Subgroups — #34089688

Order 19744.

76 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	C₂×C₂²⋊D₆₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#23754577	C₂	1	1	yes	maximal, characteristic, complemented
5	#23754577	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₁₇	C₂²	1	1	yes	characteristic
9	Dic₁₂₃₄	—	2	2	no	—
10	C₂²⋊D₆₁₇	—	2	2	no	—
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	—	2	1	no	—
20	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
21	C₂×D₆₁₇	—	2	1	no	—
22	C₂×D₆₁₇	—	2	1	no	—
23	C₂×D₆₁₇	—	2	1	no	—
24	C₄×C₆₁₇	—	2	1	no	—
25	C₄×C₆₁₇	—	2	1	no	—
26	Dic₆₁₇	—	2	1	no	—
27	Dic₆₁₇	—	2	1	no	—
28	Dic₆₁₇	D₄	1	2	yes	—
29	C₂×C₆₁₇	—	4	1	no	—
30	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₆₁₇	#41	1	1	yes	characteristic
32	C₂×C₆₁₇	#41	1	1	yes	characteristic
33	D₆₁₇	—	4	1	no	—
34	C₆₁₇	#124	1	1	yes	characteristic, complemented, Sylow
35	#124	—	617	1	no	maximal, Sylow
36	C₂×Q₈	—	617	1	no	—
37	C₂×D₄	—	617	1	no	—
38	C₂²⋊C₄	—	617	1	no	—
39	C₂²⋊C₄	—	617	1	no	—
40	C₂²⋊C₄	—	617	1	no	—
41	C₂²⋊C₄	D₆₁₇	1	1	yes	characteristic, complemented
42	C₄²	—	617	1	no	—
43	C₂³	—	617	1	no	—
44	C₂³	C₂×D₆₁₇	1	1	yes	characteristic
45	Q₈	—	1234	2	no	—
46	D₄	—	1234	2	no	—
47	C₂×C₄	—	617	1	no	—
48	C₂×C₄	—	617	1	no	—
49	C₂×C₄	—	617	1	no	—
50	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
51	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	1234	1	no	—
56	C₂²	—	1234	1	no	—
57	C₂²	—	1234	1	no	—
58	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	617	2	no	—
62	C₄	—	1234	1	no	—
63	C₄	—	1234	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	2468	1	no	—
66	C₂	D₄×D₆₁₇	1	1	yes	characteristic
67	C₂	#23754599	1	1	yes	characteristic
68	C₂	#23754600	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	C₂×C₂²⋊D₆₁₇	C₂
hypercenter	C₂²⋊C₄	D₆₁₇
hypocenter	C₆₁₇	#124
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. C₂²⋊Dic₆₁₇	C₂	semidirect
4. #23754577	C₂	semidirect
5. #23754577	C₂	semidirect
6. C₄×Dic₆₁₇	C₂	semidirect
7. C₂²⋊C₄×C₆₁₇	C₂	semidirect
34. C₆₁₇	#124	semidirect
41. C₂²⋊C₄	D₆₁₇	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
C₂×C₂²⋊D₆₁₇	2
C₂²⋊Dic₆₁₇	2
#23754577	2
C₄×Dic₆₁₇	2
C₂²⋊C₄×C₆₁₇	2
#124	617