Subgroups — #27158823

Order 12416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261842	C₂	1	2	yes	maximal
2	C₁₆×#87604	C₂	1	1	yes	maximal, characteristic
3	#15690900	C₄	1	4	yes	complemented
4	#15690883	C₂²	1	1	yes	characteristic
5	C₈×#87604	C₄	1	1	yes	characteristic
6	C₁₆×D₉₇	C₄	1	1	yes	characteristic
7	#13095519	C₈	1	1	yes	characteristic
8	C₄×#87604	C₈	1	1	yes	characteristic
9	#13095514	C₂×C₄	1	2	yes	—
10	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
11	#13095486	C₈	1	1	yes	characteristic
12	C₁₆×C₉₇	C₈	1	1	yes	characteristic
13	C₂×#87604	C₁₆	1	2	yes	—
14	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
15	#11715897	C₁₆	1	2	yes	—
16	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
17	#11715895	C₄²	1	1	yes	characteristic
18	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
19	#87604	—	2	2	no	—
20	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
21	Dic₉₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
23	D₉₇	M₆₄	1	2	yes	—
24	#1397	—	97	1	no	maximal, Sylow
25	C₉₇	#1397	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃₂	—	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	—	97	4	no	—
32	C₂×C₈	—	97	1	no	—
33	C₂×C₈	—	97	1	no	—
34	C₄²	—	97	1	no	—
35	C₁₆	—	97	2	no	—
36	C₁₆	—	97	1	no	—
37	C₁₆	#11715906	1	1	yes	characteristic
38	C₂×C₄	—	97	1	no	—
39	C₂×C₄	—	97	2	no	—
40	C₈	—	97	2	no	—
41	C₈	—	97	1	no	—
42	C₈	C₂×#11715906	1	1	yes	characteristic
43	C₂²	—	97	1	no	—
44	C₄	—	97	1	no	—
45	C₄	—	194	2	no	—
46	C₄	C₄×#11715906	1	1	yes	characteristic
47	C₂	—	97	2	no	—
48	C₂	#20261828	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×#11715906
commutator, socle	C₂×C₉₇	C₄×C₁₆
Fitting	C₁₆×C₉₇	C₈
prime radical	C₂×D₉₇	C₂×C₁₆
cosocle	#15690883	C₂²
hypercenter	C₁₆	#11715906
hypocenter	C₉₇	#1397
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
3. #15690900	C₄	semidirect
25. C₉₇	#1397	semidirect

Maximal subgroups

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

subgroup	index
#20261842	2
C₁₆×#87604	2
#1397	97