Subgroups — C₂×C₄⋊#13096802 #27161100

Order 12608.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13096802	C₂	1	2	yes	maximal
2	C₄⋊#13096802	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13096802	—	2	4	no	—
5	C₂×#13096802	C₂²	1	4	yes	—
6	C₂²×C₄×C₁₉₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₁₉₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₄²×C₁₉₇	C₄	1	2	yes	—
10	C₄²×C₁₉₇	C₂²	1	2	yes	—
11	#13096802	—	2	4	no	—
12	C₂³×C₁₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₇	—	2	2	no	—
14	C₂×C₄×C₁₉₇	C₈	1	4	yes	complemented
15	C₂×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
19	C₂×C₄×C₁₉₇	Q₈	1	2	yes	—
20	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₉₇	C₂×C₈	1	2	yes	—
23	C₂²×C₁₉₇	M₁₆	1	2	yes	—
24	C₂²×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₉₇	—	2	2	no	—
26	C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
27	C₄×C₁₉₇	C₂×C₈	1	4	yes	complemented
28	C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₁₉₇	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₁₉₇	C₂×M₁₆	1	1	yes	characteristic
34	C₁₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	197	1	no	maximal, Sylow
36	C₂²×C₈	—	197	2	no	—
37	C₂×C₄²	D₁₉₇	1	1	yes	characteristic
38	C₄⋊C₈	—	197	4	no	—
39	C₂²×C₄	Dic₁₉₇	1	1	yes	characteristic
40	C₂²×C₄	Dic₁₉₇	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
42	C₂×C₈	—	394	4	no	—
43	C₂×C₈	—	197	4	no	—
44	C₄²	Dic₁₉₇	1	2	yes	—
45	C₄²	C₂×D₁₉₇	1	2	yes	—
46	C₂³	C₂×Dic₁₉₇	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	#13096802	1	4	yes	complemented
49	C₂×C₄	C₂×Dic₁₉₇	1	2	yes	—
50	C₂×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
51	C₂×C₄	C₂×Dic₁₉₇	1	2	yes	—
52	C₂×C₄	D₇₈₈	1	2	yes	—
53	C₂×C₄	Dic₃₉₄	1	2	yes	—
54	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
55	C₈	—	394	4	no	—
56	C₂²	C₄⋊Dic₁₉₇	1	2	yes	—
57	C₂²	C₂×#13096802	1	2	yes	—
58	C₂²	#15691342	1	2	yes	—
59	C₂²	C₂²×Dic₁₉₇	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₄⋊Dic₁₉₇	1	2	yes	—
62	C₄	C₂×#13096802	1	4	yes	complemented
63	C₄	C₂×D₇₈₈	1	1	yes	characteristic
64	C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
65	C₂	C₄⋊#13096802	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
67	C₂	C₂²×#13096802	1	1	yes	characteristic
68	C₂	C₂×#15691342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₂×C₄²×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×C₄
cosocle	C₂×C₄×C₁₉₇	C₂³
hypercenter	C₂×C₄²	D₁₉₇
hypocenter	C₁₉₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13096802	C₂	direct (N × H)
14. C₂×C₄×C₁₉₇	C₈	semidirect
27. C₄×C₁₉₇	C₂×C₈	semidirect
30. C₂×C₁₉₇	C₄⋊C₈	semidirect
34. C₁₉₇	C₂×C₄⋊C₈	semidirect
48. C₂×C₄	#13096802	semidirect
62. C₄	C₂×#13096802	semidirect
65. C₂	C₄⋊#13096802	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13096802	2
C₄⋊#13096802	2
C₂×C₄²×C₁₉₇	2
C₂×C₄⋊C₈	197

Subgroups — C2×C4⋊#13096802 #27161100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#13096802 #27161100