Subgroups — C₄×Q₈×D₃₁₇ #35203686

Order 20288.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×Dic₆₃₄	C₂	1	3	yes	maximal, complemented
4	#23778745	C₂	1	3	yes	maximal, complemented
5	C₄⋊C₄×D₃₁₇	C₂	1	3	yes	maximal
6	C₄²×D₃₁₇	C₂	1	3	yes	maximal
7	C₄×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₆₃₄	C₂²	1	3	yes	—
9	Q₈×D₃₁₇	C₄	1	8	yes	direct product factor
10	C₂×C₄×D₃₁₇	C₂²	1	3	yes	—
11	C₂×C₄×D₃₁₇	C₂²	1	3	yes	—
12	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₃₁₇	C₂²	1	3	yes	—
14	#19099553	C₂²	1	6	yes	—
15	C₄×Dic₃₁₇	C₂²	1	3	yes	—
16	C₄×Dic₃₁₇	C₂²	1	3	yes	—
17	C₄×Dic₃₁₇	C₂²	1	3	yes	—
18	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₁₇	C₂²	1	3	yes	—
20	C₄²×C₃₁₇	C₂²	1	3	yes	—
21	C₂²×D₃₁₇	C₂³	1	1	yes	characteristic
22	Dic₆₃₄	C₂×C₄	1	12	yes	complemented
23	C₂×Dic₃₁₇	C₂³	1	3	yes	—
24	C₂×Dic₃₁₇	C₂³	1	3	yes	—
25	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
26	C₄×D₃₁₇	—	2	6	no	—
27	C₄×D₃₁₇	C₂×C₄	1	12	yes	—
28	C₄×D₃₁₇	Q₈	1	4	yes	direct product factor
29	Q₈×C₃₁₇	C₂×C₄	1	4	yes	complemented
30	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
31	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
32	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
33	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
34	C₂×D₃₁₇	C₂²×C₄	1	2	yes	—
35	C₂×D₃₁₇	C₂×Q₈	1	2	yes	—
36	C₂×D₃₁₇	#41	1	2	yes	—
37	C₄×C₃₁₇	—	2	3	no	—
38	C₄×C₃₁₇	C₂²×C₄	1	6	yes	—
39	C₄×C₃₁₇	C₂×Q₈	1	2	yes	complemented
40	Dic₃₁₇	—	2	3	no	—
41	Dic₃₁₇	C₂²×C₄	1	6	yes	—
42	Dic₃₁₇	C₂×Q₈	1	2	yes	complemented
43	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
45	C₂×C₃₁₇	C₂×#41	1	1	yes	characteristic
46	D₃₁₇	C₄×Q₈	1	4	yes	direct product factor
47	C₃₁₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄×Q₈	—	317	1	no	maximal, Sylow
49	C₂²×Q₈	—	317	1	no	—
50	C₄×Q₈	—	317	1	no	—
51	C₄×Q₈	—	317	3	no	—
52	C₄×Q₈	—	317	3	no	—
53	C₄×Q₈	D₃₁₇	1	1	yes	characteristic, direct product factor
54	C₂×C₄⋊C₄	—	317	3	no	—
55	C₂×C₄²	—	317	3	no	—
56	C₂×Q₈	—	317	8	no	—
57	C₂×Q₈	—	317	3	no	—
58	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
59	C₂²×C₄	—	317	3	no	—
60	C₂²×C₄	—	317	1	no	—
61	C₂²×C₄	—	317	3	no	—
62	C₄⋊C₄	—	317	6	no	—
63	C₄⋊C₄	—	317	3	no	—
64	C₄⋊C₄	C₂×D₃₁₇	1	3	yes	—
65	C₄²	—	317	3	no	—
66	C₄²	—	317	3	no	—
67	C₄²	—	317	3	no	—
68	C₄²	C₂×D₃₁₇	1	3	yes	—
69	C₂³	—	317	1	no	—
70	Q₈	—	317	12	no	—
71	Q₈	C₄×D₃₁₇	1	4	yes	direct product factor
72	C₂×C₄	—	317	1	no	—
73	C₂×C₄	—	317	3	no	—
74	C₂×C₄	—	317	12	no	—
75	C₂×C₄	—	317	4	no	—
76	C₂×C₄	—	317	3	no	—
77	C₂×C₄	—	634	6	no	—
78	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
79	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
80	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
81	C₂²	—	317	2	no	—
82	C₂²	—	317	2	no	—
83	C₂²	—	317	2	no	—
84	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
85	C₄	—	2	3	no	—
86	C₄	—	317	2	no	—
87	C₄	—	317	6	no	—
88	C₄	—	634	3	no	—
89	C₄	C₂×C₄×D₃₁₇	1	6	yes	—
90	C₄	Q₈×D₃₁₇	1	2	yes	direct product factor
91	C₂	—	317	4	no	—
92	C₂	C₂²×C₄×D₃₁₇	1	1	yes	characteristic
93	C₂	C₂×Q₈×D₃₁₇	1	1	yes	characteristic
94	C₂	#41×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₇
commutator	C₂×C₃₁₇	C₂³×C₄
Frattini	C₂²	C₂³×D₃₁₇
Fitting	C₄×Q₈×C₃₁₇	C₂
socle, cosocle	C₂²×C₃₁₇	C₂⁴
prime radical	C₂²×D₃₁₇	C₂³
hypercenter	C₄×Q₈	D₃₁₇
hypocenter	C₃₁₇	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₃₁₇	C₂	semidirect
3. C₄×Dic₆₃₄	C₂	semidirect
4. #23778745	C₂	semidirect
7. C₄×Q₈×C₃₁₇	C₂	semidirect
9. Q₈×D₃₁₇	C₄	direct (N × H)
22. Dic₆₃₄	C₂×C₄	semidirect
28. C₄×D₃₁₇	Q₈	direct (N × H)
29. Q₈×C₃₁₇	C₂×C₄	semidirect
39. C₄×C₃₁₇	C₂×Q₈	semidirect
42. Dic₃₁₇	C₂×Q₈	semidirect
46. D₃₁₇	C₄×Q₈	direct (N × H)
47. C₃₁₇	C₂×C₄×Q₈	semidirect
53. C₄×Q₈	D₃₁₇	direct (N × H)
71. Q₈	C₄×D₃₁₇	direct (N × H)
90. C₄	Q₈×D₃₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₃₁₇	2
Q₈×Dic₃₁₇	2
C₄×Dic₆₃₄	2
#23778745	2
C₄⋊C₄×D₃₁₇	2
C₄²×D₃₁₇	2
C₄×Q₈×C₃₁₇	2
C₂×C₄×Q₈	317

Subgroups — C4×Q8×D317 #35203686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×D₃₁₇ #35203686