes	characteristic
10	Q₁₆×D₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764049	—	4	1	no	—
12	#14555522	—	2	2	no	—
13	#14555518	D₄	1	1	yes	characteristic
14	C₄⋊#10587739	—	2	1	no	—
15	C₄⋊#10587739	—	4	1	no	—
16	C₄×#10587739	—	4	1	no	—
17	C₄×#10587739	D₄	1	1	yes	characteristic
18	Q₈×D₁₃₇	—	2	1	no	—
19	Dic₅₄₈	C₈	1	1	yes	characteristic, complemented
20	#14555500	—	2	1	no	—
21	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	Q₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
23	#11764049	—	8	1	no	—
24	C₂×#10587739	—	4	1	no	—
25	C₂×#10587739	D₈	1	1	yes	characteristic
26	C₂×#10587739	SD₁₆	1	1	yes	characteristic
27	#11764047	—	2	1	no	—
28	Dic₂₇₄	—	2	1	no	—
29	Dic₂₇₄	—	4	1	no	—
30	C₄×D₁₃₇	—	4	1	no	—
31	C₄×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	#11764040	M₁₆	1	1	yes	characteristic
33	Q₈×C₁₃₇	—	2	1	no	—
34	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×D₁₃₇	#102	1	1	yes	characteristic
36	#10587739	—	2	1	no	—
37	#10587739	—	8	1	no	—
38	#10587739	SD₃₂	1	1	yes	characteristic
39	#10587739	Q₃₂	1	1	yes	characteristic
40	C₄×C₁₃₇	—	4	1	no	—
41	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
42	Dic₁₃₇	—	4	1	no	—
43	Dic₁₃₇	#104	1	1	yes	characteristic
44	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
45	D₁₃₇	#358	1	1	yes	characteristic
46	D₁₃₇	#359	1	1	yes	characteristic
47	C₁₃₇	#1332	1	1	yes	characteristic, complemented, Sylow
48	#1332	—	137	1	no	maximal, Sylow
49	C₄×Q₁₆	—	137	1	no	—
50	C₄×C₁₆	—	137	1	no	—
51	#335	—	137	1	no	—
52	C₂×Q₁₆	—	137	1	no	—
53	C₄×Q₈	—	274	1	no	—
54	C₂×C₁₆	—	274	1	no	—
55	#107	—	137	1	no	—
56	C₄⋊C₈	—	274	1	no	—
57	#103	—	274	1	no	—
58	C₄×C₈	—	137	1	no	—
59	C₂×Q₈	—	274	1	no	—
60	Q₁₆	—	274	1	no	—
61	Q₁₆	—	137	1	no	—
62	Q₁₆	#11764049	1	1	yes	characteristic, complemented
63	C₂×C₈	—	548	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	—	137	1	no	—
66	C₄⋊C₄	—	274	1	no	—
67	C₄⋊C₄	—	548	1	no	—
68	C₄²	—	548	1	no	—
69	C₄²	—	137	1	no	—
70	C₁₆	—	274	2	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	274	1	no	—
73	Q₈	—	548	1	no	—
74	C₂×C₄	—	548	1	no	—
75	C₂×C₄	—	548	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	—	137	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₈	—	274	1	no	—
80	C₈	—	1096	1	no	—
81	C₈	—	137	1	no	—
82	C₈	C₂×#11764049	1	1	yes	characteristic
83	C₂²	—	137	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	274	1	no	—
86	C₄	—	548	1	no	—
87	C₄	—	1096	1	no	—
88	C₄	—	137	1	no	—
89	C₄	—	137	1	no	—
90	C₄	—	137	1	no	—
91	C₄	C₂²⋊#11764049	1	1	yes	characteristic
92	C₂	—	137	1	no	—
93	C₂	—	137	1	no	—
94	C₂	D₄⋊#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄⋊#11764049
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₈	C₂×#11764049
Fitting	Q₁₆×C₁₃₇	C₈
socle	C₂×C₁₃₇	D₄⋊C₈
prime radical	C₂×D₁₃₇	#102
cosocle	C₈×#10587739	C₂²
hypercenter	Q₁₆	#11764049
hypocenter	C₁₃₇	#1332
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
19. Dic₅₄₈	C₈	semidirect
22. Q₁₆×C₁₃₇	C₈	semidirect
47. C₁₃₇	#1332	semidirect
62. Q₁₆	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#10587739⋊C₁₆	2
#22578493	2
Q₁₆×#10587739	2
#1332	137