Subgroups — #27158837

Order 12416.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261843	C₂	1	2	yes	maximal
2	C₄×#13095514	C₂	1	1	yes	maximal, characteristic
3	#15690900	—	2	2	no	—
4	C₂×#13095514	C₄	1	1	yes	characteristic
5	C₂×#13095514	C₂²	1	1	yes	characteristic
6	C₄×#11715895	C₄	1	1	yes	characteristic
7	#13095514	—	2	1	no	—
8	#13095514	D₄	1	1	yes	characteristic
9	#13095514	Q₈	1	1	yes	characteristic
10	C₂×#11715895	C₈	1	1	yes	characteristic
11	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₉₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₉₇	—	2	1	no	—
14	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
15	#11715895	C₁₆	1	2	yes	—
16	#11715895	C₄⋊C₄	1	1	yes	characteristic
17	#11715895	M₁₆	1	1	yes	characteristic
18	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
19	C₄×C₉₇	—	2	2	no	—
20	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₉₇	—	2	1	no	—
23	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
24	#1422	—	97	1	no	maximal, Sylow
25	C₉₇	#1422	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	—	97	2	no	—
27	C₄×C₁₆	—	97	1	no	—
28	C₂×C₁₆	—	97	1	no	—
29	C₂×C₁₆	—	97	1	no	—
30	C₄×C₈	—	97	1	no	—
31	C₃₂	—	194	2	no	—
32	C₂×C₈	—	97	1	no	—
33	C₂×C₈	—	97	1	no	—
34	C₄²	#11715906	1	1	yes	characteristic
35	C₁₆	—	97	1	no	—
36	C₁₆	—	97	1	no	—
37	C₁₆	—	194	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
40	C₈	—	97	1	no	—
41	C₈	—	97	1	no	—
42	C₈	—	97	2	no	—
43	C₂²	#15690906	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	C₄⋊#11715906	1	1	yes	characteristic
46	C₄	#15690905	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20261838	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11715906
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₄²×C₉₇	C₈
socle	C₂×C₉₇	C₄⋊C₁₆
prime radical	C₂²×C₉₇	M₃₂
cosocle	C₂×#13095514	C₂²
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#1422
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
25. C₉₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#20261843	2
C₄×#13095514	2
#1422	97