Subgroups — Q₈⋊#63241 #14545838

Order 2176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63241	C₂	1	1	yes	maximal, characteristic
2	C₄×#63241	C₂	1	1	yes	maximal, characteristic
3	Q₈×#3624	C₂	1	1	yes	maximal, characteristic
4	C₂×#63241	—	2	1	no	—
5	C₂×#63241	—	2	1	no	—
6	Q₈×Dic₁₇	C₄	1	1	yes	characteristic
7	C₄⋊#3624	—	2	1	no	—
8	C₄⋊#3624	C₄	1	1	yes	characteristic
9	C₄×#3624	—	2	1	no	—
10	C₄×#3624	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
12	C₂×#3624	—	2	1	no	—
13	C₂×#3624	D₄	1	1	yes	characteristic
14	C₂×#3624	D₄	1	1	yes	characteristic
15	#63241	—	2	2	no	—
16	#63241	—	4	1	no	—
17	C₄⋊Dic₁₇	—	2	1	no	—
18	C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
19	C₄×Dic₁₇	—	2	1	no	—
20	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	Q₈×C₁₇	—	2	1	no	—
22	Q₈×C₁₇	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂×Dic₁₇	—	2	1	no	—
26	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₇	M₁₆	1	1	yes	characteristic
28	#3624	—	2	1	no	—
29	#3624	—	4	1	no	—
30	#3624	SD₁₆	1	1	yes	characteristic
31	#3624	Q₁₆	1	1	yes	characteristic
32	Q₈⋊C₁₆	—	17	1	no	maximal, Sylow
33	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₁₇	—	2	2	no	—
35	C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
36	C₄×C₁₇	M₃₂	1	1	yes	characteristic
37	Dic₁₇	—	2	1	no	—
38	Dic₁₇	—	4	1	no	—
39	Dic₁₇	#103	1	1	yes	characteristic
40	Dic₁₇	#104	1	1	yes	characteristic
41	C₈×Q₈	—	17	1	no	—
42	C₄⋊C₁₆	—	17	1	no	—
43	C₄×C₁₆	—	17	1	no	—
44	C₂×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
45	C₂×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
46	C₂×C₁₇	#350	1	1	yes	characteristic
47	C₄×Q₈	—	17	1	no	—
48	C₂×C₁₆	—	34	1	no	—
49	C₂×C₁₆	—	34	1	no	—
50	C₄⋊C₈	—	17	1	no	—
51	C₄⋊C₈	—	34	1	no	—
52	C₄×C₈	—	17	1	no	—
53	C₄×C₈	—	34	1	no	—
54	C₁₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	#3633	1	1	yes	characteristic
56	C₂×C₈	—	17	1	no	—
57	C₂×C₈	—	17	1	no	—
58	C₂×C₈	—	34	1	no	—
59	C₄⋊C₄	—	17	1	no	—
60	C₄⋊C₄	—	34	1	no	—
61	C₄²	—	17	1	no	—
62	C₄²	—	34	1	no	—
63	C₁₆	—	34	2	no	—
64	C₁₆	—	68	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	#63241	1	2	yes	complemented
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	17	1	no	—
69	C₂×C₄	—	17	1	no	—
70	C₂×C₄	—	34	1	no	—
71	C₂×C₄	C₂×#3633	1	1	yes	characteristic
72	C₈	—	17	1	no	—
73	C₈	—	17	1	no	—
74	C₈	—	34	1	no	—
75	C₈	—	68	1	no	—
76	C₂²	C₂²⋊#3633	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	17	1	no	—
79	C₄	—	17	1	no	—
80	C₄	—	34	1	no	—
81	C₄	—	68	1	no	—
82	C₄	C₂×#63241	1	1	yes	characteristic
83	C₄	#10587670	1	1	yes	characteristic
84	C₂	Q₈⋊#3633	1	1	yes	characteristic
85	C₂	C₂²⋊#63241	1	1	yes	characteristic
86	C₂	#11763910	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#3633
commutator	C₄×C₁₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#3633
Fitting	C₂×Q₈×C₁₇	C₈
socle, prime radical	C₂²×C₁₇	C₂²⋊C₈
cosocle	C₄×#3624	C₂²
hypercenter	C₂×Q₈	#3633
hypocenter	C₁₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
22. Q₈×C₁₇	C₁₆	semidirect
54. C₁₇	Q₈⋊C₁₆	semidirect
66. Q₈	#63241	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#63241	2
C₄×#63241	2
Q₈×#3624	2
Q₈⋊C₁₆	17

Subgroups — Q8⋊#63241 #14545838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊#63241 #14545838