Subgroups — #27156654

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261623	C₂	1	1	yes	maximal, characteristic
2	#20261623	C₂	1	1	yes	maximal, characteristic
3	#20260457	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₄	1	2	yes	—
5	C₂×#13095514	C₄	1	2	yes	—
6	C₄×#11715895	C₄	1	1	yes	characteristic
7	C₄×#11715895	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
9	#13095514	—	2	2	no	—
10	#13095514	C₈	1	4	yes	complemented
11	C₂×#11715895	—	2	1	no	—
12	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
13	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	—	2	1	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
17	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
18	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
19	C₈×C₉₇	—	4	1	no	—
20	#11715895	—	4	1	no	—
21	#11715895	C₂×C₈	1	2	yes	—
22	#11715895	M₁₆	1	2	yes	—
23	C₂²×C₉₇	#97	1	1	yes	characteristic
24	C₄×C₉₇	—	2	1	no	—
25	C₄×C₉₇	—	2	1	no	—
26	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
27	C₄×C₉₇	#97	1	1	yes	characteristic
28	C₂×C₉₇	#322	1	1	yes	characteristic
29	C₂×C₉₇	#347	1	1	yes	characteristic
30	C₂×C₉₇	#347	1	1	yes	characteristic
31	#1313	—	97	1	no	maximal, Sylow
32	C₉₇	#1313	1	1	yes	characteristic, complemented, Sylow
33	#346	—	97	1	no	—
34	#346	—	97	1	no	—
35	#322	—	97	1	no	—
36	C₂×C₁₆	—	97	2	no	—
37	C₂×C₁₆	—	97	2	no	—
38	C₄×C₈	—	97	1	no	—
39	C₄×C₈	—	97	1	no	—
40	C₄×C₈	#87604	1	1	yes	characteristic
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	97	1	no	—
43	C₂×C₈	—	97	1	no	—
44	C₂×C₈	—	194	1	no	—
45	C₄²	C₂×#87604	1	1	yes	characteristic
46	C₁₆	—	97	4	no	—
47	C₁₆	—	194	2	no	—
48	C₂×C₄	C₄×#87604	1	1	yes	characteristic
49	C₂×C₄	#13095520	1	1	yes	characteristic
50	C₂×C₄	#13095521	1	1	yes	characteristic
51	C₈	—	4	1	no	—
52	C₈	—	97	2	no	—
53	C₈	—	97	2	no	—
54	C₈	—	388	1	no	—
55	C₂²	#15690865	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	1	no	—
58	C₄	C₈×#87604	1	1	yes	characteristic
59	C₄	#15690864	1	1	yes	characteristic
60	C₂	#20261577	1	1	yes	characteristic
61	C₂	#20261627	1	1	yes	characteristic
62	C₂	#20261628	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#87604
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₄²	C₂×#87604
Fitting	C₄×C₈×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#97
cosocle	C₄×#11715895	C₂²
hypercenter	C₄×C₈	#87604
hypocenter	C₉₇	#1313
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
10. #13095514	C₈	semidirect
32. C₉₇	#1313	semidirect

Maximal subgroups

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

subgroup	index
#20261623	2
#20260457	2
#1313	97