Subgroups — #38740784

Order 25216.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27160832	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13096802	—	2	2	no	—
4	C₂×C₁₆×C₁₉₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₉₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
7	C₂×#13096802	—	4	2	no	—
8	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₉₇	Q₈	1	1	yes	characteristic
10	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₉₇	—	2	1	no	—
12	C₁₆×C₁₉₇	C₈	1	2	yes	complemented
13	#13096802	—	8	2	no	—
14	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₉₇	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₉₇	—	2	1	no	—
18	C₈×C₁₉₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₉₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₉₇	#107	1	1	yes	characteristic
21	C₄×C₁₉₇	—	2	1	no	—
22	C₄×C₁₉₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₉₇	#108	1	1	yes	characteristic
24	C₄×C₁₉₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₉₇	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₉₇	#335	1	1	yes	characteristic
27	C₂×C₁₉₇	#366	1	1	yes	characteristic
28	C₂×C₁₉₇	#368	1	1	yes	characteristic
29	C₁₉₇	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	197	1	no	maximal, Sylow
31	C₄×C₁₆	D₁₉₇	1	1	yes	characteristic
32	#335	—	197	2	no	—
33	C₂×C₁₆	Dic₁₉₇	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₉₇	1	1	yes	characteristic
35	C₄⋊C₈	—	394	2	no	—
36	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
37	C₂×C₈	—	788	2	no	—
38	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
39	C₂×C₈	Dic₃₉₄	1	1	yes	characteristic
40	C₄²	D₇₈₈	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	#13096802	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
44	C₂×C₄	D₁₅₇₆	1	1	yes	characteristic
45	C₂×C₄	Dic₇₈₈	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1576	2	no	—
48	C₈	C₂×#13096802	1	1	yes	characteristic
49	C₈	#15691342	1	1	yes	characteristic
50	C₂²	#20263260	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊#13096802	1	1	yes	characteristic
53	C₄	#20263262	1	1	yes	characteristic
54	C₄	D₃₁₅₂	1	1	yes	characteristic
55	C₄	Dic₁₅₇₆	1	1	yes	characteristic
56	C₂	#27160832	1	1	yes	characteristic
57	C₂	#27160905	1	1	yes	characteristic
58	C₂	#27160909	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₅₇₆
commutator	C₈×C₁₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	C₄×C₁₆×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#107
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	C₄×C₁₆	D₁₉₇
hypocenter	C₁₉₇	#1371
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₉₇	C₈	semidirect
29. C₁₉₇	#1371	semidirect
42. C₁₆	#13096802	semidirect

Maximal subgroups

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

subgroup	index
#27160832	2
C₄×C₁₆×C₁₉₇	2
#1371	197