Subgroups — #34089679

Order 19744.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₃₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₆₁₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#23754579	C₂	1	1	yes	maximal, characteristic, complemented
5	#23754579	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊Dic₆₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊C₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic
8	Dic₁₂₃₄	—	2	2	no	—
9	C₂×Dic₆₁₇	—	2	2	no	—
10	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
14	C₂³×C₆₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₁₇	—	2	1	no	—
18	C₂²×C₆₁₇	—	2	1	no	—
19	C₂²×C₆₁₇	Q₈	1	2	yes	complemented
20	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
21	C₄×C₆₁₇	—	2	1	no	—
22	C₄×C₆₁₇	—	2	1	no	—
23	Dic₆₁₇	—	2	1	no	—
24	Dic₆₁₇	—	2	1	no	—
25	Dic₆₁₇	—	2	1	no	—
26	Dic₆₁₇	D₄	1	2	yes	—
27	C₂×C₆₁₇	—	2	2	no	—
28	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₆₁₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₆₁₇	#41	1	1	yes	characteristic
31	C₆₁₇	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂²⋊Q₈	—	617	1	no	maximal, Sylow
33	C₂×Q₈	—	617	1	no	—
34	C₂²×C₄	—	617	1	no	—
35	C₄⋊C₄	—	617	1	no	—
36	C₄⋊C₄	—	617	1	no	—
37	C₄⋊C₄	—	617	1	no	—
38	C₂²⋊C₄	—	617	1	no	—
39	C₂²⋊C₄	D₆₁₇	1	1	yes	characteristic
40	C₂³	C₂×D₆₁₇	1	1	yes	characteristic
41	Q₈	—	1234	2	no	—
42	C₂×C₄	—	617	1	no	—
43	C₂×C₄	—	617	1	no	—
44	C₂×C₄	—	617	1	no	—
45	C₂×C₄	—	617	1	no	—
46	C₂×C₄	—	1234	2	no	—
47	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
48	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₁₂₃₄	1	2	yes	complemented
52	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	617	2	no	—
56	C₄	—	1234	1	no	—
57	C₄	—	1234	1	no	—
58	C₄	—	1234	1	no	—
59	C₂	—	2	2	no	—
60	C₂	D₄×D₆₁₇	1	1	yes	characteristic
61	C₂	C₂×Dic₁₂₃₄	1	1	yes	characteristic
62	C₂	#23754599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁₇
commutator, socle, cosocle	C₂²×C₆₁₇	C₂³
Fitting	C₂²⋊C₄×C₆₁₇	C₂
prime radical	C₂³×C₆₁₇	C₂²
hypercenter	C₂²⋊C₄	D₆₁₇
hypocenter	C₆₁₇	C₂²⋊Q₈
LCS	G ▷ C₂²×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₂₃₄	C₂	semidirect
3. C₄⋊Dic₆₁₇	C₂	semidirect
4. #23754579	C₂	semidirect
5. #23754579	C₂	semidirect
19. C₂²×C₆₁₇	Q₈	semidirect
31. C₆₁₇	C₂²⋊Q₈	semidirect
51. C₂²	Dic₁₂₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂₃₄	2
C₂²×Dic₆₁₇	2
C₄⋊Dic₆₁₇	2
#23754579	2
C₂²⋊Dic₆₁₇	2
C₂²⋊C₄×C₆₁₇	2
C₂²⋊Q₈	617