Subgroups — #27139913

Order 12416.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690716	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₆	—	2	1	no	—
5	D₁₅₅₂	—	2	1	no	—
6	D₁₅₅₂	C₄	1	2	yes	complemented
7	#15690716	—	2	1	no	—
8	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
9	#107×C₉₇	—	2	1	no	—
10	C₂×D₃₈₈	—	4	1	no	—
11	D₇₇₆	—	2	2	no	—
12	D₇₇₆	—	4	1	no	—
13	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₉₇	—	4	1	no	—
15	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₉₇	D₄	1	1	yes	characteristic
17	C₂²×D₉₇	—	8	1	no	—
18	C₂×C₄×C₉₇	—	8	1	no	—
19	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
20	D₃₈₈	—	4	2	no	—
21	D₃₈₈	—	8	1	no	—
22	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
25	C₂×D₉₇	—	8	2	no	—
26	C₂×D₉₇	—	16	1	no	—
27	C₂×D₉₇	—	16	1	no	—
28	C₄×C₉₇	—	16	1	no	—
29	C₄×C₉₇	#102	1	1	yes	characteristic
30	C₄×C₉₇	SD₃₂	1	1	yes	characteristic
31	C₂×C₉₇	#357	1	1	yes	characteristic
32	C₂×C₉₇	D₃₂	1	1	yes	characteristic
33	C₂×C₉₇	SD₆₄	1	1	yes	characteristic
34	D₉₇	—	16	2	no	—
35	#1415	—	97	1	no	maximal, Sylow
36	C₉₇	#1415	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₁₆	—	97	1	no	—
38	C₂×C₃₂	—	97	1	no	—
39	#366	D₉₇	1	1	yes	characteristic, complemented
40	C₂×D₈	—	194	1	no	—
41	D₁₆	—	97	2	no	—
42	D₁₆	—	194	1	no	—
43	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	194	1	no	—
46	C₂×D₄	—	388	1	no	—
47	D₈	—	388	1	no	—
48	D₈	—	194	2	no	—
49	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₉₇	1	1	yes	characteristic
52	C₁₆	D₃₈₈	1	1	yes	characteristic
53	C₂³	—	776	1	no	—
54	D₄	—	776	1	no	—
55	D₄	—	388	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#13095496	1	1	yes	characteristic
58	C₈	#13095488	1	1	yes	characteristic
59	C₈	#13095498	1	1	yes	characteristic
60	C₂²	—	776	2	no	—
61	C₂²	—	1552	1	no	—
62	C₂²	—	1552	1	no	—
63	C₂²	#15690751	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#15690730	1	1	yes	characteristic
66	C₄	#15690753	1	1	yes	characteristic
67	C₂	—	1552	2	no	—
68	C₂	#20260535	1	1	yes	characteristic
69	C₂	#20260570	1	1	yes	characteristic
70	C₂	#20260572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690751
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	#366×C₉₇	C₂
socle	C₂²×C₉₇	D₁₆
prime radical	C₂×D₁₅₅₂	C₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#366	D₉₇
hypocenter	C₉₇	#1415
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
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×#15690716	C₂	semidirect
3. #366×C₉₇	C₂	semidirect
6. D₁₅₅₂	C₄	semidirect
36. C₉₇	#1415	semidirect
39. #366	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₅₂	2
C₂×#15690716	2
#366×C₉₇	2
#1415	97