Subgroups — C₂×#23754599 #34089744

Order 19744.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754599	C₂	1	8	yes	maximal, direct product factor
2	C₂×Dic₁₂₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₆₁₇	C₂	1	2	yes	maximal, complemented
4	C₂²×Dic₆₁₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₁₇	C₂²	1	1	yes	characteristic
8	Dic₁₂₃₄	C₂²	1	4	yes	complemented
9	C₂²⋊D₆₁₇	C₂²	1	8	yes	complemented
10	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₁₇	C₂²	1	2	yes	—
12	C₂×Dic₆₁₇	C₂²	1	8	yes	complemented
13	C₄×D₆₁₇	C₂²	1	4	yes	complemented
14	C₂³×C₆₁₇	C₂²	1	2	yes	—
15	D₄×C₆₁₇	C₂²	1	4	yes	complemented
16	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₁₇	—	2	4	no	—
18	C₂²×C₆₁₇	C₂³	1	4	yes	—
19	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
20	C₂×D₆₁₇	—	2	2	no	—
21	C₂×D₆₁₇	C₂³	1	2	yes	—
22	C₄×C₆₁₇	C₂³	1	2	yes	—
23	Dic₆₁₇	C₂³	1	2	yes	—
24	Dic₆₁₇	C₂³	1	4	yes	—
25	C₂×C₆₁₇	—	2	4	no	—
26	C₂×C₆₁₇	#41	1	2	yes	complemented
27	C₂×C₆₁₇	C₂⁴	1	1	yes	characteristic
28	D₆₁₇	—	2	2	no	—
29	C₆₁₇	C₂×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂×#41	—	617	1	no	maximal, Sylow
31	#41	—	617	8	no	—
32	C₂×Q₈	—	617	1	no	—
33	C₂×D₄	—	617	2	no	—
34	C₂×D₄	D₆₁₇	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	617	2	no	—
36	C₂²×C₄	—	617	1	no	—
37	C₂³	—	617	1	no	—
38	C₂³	C₂×D₆₁₇	1	2	yes	—
39	Q₈	—	617	4	no	—
40	D₄	—	617	8	no	—
41	D₄	C₂×D₆₁₇	1	4	yes	complemented
42	C₂×C₄	—	617	8	no	—
43	C₂×C₄	—	617	1	no	—
44	C₂×C₄	—	617	2	no	—
45	C₂×C₄	—	617	4	no	—
46	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
47	C₂²	—	2	4	no	—
48	C₂²	—	617	2	no	—
49	C₂²	—	1234	2	no	—
50	C₂²	C₂²×D₆₁₇	1	4	yes	—
51	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
52	C₄	—	617	2	no	—
53	C₄	—	617	4	no	—
54	C₄	C₂²×D₆₁₇	1	2	yes	—
55	C₂	—	2	4	no	—
56	C₂	—	1234	2	no	—
57	C₂	#23754599	1	2	yes	direct product factor
58	C₂	C₂³×D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₁₇
commutator, cosocle	C₂×C₆₁₇	C₂⁴
Frattini	C₂	C₂³×D₆₁₇
Fitting	C₂×D₄×C₆₁₇	C₂
socle	C₂²×C₆₁₇	C₂³
hypercenter	C₂×D₄	D₆₁₇
hypocenter	C₆₁₇	C₂×#41
LCS	G ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #23754599	C₂	direct (N × H)
2. C₂×Dic₁₂₃₄	C₂	semidirect
3. C₂×C₂²⋊D₆₁₇	C₂	semidirect
4. C₂²×Dic₆₁₇	C₂	semidirect
5. C₂×C₄×D₆₁₇	C₂	semidirect
6. C₂×D₄×C₆₁₇	C₂	semidirect
8. Dic₁₂₃₄	C₂²	semidirect
9. C₂²⋊D₆₁₇	C₂²	semidirect
12. C₂×Dic₆₁₇	C₂²	semidirect
13. C₄×D₆₁₇	C₂²	semidirect
15. D₄×C₆₁₇	C₂²	semidirect
26. C₂×C₆₁₇	#41	semidirect
29. C₆₁₇	C₂×#41	semidirect
34. C₂×D₄	D₆₁₇	semidirect
41. D₄	C₂×D₆₁₇	semidirect
57. C₂	#23754599	direct (N × H)

Maximal subgroups

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

subgroup	index
#23754599	2
C₂×Dic₁₂₃₄	2
C₂×C₂²⋊D₆₁₇	2
C₂²×Dic₆₁₇	2
C₂×C₄×D₆₁₇	2
C₂×D₄×C₆₁₇	2
C₂×#41	617

Subgroups — C2×#23754599 #34089744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23754599 #34089744