Subgroups — #38740889

Order 25216.

82 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	C₂×#20263232	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×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	complemented
7	#20263232	—	2	1	no	—
8	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
9	#107×C₁₉₇	—	2	1	no	—
10	C₂×D₇₈₈	—	4	1	no	—
11	D₁₅₇₆	—	2	2	no	—
12	D₁₅₇₆	—	4	1	no	—
13	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₇	—	4	1	no	—
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	C₂×C₄×C₁₉₇	—	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	D₁₆	1	1	yes	characteristic
25	C₂×D₁₉₇	—	8	2	no	—
26	C₂×D₁₉₇	—	16	1	no	—
27	C₂×D₁₉₇	—	16	1	no	—
28	C₄×C₁₉₇	—	16	1	no	—
29	C₄×C₁₉₇	#102	1	1	yes	characteristic
30	C₄×C₁₉₇	SD₃₂	1	1	yes	characteristic
31	C₂×C₁₉₇	#357	1	1	yes	characteristic
32	C₂×C₁₉₇	D₃₂	1	1	yes	characteristic
33	C₂×C₁₉₇	SD₆₄	1	1	yes	characteristic
34	D₁₉₇	—	16	2	no	—
35	C₁₉₇	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	197	1	no	maximal, Sylow
37	C₂×D₁₆	—	197	1	no	—
38	C₂×C₃₂	—	197	1	no	—
39	#366	D₁₉₇	1	1	yes	characteristic, complemented
40	C₂×D₈	—	394	1	no	—
41	D₁₆	—	394	1	no	—
42	D₁₆	—	197	2	no	—
43	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	394	1	no	—
46	C₂×D₄	—	788	1	no	—
47	D₈	—	788	1	no	—
48	D₈	—	394	2	no	—
49	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₁₉₇	1	1	yes	characteristic
52	C₁₆	D₇₈₈	1	1	yes	characteristic
53	C₂³	—	1576	1	no	—
54	D₄	—	788	2	no	—
55	D₄	—	1576	1	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#15691343	1	1	yes	characteristic
58	C₈	#15691335	1	1	yes	characteristic
59	C₈	#15691345	1	1	yes	characteristic
60	C₂²	—	1576	2	no	—
61	C₂²	—	3152	1	no	—
62	C₂²	—	3152	1	no	—
63	C₂²	#20263267	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#20263246	1	1	yes	characteristic
66	C₄	#20263269	1	1	yes	characteristic
67	C₂	—	3152	2	no	—
68	C₂	#27160879	1	1	yes	characteristic
69	C₂	#27160914	1	1	yes	characteristic
70	C₂	#27160916	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263267
commutator	C₁₆×C₁₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₇
Fitting	#366×C₁₉₇	C₂
socle	C₂²×C₁₉₇	D₁₆
prime radical	C₂×D₃₁₅₂	C₂
cosocle	C₂×C₁₆×C₁₉₇	C₂²
hypercenter	#366	D₁₉₇
hypocenter	C₁₉₇	#1415
LCS	G ▷ C₁₆×C₁₉₇ ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×#20263232	C₂	semidirect
3. #366×C₁₉₇	C₂	semidirect
6. D₃₁₅₂	C₄	semidirect
35. C₁₉₇	#1415	semidirect
39. #366	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
C₂×#20263232	2
#366×C₁₉₇	2
#1415	197