Subgroups — Q₈×#14555522 #31798012

Order 17536.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555522	C₂	1	3	yes	maximal
2	C₄×#14555522	C₂	1	3	yes	maximal
3	Q₈×#11764047	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555522	C₂²	1	3	yes	—
5	C₂×#14555522	C₂²	1	1	yes	characteristic
6	C₄⋊#11764047	C₄	1	3	yes	—
7	C₄×#11764047	C₂²	1	3	yes	—
8	Q₈×Dic₁₃₇	C₄	1	1	yes	characteristic
9	#14555522	—	2	3	no	—
10	#14555522	Q₈	1	2	yes	direct product factor
11	C₂×#11764047	C₂×C₄	1	3	yes	—
12	C₂×#11764047	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₁₃₇	C₈	1	3	yes	—
14	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
15	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
16	#11764047	—	2	3	no	—
17	#11764047	C₂×Q₈	1	1	yes	characteristic
18	#11764047	#41	1	1	yes	characteristic
19	C₂×Dic₁₃₇	C₂×C₈	1	3	yes	—
20	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	Q₈×C₁₃₇	C₁₆	1	4	yes	complemented
22	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
23	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₁₃₇	C₂×C₁₆	1	6	yes	—
25	Dic₁₃₇	—	2	3	no	—
26	Dic₁₃₇	C₄×Q₈	1	1	yes	characteristic
27	Dic₁₃₇	#131	1	1	yes	characteristic
28	C₂×C₁₃₇	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₁₃₇	#504	1	1	yes	characteristic
31	C₁₃₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	137	1	no	maximal, Sylow
33	C₈×Q₈	—	137	1	no	—
34	C₄⋊C₁₆	—	137	3	no	—
35	C₄×C₁₆	—	137	3	no	—
36	C₄×Q₈	—	137	1	no	—
37	C₂×C₁₆	—	137	3	no	—
38	C₂×C₁₆	—	137	1	no	—
39	C₄⋊C₈	—	137	3	no	—
40	C₄×C₈	—	137	3	no	—
41	C₂×Q₈	#11764049	1	1	yes	characteristic
42	C₂×C₈	—	137	3	no	—
43	C₂×C₈	—	137	1	no	—
44	C₄⋊C₄	—	137	3	no	—
45	C₄²	—	137	3	no	—
46	C₁₆	—	274	3	no	—
47	C₁₆	—	137	2	no	—
48	Q₈	#14555522	1	4	yes	direct product factor
49	C₂×C₄	—	137	3	no	—
50	C₂×C₄	—	137	1	no	—
51	C₂×C₄	C₂×#11764049	1	3	yes	—
52	C₈	—	274	3	no	—
53	C₈	—	137	1	no	—
54	C₈	—	137	1	no	—
55	C₂²	C₂²×#11764049	1	1	yes	characteristic
56	C₄	—	274	3	no	—
57	C₄	—	137	1	no	—
58	C₄	—	137	1	no	—
59	C₄	C₂×#14555522	1	6	yes	—
60	C₂	Q₈×#11764049	1	1	yes	characteristic
61	C₂	C₂²×#14555522	1	1	yes	characteristic
62	C₂	#22578525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator	C₂×C₁₃₇	C₂²×C₁₆
Fitting	C₂×Q₈×C₁₃₇	C₈
socle, prime radical	C₂²×C₁₃₇	C₂²×C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
10. #14555522	Q₈	direct (N × H)
21. Q₈×C₁₃₇	C₁₆	semidirect
31. C₁₃₇	Q₈×C₁₆	semidirect
48. Q₈	#14555522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊#14555522	2
C₄×#14555522	2
Q₈×#11764047	2
Q₈×C₁₆	137

Subgroups — Q8×#14555522 #31798012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14555522 #31798012