Subgroups — #38762279

Order 25408.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163055	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₃₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₃₉₇	—	2	2	no	—
4	C₂²⋊Dic₃₉₇	—	2	2	no	—
5	C₄×Dic₃₉₇	C₄	1	4	yes	complemented
6	C₂²×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
7	M₁₆×C₃₉₇	—	2	2	no	—
8	C₂×C₈×C₃₉₇	C₄	1	2	yes	complemented
9	C₂×Dic₃₉₇	—	2	4	no	—
10	C₂³×C₃₉₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₃₉₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₃₉₇	D₄	1	2	yes	—
14	C₂×C₄×C₃₉₇	Q₈	1	1	yes	characteristic
15	C₈×C₃₉₇	—	2	2	no	—
16	C₂²×C₃₉₇	—	4	1	no	—
17	C₂²×C₃₉₇	C₂²⋊C₄	1	2	yes	—
18	C₂²×C₃₉₇	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₃₉₇	C₄²	1	1	yes	characteristic
20	C₄×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₉₇	C₄⋊C₄	1	2	yes	—
22	Dic₃₉₇	—	4	4	no	—
23	C₂×C₃₉₇	—	2	2	no	—
24	C₂×C₃₉₇	—	2	1	no	—
25	C₂×C₃₉₇	#95	1	1	yes	characteristic
26	C₃₉₇	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
27	H(3,ℤ/4)	—	397	1	no	maximal, Sylow
28	C₂×M₁₆	D₃₉₇	1	1	yes	characteristic
29	#117	—	397	2	no	—
30	C₂²×C₄	C₂×D₃₉₇	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	Dic₃₉₇	1	2	yes	complemented
33	C₄⋊C₄	—	794	2	no	—
34	C₂²⋊C₄	—	794	2	no	—
35	C₄²	—	397	4	no	—
36	C₂³	D₁₅₈₈	1	1	yes	characteristic
37	C₂×C₄	—	794	4	no	—
38	C₂×C₄	C₄×D₃₉₇	1	2	yes	—
39	C₂×C₄	C₂×Dic₃₉₇	1	1	yes	characteristic
40	C₂×C₄	C₂²⋊D₃₉₇	1	2	yes	—
41	C₂×C₄	Dic₇₉₄	1	1	yes	characteristic
42	C₈	—	2	2	no	—
43	C₂²	—	4	1	no	—
44	C₂²	#20263708	1	2	yes	—
45	C₂²	C₄⋊Dic₃₉₇	1	1	yes	characteristic
46	C₄	—	1588	4	no	—
47	C₄	C₄×Dic₃₉₇	1	1	yes	characteristic
48	C₄	C₂²⋊Dic₃₉₇	1	1	yes	characteristic
49	C₄	#20263710	1	2	yes	—
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	#27163003	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₃₉₇
commutator	C₄×C₃₉₇	C₄²
Frattini	C₂²×C₄	C₂×D₃₉₇
Fitting	C₂×M₁₆×C₃₉₇	C₂
socle	C₂×C₃₉₇	#95
prime radical	C₂³×C₃₉₇	D₄
cosocle	C₂²×C₄×C₃₉₇	C₂²
hypercenter	C₂×M₁₆	D₃₉₇
hypocenter	C₃₉₇	H(3,ℤ/4)
LCS	G ▷ C₄×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄×Dic₃₉₇	C₄	semidirect
8. C₂×C₈×C₃₉₇	C₄	semidirect
26. C₃₉₇	H(3,ℤ/4)	semidirect
32. C₂×C₈	Dic₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27163055	2
C₂×M₁₆×C₃₉₇	2
H(3,ℤ/4)	397