Subgroups — C₂×C₄⋊Q₈×C₁₉₇ #27160742

Order 12608.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₉₇	C₂	1	2	yes	maximal
2	C₄⋊Q₈×C₁₉₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×C₁₉₇	C₂	1	4	yes	maximal
4	C₂×C₄²×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₁₉₇	—	2	8	no	—
6	C₂×Q₈×C₁₉₇	C₂²	1	8	yes	—
7	C₂²×C₄×C₁₉₇	C₂²	1	4	yes	—
8	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₁₉₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₉₇	C₂²	1	16	yes	—
11	C₄²×C₁₉₇	C₂²	1	4	yes	—
12	C₂³×C₁₉₇	C₂³	1	1	yes	characteristic
13	Q₈×C₁₉₇	—	2	16	no	—
14	C₂×C₄×C₁₉₇	—	2	8	no	—
15	C₂×C₄×C₁₉₇	D₄	1	4	yes	—
16	C₂×C₄×C₁₉₇	Q₈	1	8	yes	complemented
17	C₂×C₄×C₁₉₇	C₂³	1	8	yes	—
18	C₂×C₄×C₁₉₇	C₂³	1	2	yes	—
19	C₂×C₄×C₁₉₇	C₂³	1	4	yes	—
20	C₂²×C₁₉₇	C₂×D₄	1	2	yes	—
21	C₂²×C₁₉₇	C₂×Q₈	1	4	yes	—
22	C₂²×C₁₉₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₁₉₇	—	2	8	no	—
24	C₄×C₁₉₇	C₂×D₄	1	4	yes	—
25	C₄×C₁₉₇	C₂×Q₈	1	8	yes	complemented
26	C₂×C₁₉₇	C₄⋊Q₈	1	4	yes	direct product factor
27	C₂×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₁₉₇	C₂²×Q₈	1	2	yes	—
29	C₁₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₄⋊Q₈	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×Q₈	C₂×C₁₉₇	1	2	yes	—
32	C₄⋊Q₈	C₂×C₁₉₇	1	8	yes	direct product factor
33	C₂×C₄⋊C₄	C₂×C₁₉₇	1	4	yes	—
34	C₂×C₄²	C₂×C₁₉₇	1	1	yes	characteristic
35	C₂×Q₈	—	2	8	no	—
36	C₂×Q₈	C₂²×C₁₉₇	1	8	yes	—
37	C₂²×C₄	C₂²×C₁₉₇	1	4	yes	—
38	C₂²×C₄	C₂²×C₁₉₇	1	1	yes	characteristic
39	C₂²×C₄	C₂²×C₁₉₇	1	2	yes	—
40	C₄⋊C₄	C₂²×C₁₉₇	1	16	yes	—
41	C₄²	C₂²×C₁₉₇	1	4	yes	—
42	C₂³	C₂³×C₁₉₇	1	1	yes	characteristic
43	Q₈	—	2	16	no	—
44	C₂×C₄	—	2	8	no	—
45	C₂×C₄	D₄×C₁₉₇	1	4	yes	—
46	C₂×C₄	Q₈×C₁₉₇	1	8	yes	complemented
47	C₂×C₄	C₂³×C₁₉₇	1	8	yes	—
48	C₂×C₄	C₂³×C₁₉₇	1	2	yes	—
49	C₂×C₄	C₂³×C₁₉₇	1	4	yes	—
50	C₂²	C₂×D₄×C₁₉₇	1	2	yes	—
51	C₂²	C₂×Q₈×C₁₉₇	1	4	yes	—
52	C₂²	C₂⁴×C₁₉₇	1	1	yes	characteristic
53	C₄	—	2	8	no	—
54	C₄	C₂×D₄×C₁₉₇	1	4	yes	—
55	C₄	C₂×Q₈×C₁₉₇	1	8	yes	complemented
56	C₂	C₄⋊Q₈×C₁₉₇	1	4	yes	direct product factor
57	C₂	C₂²×D₄×C₁₉₇	1	1	yes	characteristic
58	C₂	C₂²×Q₈×C₁₉₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄⋊Q₈×C₁₉₇	C₂	direct (N × H)
16. C₂×C₄×C₁₉₇	Q₈	semidirect
25. C₄×C₁₉₇	C₂×Q₈	semidirect
26. C₂×C₁₉₇	C₄⋊Q₈	direct (N × H)
29. C₁₉₇	C₂×C₄⋊Q₈	direct (N × H)
30. C₂×C₄⋊Q₈	C₁₉₇	direct (N × H)
32. C₄⋊Q₈	C₂×C₁₉₇	direct (N × H)
46. C₂×C₄	Q₈×C₁₉₇	semidirect
55. C₄	C₂×Q₈×C₁₉₇	semidirect
56. C₂	C₄⋊Q₈×C₁₉₇	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×Q₈×C₁₉₇	2
C₄⋊Q₈×C₁₉₇	2
C₂×C₄⋊C₄×C₁₉₇	2
C₂×C₄²×C₁₉₇	2
C₂×C₄⋊Q₈	197

Subgroups — C2×C4⋊Q8×C197 #27160742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊Q₈×C₁₉₇ #27160742