Subgroups — #27139877

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261151	C₂	1	1	yes	maximal, characteristic
2	#20260569	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690816	C₄	1	1	yes	characteristic
5	#15690750	C₄	1	1	yes	characteristic
6	#15690749	—	2	1	no	—
7	C₁₆×D₉₇	—	2	1	no	—
8	#15690716	—	2	1	no	—
9	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₉₇	—	2	1	no	—
11	#13095511	C₈	1	1	yes	characteristic
12	#13095495	C₈	1	1	yes	characteristic
13	#13095494	—	2	1	no	—
14	C₈×D₉₇	—	2	1	no	—
15	#13095486	—	2	1	no	—
16	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₉₇	D₄	1	1	yes	characteristic
18	C₁₆×C₉₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₉₇	—	2	1	no	—
21	D₃₈₈	C₁₆	1	1	yes	characteristic
22	C₄×D₉₇	—	2	1	no	—
23	Dic₁₉₄	C₁₆	1	1	yes	characteristic
24	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₉₇	M₁₆	1	1	yes	characteristic
26	#11715895	—	2	1	no	—
27	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
28	C₂×D₉₇	—	2	1	no	—
29	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₉₇	—	2	1	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₉₇	—	4	1	no	—
35	#1401	—	97	1	no	maximal, Sylow
36	C₉₇	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	97	1	no	—
38	M₆₄	—	97	1	no	—
39	C₂×C₃₂	D₉₇	1	1	yes	characteristic, complemented
40	#131	—	97	1	no	—
41	M₃₂	—	97	1	no	—
42	M₃₂	—	194	1	no	—
43	C₂×C₁₆	—	194	1	no	—
44	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	194	1	no	—
47	#41	—	97	1	no	—
48	M₁₆	—	97	1	no	—
49	M₁₆	—	194	1	no	—
50	C₂×C₈	—	194	1	no	—
51	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
52	C₁₆	—	194	1	no	—
53	C₁₆	D₃₈₈	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
55	Q₈	—	97	1	no	—
56	D₄	—	97	1	no	—
57	D₄	—	194	1	no	—
58	C₂×C₄	—	194	1	no	—
59	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
60	C₈	—	194	1	no	—
61	C₈	#13095488	1	1	yes	characteristic
62	C₈	#13095494	1	1	yes	characteristic
63	C₂²	—	194	1	no	—
64	C₂²	#15690749	1	1	yes	characteristic
65	C₄	—	194	1	no	—
66	C₄	#15690722	1	1	yes	characteristic
67	C₄	C₁₆×D₉₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	388	1	no	—
70	C₂	#20260514	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₃₈₈
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	C₂×C₃₂×C₉₇	C₂
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	#13095511	C₈
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	#1401
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #20260569	C₂	semidirect
3. C₂×C₃₂×C₉₇	C₂	semidirect
36. C₉₇	#1401	semidirect
39. C₂×C₃₂	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261151	2
#20260569	2
C₂×C₃₂×C₉₇	2
#1401	97