Subgroups — #38740903

Order 25216.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#27160909	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₅₇₆	—	2	1	no	—
5	D₃₁₅₂	—	2	1	no	—
6	D₃₁₅₂	C₄	1	2	yes	—
7	#20263262	—	2	1	no	—
8	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₉₇	—	2	1	no	—
10	C₂×D₇₈₈	—	4	1	no	—
11	D₁₅₇₆	—	2	2	no	—
12	D₁₅₇₆	—	4	1	no	—
13	#15691342	—	4	1	no	—
14	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₉₇	D₄	1	1	yes	characteristic
17	C₂²×D₁₉₇	—	8	1	no	—
18	D₇₈₈	—	4	2	no	—
19	D₇₈₈	—	8	1	no	—
20	#13096802	—	8	1	no	—
21	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
22	C₈×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₉₇	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₉₇	—	8	2	no	—
26	C₂×D₁₉₇	—	16	2	no	—
27	C₄×C₁₉₇	#102	1	1	yes	characteristic
28	C₄×C₁₉₇	D₁₆	1	1	yes	characteristic
29	C₂×C₁₉₇	—	2	1	no	—
30	C₂×C₁₉₇	#357	1	1	yes	characteristic
31	D₁₉₇	—	16	2	no	—
32	C₁₉₇	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	197	1	no	maximal, Sylow
34	C₂×D₁₆	—	197	1	no	—
35	M₆₄	D₁₉₇	1	1	yes	characteristic, complemented
36	#368	—	197	1	no	—
37	C₂×D₈	—	394	1	no	—
38	D₁₆	—	394	1	no	—
39	D₁₆	—	197	2	no	—
40	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
41	#108	—	394	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	788	1	no	—
44	D₈	—	788	1	no	—
45	D₈	—	394	2	no	—
46	M₁₆	—	788	1	no	—
47	C₂×C₈	D₇₈₈	1	1	yes	characteristic
48	C₁₆	C₄×D₁₉₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₉₇	1	1	yes	characteristic
50	C₂³	—	1576	1	no	—
51	D₄	—	788	2	no	—
52	D₄	—	1576	1	no	—
53	C₂×C₄	D₁₅₇₆	1	1	yes	characteristic
54	C₈	—	1576	1	no	—
55	C₈	#15691335	1	1	yes	characteristic
56	C₈	#15691347	1	1	yes	characteristic
57	C₂²	—	1576	2	no	—
58	C₂²	—	3152	2	no	—
59	C₂²	#20263271	1	1	yes	characteristic
60	C₄	#20263248	1	1	yes	characteristic
61	C₄	D₃₁₅₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	3152	2	no	—
64	C₂	#27160881	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27160881
commutator	C₁₆×C₁₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₇
Fitting	M₆₄×C₁₉₇	C₂
socle	C₂×C₁₉₇	#357
prime radical	C₂×D₃₁₅₂	C₂
cosocle	C₂×C₁₆×C₁₉₇	C₂²
hypercenter	M₆₄	D₁₉₇
hypocenter	C₁₉₇	#1419
LCS	G ▷ C₁₆×C₁₉₇ ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #27160909	C₂	semidirect
3. M₆₄×C₁₉₇	C₂	semidirect
32. C₁₉₇	#1419	semidirect
35. M₆₄	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₁₅₂	2
#27160909	2
M₆₄×C₁₉₇	2
#1419	197