Subgroups — C₂×#20259842 #27135682

Order 12352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₄₄	C₂	1	1	yes	maximal, characteristic
2	#20259842	C₂	1	4	yes	maximal, direct product factor
3	C₂×#15690530	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₁₆×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₇₂	—	2	1	no	—
6	D₁₅₄₄	—	2	1	no	—
7	D₁₅₄₄	C₂²	1	2	yes	—
8	#15690530	C₂²	1	2	yes	complemented
9	C₂×Q₈×C₁₉₃	—	2	1	no	—
10	Q₁₆×C₁₉₃	—	2	1	no	—
11	Q₁₆×C₁₉₃	C₂²	1	2	yes	complemented
12	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₂²×D₁₉₃	—	4	1	no	—
14	D₇₇₂	—	2	2	no	—
15	D₇₇₂	—	4	1	no	—
16	Q₈×C₁₉₃	—	2	2	no	—
17	Q₈×C₁₉₃	—	4	1	no	—
18	C₂×C₄×C₁₉₃	—	4	1	no	—
19	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
20	C₈×C₁₉₃	D₄	1	1	yes	characteristic
21	C₈×C₁₉₃	C₂³	1	1	yes	characteristic
22	C₂²×C₁₉₃	D₈	1	1	yes	characteristic
23	C₂×D₁₉₃	—	4	2	no	—
24	C₂×D₁₉₃	—	8	2	no	—
25	C₄×C₁₉₃	—	4	2	no	—
26	C₄×C₁₉₃	D₈	1	1	yes	characteristic
27	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₁₉₃	SD₃₂	1	2	yes	complemented
29	C₂×C₁₉₃	C₂×D₈	1	1	yes	characteristic
30	D₁₉₃	—	8	2	no	—
31	C₁₉₃	C₂×SD₃₂	1	1	yes	characteristic, complemented, Sylow
32	C₂×SD₃₂	—	193	1	no	maximal, Sylow
33	C₂×Q₁₆	D₁₉₃	1	1	yes	characteristic, complemented
34	C₂×D₈	—	193	1	no	—
35	SD₃₂	—	193	4	no	—
36	C₂×C₁₆	—	193	1	no	—
37	C₂×Q₈	—	2	1	no	—
38	C₂×D₄	—	386	1	no	—
39	Q₁₆	—	2	1	no	—
40	Q₁₆	C₂×D₁₉₃	1	2	yes	complemented
41	D₈	—	386	1	no	—
42	D₈	—	193	2	no	—
43	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
44	C₁₆	—	193	2	no	—
45	C₂³	—	772	1	no	—
46	Q₈	—	2	2	no	—
47	Q₈	—	4	1	no	—
48	D₄	—	772	1	no	—
49	D₄	—	386	2	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
52	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
53	C₈	C₂²×D₁₉₃	1	1	yes	characteristic
54	C₂²	—	772	2	no	—
55	C₂²	—	1544	2	no	—
56	C₂²	#15690540	1	1	yes	characteristic
57	C₄	—	4	2	no	—
58	C₄	#15690540	1	1	yes	characteristic
59	C₄	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
60	C₂	—	1544	2	no	—
61	C₂	#20259842	1	2	yes	direct product factor
62	C₂	C₂×#15690540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690540
commutator, cosocle	C₈×C₁₉₃	C₂³
Frattini	C₈	C₂²×D₁₉₃
Fitting	C₂×Q₁₆×C₁₉₃	C₂
socle	C₂²×C₁₉₃	D₈
prime radical	C₂×D₁₅₄₄	C₂
hypercenter	C₂×Q₁₆	D₁₉₃
hypocenter	C₁₉₃	C₂×SD₃₂
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20259842	C₂	direct (N × H)
3. C₂×#15690530	C₂	semidirect
4. C₂×Q₁₆×C₁₉₃	C₂	semidirect
8. #15690530	C₂²	semidirect
11. Q₁₆×C₁₉₃	C₂²	semidirect
28. C₂×C₁₉₃	SD₃₂	semidirect
31. C₁₉₃	C₂×SD₃₂	semidirect
33. C₂×Q₁₆	D₁₉₃	semidirect
40. Q₁₆	C₂×D₁₉₃	semidirect
61. C₂	#20259842	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×D₁₅₄₄	2
#20259842	2
C₂×#15690530	2
C₂×Q₁₆×C₁₉₃	2
C₂×SD₃₂	193

Subgroups — C2×#20259842 #27135682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20259842 #27135682