Subgroups — #27158819

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690900	C₂	1	2	yes	maximal
2	#20261621	C₂	1	1	yes	maximal, characteristic
3	#15690900	C₄	1	4	yes	complemented
4	C₂×#13095514	C₄	1	1	yes	characteristic
5	C₂×#13095514	C₂²	1	1	yes	characteristic
6	C₈×Dic₉₇	C₄	1	1	yes	characteristic
7	#13095514	C₈	1	2	yes	—
8	#13095514	C₂×C₄	1	2	yes	—
9	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
10	C₄×Dic₉₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₉₇	C₈	1	1	yes	characteristic
12	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
13	C₂×Dic₉₇	C₁₆	1	2	yes	—
14	C₈×C₉₇	C₁₆	1	2	yes	—
15	#11715895	C₄²	1	1	yes	characteristic
16	#11715895	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
20	Dic₉₇	C₃₂	1	4	yes	complemented
21	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₉₇	C₂×C₃₂	1	2	yes	—
23	C₄×C₃₂	—	97	1	no	maximal, Sylow
24	C₉₇	C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	97	2	no	—
26	C₄×C₁₆	—	97	1	no	—
27	C₂×C₁₆	—	97	1	no	—
28	C₂×C₁₆	—	97	1	no	—
29	C₄×C₈	—	97	1	no	—
30	C₃₂	—	97	4	no	—
31	C₂×C₈	—	97	1	no	—
32	C₂×C₈	#11715906	1	1	yes	characteristic
33	C₄²	—	97	1	no	—
34	C₁₆	—	97	2	no	—
35	C₁₆	—	97	2	no	—
36	C₂×C₄	—	97	2	no	—
37	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
38	C₈	—	97	1	no	—
39	C₈	—	97	1	no	—
40	C₈	#13095523	1	2	yes	—
41	C₂²	#15690905	1	1	yes	characteristic
42	C₄	—	97	4	no	—
43	C₄	C₄×#11715906	1	1	yes	characteristic
44	C₄	C₂×#13095523	1	1	yes	characteristic
45	C₂	C₄×#13095523	1	1	yes	characteristic
46	C₂	#20261842	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₈	#11715906
commutator, hypocenter	C₉₇	C₄×C₃₂
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₂×C₈×C₉₇	C₈
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×#13095514	C₂²
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #15690900	C₄	semidirect
20. Dic₉₇	C₃₂	semidirect
24. C₉₇	C₄×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690900	2
#20261621	2
C₄×C₃₂	97