Subgroups — #22578452

Order 8768.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984155	C₂	1	2	yes	maximal
2	C₄⋊#11764047	C₂	1	1	yes	maximal, characteristic
3	C₄×#11764047	C₂	1	1	yes	maximal, characteristic
4	C₄×#11764047	C₂	1	2	yes	maximal
5	#17984053	C₂	1	1	yes	maximal, characteristic
6	C₂×#11764047	C₂²	1	2	yes	—
7	C₂×#11764047	C₂²	1	1	yes	characteristic
8	C₂×#11764047	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₇₄	C₄	1	1	yes	characteristic
10	#14555489	C₄	1	2	yes	—
11	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₃₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
14	#11764047	—	2	2	no	—
15	#11764047	—	2	1	no	—
16	#11764047	Q₈	1	2	yes	complemented
17	Dic₂₇₄	C₈	1	4	yes	complemented
18	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₃₇	—	2	2	no	—
25	C₄×C₁₃₇	C₂×C₈	1	2	yes	—
26	Dic₁₃₇	—	2	1	no	—
27	Dic₁₃₇	C₂×C₈	1	4	yes	—
28	Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	Dic₁₃₇	#41	1	1	yes	characteristic
30	C₂×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₃₇	#131	1	1	yes	characteristic
33	C₁₃₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	137	1	no	maximal, Sylow
35	C₄×Q₈	—	137	1	no	—
36	C₄⋊C₈	—	137	1	no	—
37	C₄⋊C₈	—	137	2	no	—
38	C₄×C₈	—	137	1	no	—
39	C₄×C₈	—	137	2	no	—
40	C₂×Q₈	—	137	1	no	—
41	C₂×C₈	—	137	2	no	—
42	C₂×C₈	—	137	1	no	—
43	C₂×C₈	—	137	1	no	—
44	C₄⋊C₄	—	137	2	no	—
45	C₄⋊C₄	#10587739	1	1	yes	characteristic
46	C₄²	—	137	1	no	—
47	C₄²	—	137	2	no	—
48	Q₈	—	137	4	no	—
49	C₂×C₄	—	137	1	no	—
50	C₂×C₄	—	137	1	no	—
51	C₂×C₄	—	137	2	no	—
52	C₂×C₄	C₂×#10587739	1	2	yes	—
53	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
54	C₈	—	274	2	no	—
55	C₈	—	274	1	no	—
56	C₈	—	137	2	no	—
57	C₂²	C₂²×#10587739	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	274	1	no	—
60	C₄	—	137	1	no	—
61	C₄	—	137	1	no	—
62	C₄	—	137	4	no	—
63	C₄	#14555518	1	2	yes	—
64	C₂	Q₈×#10587739	1	1	yes	characteristic
65	C₂	C₂×#14555518	1	1	yes	characteristic
66	C₂	#17984175	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂²×C₈
Fitting	C₄⋊C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₄
hypercenter	C₄⋊C₄	#10587739
hypocenter	C₁₃₇	C₈×Q₈
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #11764047	Q₈	semidirect
17. Dic₂₇₄	C₈	semidirect
33. C₁₃₇	C₈×Q₈	semidirect