Subgroups — C₂×#20263896 #27164438

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₉₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263896	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₉₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₉₆	—	2	1	no	—
6	D₁₅₉₂	—	2	1	no	—
7	D₁₅₉₂	C₂²	1	2	yes	complemented
8	C₁₉₉⋊C₁₆	C₂²	1	2	yes	complemented
9	C₂×D₄×C₁₉₉	—	2	1	no	—
10	D₈×C₁₉₉	—	2	1	no	—
11	D₈×C₁₉₉	C₂²	1	2	yes	complemented
12	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂²×D₁₉₉	—	4	1	no	—
14	D₇₉₆	—	2	2	no	—
15	D₇₉₆	—	4	1	no	—
16	C₂³×C₁₉₉	—	4	1	no	—
17	D₄×C₁₉₉	—	2	2	no	—
18	D₄×C₁₉₉	—	4	1	no	—
19	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₈×C₁₉₉	D₄	1	1	yes	characteristic
21	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
22	C₂²×C₁₉₉	—	4	2	no	—
23	C₂²×C₁₉₉	—	8	2	no	—
24	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
25	C₂×D₁₉₉	—	4	2	no	—
26	C₂×D₁₉₉	—	8	2	no	—
27	C₄×C₁₉₉	D₈	1	1	yes	characteristic
28	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₁₉₉	—	8	2	no	—
30	C₂×C₁₉₉	D₁₆	1	2	yes	complemented
31	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
32	D₁₉₉	—	8	2	no	—
33	C₁₉₉	C₂×D₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	199	1	no	maximal, Sylow
35	C₂×D₈	—	199	1	no	—
36	C₂×D₈	D₁₉₉	1	1	yes	characteristic, complemented
37	D₁₆	—	199	4	no	—
38	C₂×C₁₆	—	199	1	no	—
39	C₂×D₄	—	2	1	no	—
40	C₂×D₄	—	398	1	no	—
41	D₈	—	2	1	no	—
42	D₈	—	398	1	no	—
43	D₈	—	199	2	no	—
44	D₈	C₂×D₁₉₉	1	2	yes	complemented
45	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
46	C₁₆	—	199	2	no	—
47	C₂³	—	4	1	no	—
48	C₂³	—	796	1	no	—
49	D₄	—	2	2	no	—
50	D₄	—	4	1	no	—
51	D₄	—	796	1	no	—
52	D₄	—	398	2	no	—
53	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
54	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
55	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
56	C₂²	—	4	2	no	—
57	C₂²	—	8	2	no	—
58	C₂²	—	796	2	no	—
59	C₂²	—	1592	2	no	—
60	C₂²	#15691871	1	1	yes	characteristic
61	C₄	#15691871	1	1	yes	characteristic
62	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₂	—	8	2	no	—
64	C₂	—	1592	2	no	—
65	C₂	#20263896	1	2	yes	direct product factor
66	C₂	C₂×#15691871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15691871
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	C₂×D₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	D₈
hypercenter	C₂×D₈	D₁₉₉
hypocenter	C₁₉₉	C₂×D₁₆
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₉₂	C₂	semidirect
2. #20263896	C₂	direct (N × H)
3. C₂×C₁₉₉⋊C₁₆	C₂	semidirect
4. C₂×D₈×C₁₉₉	C₂	semidirect
7. D₁₅₉₂	C₂²	semidirect
8. C₁₉₉⋊C₁₆	C₂²	semidirect
11. D₈×C₁₉₉	C₂²	semidirect
30. C₂×C₁₉₉	D₁₆	semidirect
33. C₁₉₉	C₂×D₁₆	semidirect
36. C₂×D₈	D₁₉₉	semidirect
44. D₈	C₂×D₁₉₉	semidirect
65. C₂	#20263896	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×D₁₅₉₂	2
#20263896	2
C₂×C₁₉₉⋊C₁₆	2
C₂×D₈×C₁₉₉	2
C₂×D₁₆	199

Subgroups — C2×#20263896 #27164438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263896 #27164438