Subgroups — #27139878

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260569	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690716	C₂	1	1	yes	maximal, characteristic, complemented
3	#504×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690716	—	2	1	no	—
5	#15690716	—	2	1	no	—
6	#131×C₉₇	C₄	1	1	yes	characteristic
7	M₃₂×C₉₇	—	2	1	no	—
8	M₃₂×C₉₇	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₉₇	—	2	1	no	—
10	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	#41×C₉₇	C₈	1	1	yes	characteristic
12	M₁₆×C₉₇	—	2	1	no	—
13	M₁₆×C₉₇	C₈	1	1	yes	characteristic
14	C₂×C₈×C₉₇	—	2	1	no	—
15	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₉₇	—	2	1	no	—
17	C₁₆×C₉₇	D₄	1	1	yes	characteristic
18	C₁₆×C₉₇	D₄	1	1	yes	characteristic
19	Q₈×C₉₇	C₁₆	1	1	yes	characteristic
20	D₄×C₉₇	—	2	1	no	—
21	D₄×C₉₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
24	C₈×C₉₇	—	2	1	no	—
25	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₉₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₉₇	—	2	1	no	—
28	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
29	C₄×C₉₇	—	2	1	no	—
30	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	—	4	1	no	—
34	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	#1401	—	97	1	no	maximal, Sylow
36	C₉₇	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	D₉₇	1	1	yes	characteristic
38	M₆₄	—	97	1	no	—
39	C₂×C₃₂	—	97	1	no	—
40	#131	Dic₉₇	1	1	yes	characteristic
41	M₃₂	—	2	1	no	—
42	M₃₂	Dic₉₇	1	1	yes	characteristic
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
45	C₃₂	—	194	1	no	—
46	C₃₂	—	194	1	no	—
47	#41	#11715895	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	#11715895	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
55	Q₈	#13095486	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	#13095486	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×#11715895	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
62	C₈	#13095495	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#15690750	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊#11715895	1	1	yes	characteristic
67	C₄	C₂×#13095486	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊#13095486	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	#504×C₉₇	C₂
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	#41×C₉₇	C₈
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#504	D₉₇
hypocenter	C₉₇	#1401
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20260569	C₂	semidirect
2. C₂×#15690716	C₂	semidirect
36. C₉₇	#1401	semidirect

Maximal subgroups

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

subgroup	index
#20260569	2
C₂×#15690716	2
#504×C₉₇	2
#1401	97